https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Valko&feedformat=atomUW-Math Wiki - User contributions [en]2024-03-19T13:19:28ZUser contributionsMediaWiki 1.39.5https://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=26057Probability Seminar2024-02-06T13:30:32Z<p>Valko: /* February 15, 2024: Brian Rider (Temple) */</p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
[[Past Seminars]]<br />
<br />
= Spring 2024 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== January 25, 2024: Tatyana Shcherbina (UW-Madison) ==<br />
'''Characteristic polynomials of sparse non-Hermitian random matrices'''<br />
<br />
We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalized iid Bernoulli$(p)$ $d_{jk}$. If $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for Ginibre ensemble of non-Hermitian matrices with iid Gaussian entries: it converges to a determinant of the Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. It appears, however, that for the finite $p>0$, the behavior is different and it exhibits the transition between three different regimes depending on values $p$ and $|z_0|^2$. This is the joint work with Ie. Afanasiev. <br />
<br />
== February 1, 2024: [https://lopat.to/index.html Patrick Lopatto (Brown)] ==<br />
'''Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices'''<br />
<br />
We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.<br />
== February 8, 2024: Benoit Dagallier (NYU), online talk: https://uwmadison.zoom.us/j/95724628357 ==<br />
'''Stochastic dynamics and the Polchinski equation'''<br />
<br />
I will discuss a general framework to obtain large scale information in statistical mechanics and field theory models. The basic, well known idea is to build a dynamics that samples from the model and control its long time behaviour. There are many ways to build such a dynamics, the Langevin dynamics being a typical example. In this talk I will introduce another, the Polchinski dynamics, based on renormalisation group ideas. The dynamics is parametrised by a parameter representing a certain notion of scale in the model under consideration. The Polchinski dynamics has a number of interesting properties that make it well suited to study large-dimensional models. It is also known under the name stochastic localisation. I will mention a number of recent applications of this dynamics, in particular to prove functional inequalities via a generalisation of Bakry and Emery's convexity-based argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau and the recent review paper <nowiki>https://arxiv.org/abs/2307.07619</nowiki> .<br />
<br />
== February 15, 2024: [https://math.temple.edu/~tue86896/ Brian Rider (Temple)] ==<br />
'''A matrix model for conditioned Stochastic Airy'''<br />
<br />
There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. Here I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level. I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).<br />
<br />
== February 22, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== February 29, 2024: Zongrui Yang (Columbia) ==<br />
'''TBA'''<br />
<br />
== March 7, 2024: Atilla Yilmaz (Temple) ==<br />
'''TBA'''<br />
<br />
== March 14, 2024: Eric Foxall (UBC Okanagan) ==<br />
'''TBA'''<br />
<br />
== March 21, 2024: Semon Rezchikov (Princeton) ==<br />
'''TBA'''<br />
<br />
== March 28, 2024: Spring Break ==<br />
'''TBA'''<br />
<br />
== April 4, 2024: Christopher Janjigian (Purdue) ==<br />
'''TBA'''<br />
<br />
== April 11, 2024: Bjoern Bringman (Princeton) ==<br />
'''TBA'''<br />
<br />
== April 18, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== April 25, 2024: Colin McSwiggen (NYU) ==<br />
'''TBA'''<br />
<br />
== May 2, 2024: Anya Katsevich (MIT) ==<br />
'''TBA'''</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=25941Probability Seminar2024-01-19T19:29:14Z<p>Valko: /* January 25, 2024: Tatyana Shcherbina (UW-Madison) */</p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
[[Past Seminars]]<br />
<br />
= Spring 2024 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== January 25, 2024: Tatyana Shcherbina (UW-Madison) ==<br />
'''Characteristic polynomials of sparse non-Hermitian random matrices'''<br />
<br />
We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalized iid Bernoulli$(p)$ $d_{jk}$. If $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for Ginibre ensemble of non-Hermitian matrices with iid Gaussian entries: it converges to a determinant of the Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. It appears, however, that for the finite $p>0$, the behavior is different and it exhibits the transition between three different regimes depending on values $p$ and $|z_0|^2$. This is the joint work with Ie. Afanasiev. <br />
<br />
== February 1, 2024: [https://lopat.to/index.html Patrick Lopatto (Brown)] ==<br />
'''Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices'''<br />
<br />
We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.<br />
== February 8, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== February 15, 2024: Brian Rider (Temple) ==<br />
'''TBA'''<br />
<br />
== February 22, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== February 29, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== March 7, 2024: Atilla Yilmaz (Temple) ==<br />
'''TBA'''<br />
<br />
== March 14, 2024: Eric Foxall (UBC Okanagan) ==<br />
'''TBA'''<br />
<br />
== March 21, 2024: Semon Rezchikov (Princeton) ==<br />
'''TBA'''<br />
<br />
== March 28, 2024: Spring Break ==<br />
'''TBA'''<br />
<br />
== April 4, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== April 11, 2024: Bjoern Bringman (Princeton) ==<br />
'''TBA'''<br />
<br />
== April 18, 2024: TBA ==<br />
'''TBA'''<br />
<br />
== April 25, 2024: Colin McSwiggen (NYU) ==<br />
'''TBA'''<br />
<br />
== May 2, 2024: TBA ==<br />
'''TBA'''</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=25788Probability Seminar2024-01-03T17:14:57Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
[[Past Seminars]]<br />
<br />
= Spring 2024 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== January 25, 2024: Tatyana Shcherbina (UW-Madison) ==<br />
'''TBA'''<br />
<br />
== February 1, 2024: [https://lopat.to/index.html Patrick Lopatto (Brown)] ==<br />
'''Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices'''<br />
<br />
We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=25764Probability Seminar2024-01-02T16:47:33Z<p>Valko: Replaced content with "__NOTOC__ Back to Probability Group Past Seminars = Spring 2024 = <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> We usually end for questions at 3:20 PM. == January 25, 2024: Tatyana Shcherbina (UW-Madison) == '''TBA'''"</p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
[[Past Seminars]]<br />
<br />
= Spring 2024 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== January 25, 2024: Tatyana Shcherbina (UW-Madison) ==<br />
'''TBA'''</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Past_Probability_Seminars_Fall_2023&diff=25763Past Probability Seminars Fall 20232024-01-02T16:45:23Z<p>Valko: Created page with "Past Seminars = Fall 2023 = <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> We usually end for questions at 3:20 PM. == September 14, 2023: [https://www.mathjunge.com/ Matthew Junge] (CUNY) == '''The frog model on trees''' The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on ''d''-ary trees for ten years. I will discuss our progress and what remains to be done. == September 2..."</p>
<hr />
<div>[[Past Seminars]]<br />
<br />
= Fall 2023 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== September 14, 2023: [https://www.mathjunge.com/ Matthew Junge] (CUNY) ==<br />
'''The frog model on trees'''<br />
<br />
The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on ''d''-ary trees for ten years. I will discuss our progress and what remains to be done.<br />
<br />
== September 21, 2023: [https://yierlin.me/ Yier Lin] (U. Chicago) ==<br />
'''Large Deviations of the KPZ Equation and Most Probable Shapes'''<br />
<br />
<br />
The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.<br />
<br />
== September 28, 2023: [https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/rosati/ Tommaso Rosati] (U. Warwick) ==<br />
'''The Allen-Cahn equation with weakly critical initial datum'''<br />
<br />
We study the 2D Allen-Cahn with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint work with Simon Gabriel and Nikolaos Zygouras.<br />
<br />
== October 5, 2023: ==<br />
'''Abstract, title: TBA'''<br />
<br />
== October 12, 2023: No Seminar ([https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium]) ==<br />
<br />
== October 19, 2023: [https://www.paulduncan.net/ Paul Duncan] (Hebrew University of Jerusalem) ==<br />
'''Deconfinement in Ising Lattice Gauge Theory'''<br />
<br />
A lattice gauge theory is a random assignment of spins to edges of a lattice that offers a more tractable model in which to study path integrals that appear in particle physics. We demonstrate the existence of a phase transition corresponding to deconfinement in a simplified model called Ising lattice gauge theory on the cubical lattice Z^3. Our methods involve studying the topology of a random 2-dimensional cubical complex on Z^3 called random-cluster plaquette percolation, which in turn can be reduced to the study of a random dual graph. No prior background in topology or physics will be assumed. This is based on joint work with Benjamin Schweinhart.<br />
<br />
== October 26, 2023: Yuchen Liao (UW - Madison) ==<br />
'''Large deviations for the deformed Polynuclear growth'''<br />
<br />
The polynuclear growth model (PNG) is a prototypical example of random interface growth among the Kardar-Parisi-Zhang universality class. In this talk I will discuss a q-deformation of the PNG model recently introduced by Aggarwal-Borodin-Wheeler. We are mainly interested in the large time large deviations of the one-point distribution under narrow-wedge (droplet) initial data, i.e., the rare events that the height function at time t being much larger (upper tail) or much smaller (lower tail) than its expected value. Large deviation principles with speed t and t^2 are established for the upper and lower tails, respectively. The upper tail rate function is computed explicitly and is independent of q. The lower tail rate function is described through a variational problem and shows nontrivial q-dependence. Based on joint work with Matteo Mucciconi and Sayan Das.<br />
<br />
== November 2, 2023: [http://homepages.math.uic.edu/~couyang/ Cheng Ouyang] (U. Illinois Chicago) ==<br />
'''Colored noise and parabolic Anderson model on Torus'''<br />
<br />
We construct an intrinsic family of Gaussian noises on compact Riemannian manifolds which we call the colored noise on manifolds. It consists of noises with a wide range of singularities. Using this family of noises, we study the parabolic Anderson model on compact manifolds. To begin with, we started our investigation on a flat torus and established existence and uniqueness of the solution, as well as some sharp bounds on the second moment of the solution. In particular, our methodology does not necessarily rely on Fourier analysis and can be applied to study the PAM on more general manifolds.<br />
<br />
== November 9, 2023: [https://scottandrewsmith.github.io/ Scott Smith] (Chinese Academy of Sciences) ==<br />
'''A stochastic analysis viewpoint on the master loop equation for lattice Yang-Mills''' <br />
<br />
I will discuss the master loop equation for lattice Yang-Mills, introduced in the physics literature by Makeenko/Migdal (1979). A more precise formulation and proof was given by Chatterjee (2019) for SO(N) and later by Jafarov for SU(N). I will explain how the loop equation arises naturally from the Langevin dynamic for the lattice Yang-Mills measure. Based on joint work with Hao Shen and Rongchan Zhu.<br />
<br />
== November 16, 2023: [https://math.mit.edu/~mnicolet/ Matthew Nicoletti] (MIT) ==<br />
'''Colored Interacting Particle Systems on the Ring: Stationary Measures from Yang--Baxter Equation'''<br />
<br />
Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials).<br />
<br />
In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1)~the Asymmetric Simple Exclusion Process (mASEP); (2)~the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3)~the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang--Baxter equation. We express the stationary measures as partition functions of new ``queue vertex models<nowiki>''</nowiki> on the cylinder. The stationarity property is a direct consequence of the Yang--Baxter equation. This is joint work with A. Aggarwal and L. Petrov.<br />
<br />
== November 23, 2023: No Seminar ==<br />
'''No seminar. Thanksgiving.'''<br />
<br />
== November 30, 2023: [http://web.mit.edu/youngtak/www/homepage.html Youngtak Sohn] (MIT) ==<br />
'''Geometry of random constraint satisfaction problems'''<br />
<br />
The framework of constraint satisfaction problem (CSP) captures many fundamental problems in combinatorics and computer science, such as finding a proper coloring of a graph or solving the boolean satisfiability problems. Solving a CSP can often be NP-hard in the worst-case scenario. To study the typical cases of CSPs, statistical physicists have proposed a detailed picture of the solution space for random CSPs based on non-rigorous methods from spin glass theory. In this talk, I will first survey the conjectured rich phase diagrams of random CSPs in the one-step replica symmetry breaking (1RSB) universality class. Then, I will describe the recent progress in understanding the global and local geometry of solutions, particularly in random regular NAE-SAT problem. <br />
<br />
This talk is based on joint works with Danny Nam and Allan Sly.<br />
<br />
== December 7, 2023: Minjae Park (U. Chicago) ==<br />
'''Yang-Mills theory and random surfaces'''<br />
<br />
I will talk about some recent work on Yang-Mills theory for classical Lie groups and its relationship to the theory of random surfaces. In particular, I will explain how Wilson loop expectations in lattice Yang-Mills can be expressed as sums over embedded planar maps for any matrix dimension N ≥ 1, any inverse temperature β > 0, and any lattice dimension d ≥ 2. The main idea is from my similar result for 2D continuum Yang-Mills (with Joshua Pfeffer, Scott Sheffield, and Pu Yu), and it gives alternative derivations and interpretations of several recent theorems including Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov), and surface sums (Magee and Puder). Based on joint work with Sky Cao and Scott Sheffield.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Past_Probability_Seminars_Fall_2024&diff=25762Past Probability Seminars Fall 20242024-01-02T16:45:11Z<p>Valko: Replaced content with "Past Seminars = Fall 2024 = <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> We usually end for questions at 3:20 PM."</p>
<hr />
<div>[[Past Seminars]]<br />
<br />
= Fall 2024 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Past_Seminars&diff=25761Past Seminars2024-01-02T16:44:50Z<p>Valko: </p>
<hr />
<div>[[Probability | Back to Probability Group]]<br />
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<!-- [http://www.math.wisc.edu/~probsem/list-old-sem.html Webpage for older past probability seminars] --></div>Valkohttps://wiki.math.wisc.edu/index.php?title=Past_Probability_Seminars_Fall_2024&diff=25760Past Probability Seminars Fall 20242024-01-02T16:44:17Z<p>Valko: Created page with "Past Seminars = Fall 2023 = <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> We usually end for questions at 3:20 PM. == September 14, 2023: [https://www.mathjunge.com/ Matthew Junge] (CUNY) == '''The frog model on trees''' The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on ''d''-ary trees for ten years. I will discuss our progress and what remains to be done. == September 2..."</p>
<hr />
<div>[[Past Seminars]]<br />
<br />
= Fall 2023 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== September 14, 2023: [https://www.mathjunge.com/ Matthew Junge] (CUNY) ==<br />
'''The frog model on trees'''<br />
<br />
The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on ''d''-ary trees for ten years. I will discuss our progress and what remains to be done.<br />
<br />
== September 21, 2023: [https://yierlin.me/ Yier Lin] (U. Chicago) ==<br />
'''Large Deviations of the KPZ Equation and Most Probable Shapes'''<br />
<br />
<br />
The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.<br />
<br />
== September 28, 2023: [https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/rosati/ Tommaso Rosati] (U. Warwick) ==<br />
'''The Allen-Cahn equation with weakly critical initial datum'''<br />
<br />
We study the 2D Allen-Cahn with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint work with Simon Gabriel and Nikolaos Zygouras.<br />
<br />
== October 5, 2023: ==<br />
'''Abstract, title: TBA'''<br />
<br />
== October 12, 2023: No Seminar ([https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium]) ==<br />
<br />
== October 19, 2023: [https://www.paulduncan.net/ Paul Duncan] (Hebrew University of Jerusalem) ==<br />
'''Deconfinement in Ising Lattice Gauge Theory'''<br />
<br />
A lattice gauge theory is a random assignment of spins to edges of a lattice that offers a more tractable model in which to study path integrals that appear in particle physics. We demonstrate the existence of a phase transition corresponding to deconfinement in a simplified model called Ising lattice gauge theory on the cubical lattice Z^3. Our methods involve studying the topology of a random 2-dimensional cubical complex on Z^3 called random-cluster plaquette percolation, which in turn can be reduced to the study of a random dual graph. No prior background in topology or physics will be assumed. This is based on joint work with Benjamin Schweinhart.<br />
<br />
== October 26, 2023: Yuchen Liao (UW - Madison) ==<br />
'''Large deviations for the deformed Polynuclear growth'''<br />
<br />
The polynuclear growth model (PNG) is a prototypical example of random interface growth among the Kardar-Parisi-Zhang universality class. In this talk I will discuss a q-deformation of the PNG model recently introduced by Aggarwal-Borodin-Wheeler. We are mainly interested in the large time large deviations of the one-point distribution under narrow-wedge (droplet) initial data, i.e., the rare events that the height function at time t being much larger (upper tail) or much smaller (lower tail) than its expected value. Large deviation principles with speed t and t^2 are established for the upper and lower tails, respectively. The upper tail rate function is computed explicitly and is independent of q. The lower tail rate function is described through a variational problem and shows nontrivial q-dependence. Based on joint work with Matteo Mucciconi and Sayan Das.<br />
<br />
== November 2, 2023: [http://homepages.math.uic.edu/~couyang/ Cheng Ouyang] (U. Illinois Chicago) ==<br />
'''Colored noise and parabolic Anderson model on Torus'''<br />
<br />
We construct an intrinsic family of Gaussian noises on compact Riemannian manifolds which we call the colored noise on manifolds. It consists of noises with a wide range of singularities. Using this family of noises, we study the parabolic Anderson model on compact manifolds. To begin with, we started our investigation on a flat torus and established existence and uniqueness of the solution, as well as some sharp bounds on the second moment of the solution. In particular, our methodology does not necessarily rely on Fourier analysis and can be applied to study the PAM on more general manifolds.<br />
<br />
== November 9, 2023: [https://scottandrewsmith.github.io/ Scott Smith] (Chinese Academy of Sciences) ==<br />
'''A stochastic analysis viewpoint on the master loop equation for lattice Yang-Mills''' <br />
<br />
I will discuss the master loop equation for lattice Yang-Mills, introduced in the physics literature by Makeenko/Migdal (1979). A more precise formulation and proof was given by Chatterjee (2019) for SO(N) and later by Jafarov for SU(N). I will explain how the loop equation arises naturally from the Langevin dynamic for the lattice Yang-Mills measure. Based on joint work with Hao Shen and Rongchan Zhu.<br />
<br />
== November 16, 2023: [https://math.mit.edu/~mnicolet/ Matthew Nicoletti] (MIT) ==<br />
'''Colored Interacting Particle Systems on the Ring: Stationary Measures from Yang--Baxter Equation'''<br />
<br />
Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials).<br />
<br />
In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1)~the Asymmetric Simple Exclusion Process (mASEP); (2)~the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3)~the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang--Baxter equation. We express the stationary measures as partition functions of new ``queue vertex models<nowiki>''</nowiki> on the cylinder. The stationarity property is a direct consequence of the Yang--Baxter equation. This is joint work with A. Aggarwal and L. Petrov.<br />
<br />
== November 23, 2023: No Seminar ==<br />
'''No seminar. Thanksgiving.'''<br />
<br />
== November 30, 2023: [http://web.mit.edu/youngtak/www/homepage.html Youngtak Sohn] (MIT) ==<br />
'''Geometry of random constraint satisfaction problems'''<br />
<br />
The framework of constraint satisfaction problem (CSP) captures many fundamental problems in combinatorics and computer science, such as finding a proper coloring of a graph or solving the boolean satisfiability problems. Solving a CSP can often be NP-hard in the worst-case scenario. To study the typical cases of CSPs, statistical physicists have proposed a detailed picture of the solution space for random CSPs based on non-rigorous methods from spin glass theory. In this talk, I will first survey the conjectured rich phase diagrams of random CSPs in the one-step replica symmetry breaking (1RSB) universality class. Then, I will describe the recent progress in understanding the global and local geometry of solutions, particularly in random regular NAE-SAT problem. <br />
<br />
This talk is based on joint works with Danny Nam and Allan Sly.<br />
<br />
== December 7, 2023: Minjae Park (U. Chicago) ==<br />
'''Yang-Mills theory and random surfaces'''<br />
<br />
I will talk about some recent work on Yang-Mills theory for classical Lie groups and its relationship to the theory of random surfaces. In particular, I will explain how Wilson loop expectations in lattice Yang-Mills can be expressed as sums over embedded planar maps for any matrix dimension N ≥ 1, any inverse temperature β > 0, and any lattice dimension d ≥ 2. The main idea is from my similar result for 2D continuum Yang-Mills (with Joshua Pfeffer, Scott Sheffield, and Pu Yu), and it gives alternative derivations and interpretations of several recent theorems including Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov), and surface sums (Magee and Puder). Based on joint work with Sky Cao and Scott Sheffield.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Past_Seminars&diff=25759Past Seminars2024-01-02T16:43:49Z<p>Valko: </p>
<hr />
<div>[[Probability | Back to Probability Group]]<br />
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<!-- [http://www.math.wisc.edu/~probsem/list-old-sem.html Webpage for older past probability seminars] --></div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=25400Probability Seminar2023-10-09T19:37:19Z<p>Valko: /* October 12, 2023: No Seminar */</p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
[[Past Seminars]]<br />
<br />
= Fall 2023 =<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b><br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
== September 14, 2023: [https://www.mathjunge.com/ Matthew Junge] (CUNY) ==<br />
'''The frog model on trees'''<br />
<br />
The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on ''d''-ary trees for ten years. I will discuss our progress and what remains to be done.<br />
<br />
== September 21, 2023: [https://yierlin.me/ Yier Lin] (U. Chicago) ==<br />
'''Large Deviations of the KPZ Equation and Most Probable Shapes'''<br />
<br />
<br />
The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.<br />
<br />
== September 28, 2023: [https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/rosati/ Tommaso Rosati] (U. Warwick) ==<br />
'''The Allen-Cahn equation with weakly critical initial datum'''<br />
<br />
We study the 2D Allen-Cahn with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint work with Simon Gabriel and Nikolaos Zygouras.<br />
<br />
== October 5, 2023: ==<br />
'''Abstract, title: TBA'''<br />
<br />
== October 12, 2023: No Seminar ([https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium]) ==<br />
<br />
== October 19, 2023: ==<br />
<br />
== October 26, 2023: Yuchen Liao (UW - Madison) ==<br />
'''Abstract, title: TBA'''<br />
<br />
== November 2, 2023: [http://homepages.math.uic.edu/~couyang/ Cheng Ouyang] (U. Illinois Chicago) ==<br />
'''Abstract, title: TBA'''<br />
<br />
== November 9, 2023: [https://scottandrewsmith.github.io/ Scott Smith] (Chinese Academy of Sciences) ==<br />
'''Abstract, title: TBA'''<br />
<br />
== November 16, 2023: Matthew Nicoletti (MIT) ==<br />
'''Abstract, title: TBA'''<br />
<br />
== November 23, 2023: No Seminar ==<br />
'''No seminar. Thanksgiving.'''<br />
<br />
== November 30, 2023: [http://web.mit.edu/youngtak/www/homepage.html Youngtak Sohn] (MIT) ==<br />
'''Abstract, title: TBA'''<br />
<br />
== December 7, 2023: Minjae Park (U. Chicago) ==<br />
'''Abstract, title: TBA'''</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=25201Probability2023-09-11T14:29:05Z<p>Valko: /* Postdocs */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[https://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[https://hanbaeklyu.com/ Hanbaek Lyu] (Ohio State, 2018) discrete probability, dynamical systems, networks, optimization, machine learning <br />
<br />
[https://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied discrete probability, mathematical and computational biology, networks.<br />
<br />
[https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[https://math.wisc.edu/staff/shcherbyna-tatiana/ Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[https://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[https://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
== Postdocs ==<br />
<br />
David Keating (UC Berkeley, 2021)<br />
<br />
David Clancy (UWashington, 2022)<br />
<br />
Yuchen Liao (Michigan, 2021)<br />
<br />
== Graduate students ==<br />
<br />
Max Bacharach<br />
<br />
Yu Sun<br />
<br />
Jiaming Xu<br />
<br />
Shuqi Yu<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2023 Spring'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
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Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
Math/ECE/Stat 888 Topics in Mathematical Data Science<br />
<br />
<br />
'''2023 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Undergraduate_courses_in_probability&diff=24806Undergraduate courses in probability2023-04-21T19:30:50Z<p>Valko: </p>
<hr />
<div>'''431 - Introduction to the theory of probability'''<br />
<br />
Math 431 is an introduction to probability theory, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include methods of counting (combinatorics), axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.<br />
<br />
Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.<br />
<br />
The course is offered every semester, including the summer. <br />
<br />
''Prerequisite'': Math 234. <br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> A well rounded undergraduate experience in math should include some probability theory. Math 431 is our introductory probability class with no high level prerequisites. <br />
<br />
<br />
<br />
'''531 - Probability theory'''<br />
<br />
The course is a rigorous introduction to probability theory on an advanced undergraduate level. Only a minimal amount of measure theory is used (in particular, Lebesgue integrals will not be needed). The course gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some of the classical results of probability theory with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on Z, applications of generating functions).<br />
<br />
The course is offered every spring.<br />
<br />
''Prerequisite'': a proof based analysis course (Math 376, Math 421 or Math 521). <br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Students who would like to get a rigorous introduction to probability. It could also provide a stepping stone for our 600 level stochastic processes courses. (The course can be taken even after taking Math 431.)<br />
<br />
<br />
<br />
<br />
<br />
'''632 - Introduction to stochastic processes'''<br />
<br />
Math 632 gives an introduction to Markov chains and Markov processes with discrete state spaces and their applications. Particular models studied include birth-death chains, queuing models, random walks and branching processes. Selected topics from renewal theory, martingales, and Brownian motion are also included, but vary from semester to semester to meet the needs of different audiences. <br />
<br />
''Prerequisite'': Intro to probability (Math 309, 431 or 531)+ a linear algebra or an intro to proofs class (320, 340, 341, 375, 421)<br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Math 632 is the natural next step after an introductory probability course. It could be useful for an Option 1 math major interested in higher level probability and it is also a great fit for many of our [https://www.math.wisc.edu/undergraduate/option-2-sample-packages Option 2 packages]. <br />
<br />
<br />
<br />
'''635 - Introduction to Brownian motion and stochastic calculus'''<br />
<br />
Math 635 is an introduction to Brownian motion and stochastic calculus without a measure theory prerequisite. Topics touched upon include sample path properties of Brownian motion, Itô stochastic integrals, Itô's formula, stochastic differential equations and their solutions. As an application we will discuss the Black-Scholes formula of mathematical finance.<br />
<br />
The course is offered every two years in the spring semester. <br />
<br />
''Prerequisite'': Math 521 and Math 632<br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Anybody with an interest in higher level probability. It is especially useful for those who are planning to study financial math on a graduate level. <br />
<br />
<br />
<!--[[File:Probability_courses_1.jpg|600px]]--></div>Valkohttps://wiki.math.wisc.edu/index.php?title=Colloquia&diff=24683Colloquia2023-03-21T21:03:05Z<p>Valko: /* March 31, 2023 , Friday at 4pm Bálint Virág (University of Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==<br />
(host: Lyu)<br />
<br />
The Gromov-Hausdorff distance between spheres.<br />
<br />
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.<br />
<br />
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.<br />
<br />
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.<br />
<br />
== February 24, 2023, Cancelled/available ==<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
Title: How curved is a combinatorial graph?<br />
<br />
Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open. No prior knowledge of differential geometry (or graphs) is required.<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: Surfaces and foliations in hyperbolic 3-manifolds<br />
<br />
Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.<br />
<br />
(host: Kent)<br />
<br />
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: End-periodic maps, via fibered 3-manifolds<br />
<br />
Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning<nowiki>''</nowiki> construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.<br />
<br />
(host: Kent)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
'''Title''': Boundaries, boundaries, and more boundaries <br />
<br />
'''Abstract:''' It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell. <br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
'''Title:''' Random plane geometry -- a gentle introduction<br />
<br />
'''Abstract:''' Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percolation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", and its relation to traffic jams, tetris, coffee stains and random matrices.<br />
<br />
(host: Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==<br />
<br />
'''''Wasow lecture'''''<br />
<br />
Title: Mathematics: A key to climate research<br />
<br />
Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:<br />
<br />
1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.<br />
<br />
2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.<br />
<br />
3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.<br />
<br />
(hosts: Smith, Stechmann)<br />
<br />
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
<br />
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le] (Indiana University) ==<br />
<br />
== Future Colloquia ==<br />
<br />
[[Colloquia/Fall2023|Fall 2023]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
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[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=24682Probability Seminar2023-03-21T21:00:15Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
[[Past Seminars]]<br />
<br />
= Spring 2023 =<br />
<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> <br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
<br />
== January 26, 2023, in person: [https://sites.google.com/wisc.edu/evan-sorensen?pli=1 Evan Sorensen] (UW-Madison) ==<br />
'''The stationary horizon as a universal object for KPZ models'''<br />
<br />
The last 5-10 years has seen remarkable progress in constructing the central objects of the KPZ universality class, namely the KPZ fixed point and directed landscape. In this talk, I will discuss a third central object known as the stationary horizon (SH). The SH is a coupling of Brownian motions with drifts, indexed by the real line, and it describes the unique coupled invariant measures for the directed landscape. I will talk about how the SH appears as the scaling limit of several models, including Busemann processes in last-passage percolation and the TASEP speed process. I will also discuss how the SH helps to describe the collection of infinite geodesics in all directions for the directed landscape. Based on joint work with Timo Seppäläinen and Ofer Busani.<br />
<br />
== February 2, 2023, in person: [https://mathjinsukim.com/ Jinsu Kim] (POSTECH) ==<br />
'''Fast and slow mixing of continuous-time Markov chains with polynomial rates'''<br />
<br />
Continuous-time Markov chains on infinite positive integer grids with polynomial rates are often used in modeling queuing systems, molecular counts of small-size biological systems, etc. In this talk, we will discuss continuous-time Markov chains that admit either fast or slow mixing behaviors. For a positive recurrent continuous-time Markov chain, the convergence rate to its stationary distribution is typically investigated with the Lyapunov function method and canonical path method. Recently, we discovered examples that do not lend themselves easily to analysis via those two methods but are shown to have either fast mixing or slow mixing with our new technique. The main ideas of the new methodologies are presented in this talk along with their applications to stochastic biochemical reaction network theory.<br />
<br />
== February 9, 2023, in person: [https://www.math.tamu.edu/~jkuan/ Jeffrey Kuan] (Texas A&M) ==<br />
'''Shift invariance for the multi-species q-TAZRP on the infinite line'''<br />
<br />
We prove a shift--invariance for the multi-species q-TAZRP (totally asymmetric zero range process) on the infinite line. Similar-looking results had appeared in works by [Borodin-Gorin-Wheeler] and [Galashin], using integrability, but are on the quadrant. The proof in this talk relies instead on a combinatorial approach, in which the state space is generalized to a poset, and the totally asymmetric process is generalized to a monotone process on a poset. The continuous-time process is decomposed into its discrete embedded Markov chain and its exponential holding times, and the shift-invariance is proved using explicit contour integral formulas. Open problems about multi-species ASEP will be discussed as well.<br />
<br />
== February 16, 2023, in person: [http://math.columbia.edu/~milind/ Milind Hegde] (Columbia) ==<br />
'''Understanding the upper tail behaviour of the KPZ equation via the tangent method'''<br />
<br />
The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data. <br />
<br />
== February 23, 2023, in person: [https://sites.math.rutgers.edu/~sc2518/ Swee Hong Chan] (Rutgers) ==<br />
'''Log-concavity and cross product inequalities in order theory'''<br />
<br />
Given a finite poset that is not completely ordered, is it always possible find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.<br />
<br />
== March 2, 2023, in person: Max Hill (UW-Madison) ==<br />
'''On the Effect of Intralocus Recombination on Triplet-Based Species Tree Estimation'''<br />
<br />
My talk will introduce some key topics in mathematical phylogenetics and is intended to be accessible for those not familiar with the field. I will discuss joint work with Sebastien Roch on the subject of species tree estimation from multiple loci subject to intralocus recombination. The focus is on R*, a summary coalescent-based method using rooted triplets. I will present a result showing how intralocus recombination can give rise to an "inconsistency zone," in which correct inference using R* is not assured even in the limit of infinite amount of data.<br />
<br />
== March 9, 2023, in person: [https://math.uchicago.edu/~xuanw/ Xuan Wu] (U. Chicago) ==<br />
'''From the KPZ equation to the directed landscape'''<br />
<br />
This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.<br />
<br />
== March 23, 2023, in person: Jiaming Xu (UW-Madison) ==<br />
<br />
'''Rectangular Matrix addition in low and high temperatures'''<br />
<br />
We study the addition of two <math>{\scriptsize M \times N}</math> rectangular random matrices with certain<br />
invariant distributions in two limit regimes, where the parameter <math>{\scriptsize \beta}</math> (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers of the empirical measures. Our proof uses the so-called type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and<br />
degenerate to the classical or free cumulants in special cases.<br />
<br />
== March 30, 2023, in person: [http://www.math.toronto.edu/balint/ Bálint Virág] (Toronto) ==<br />
'''The planar stochastic heat equation and the directed landscape'''<br />
<br />
The planar stochastic heat equation describes heat flow or random polymers on an inhomogeneous surface. It is a finite-temperature version of planar first passage percolation such as the Eden growth model. It is the first model with plane symmetries for which we can show convergence to the directed landscape. The methods use a Skorokhod integral representation and Gaussian multiplicative chaos on path space. <br />
<br />
Joint work with Jeremy Quastel and Alejandro Ramirez.<br />
<br />
== April 6, 2023, in person: [https://shankarbhamidi.web.unc.edu/ Shankar Bhamidi] (UNC-Chapel Hill) ==<br />
<br />
'''Disorder models for random graphs, Erdos’s leader problem, and power of limited choice models for network evolution'''<br />
<br />
First passage percolation, and more generally the study of diffusion of material through disordered systems is a fundamental area in probabilistic combinatorics with a vast body of work especially in the context of spatial systems.<br />
<br />
The goal of this talk is to survey a slightly different setting for such questions namely the more “mean-field” setting of random graph models. We will describe the state of the art of this field, with the final goal of describing one of the main conjectures in this area namely the conjectured scaling limit of the minimal spanning tree and its dependence on the degree exponent of the corresponding network model. We will describe recent progress in this area, its connection to questions in dynamic network models, in particular Erdos’s leader problem for the identity of the maximal component for critical random graphs, and the intuition for understanding the evolution of maximal components through the critical scaling window from a different area of probabilistic combinatorics, namely the study of limited choice models for network evolution. <br />
<br />
== April 13, 2023, in person: [https://msellke.com/ Mark Sellke] (Amazon) ==<br />
<br />
== April 20, 2023, in person: [http://www.math.columbia.edu/~remy/ Guillaume Remy] (IAS) ==<br />
<br />
== April 27, 2023, in person: [http://www.math.tau.ac.il/~peledron/ Ron Peled] (Tel Aviv/IAS) ==<br />
<br />
== May 4, 2023, in person: [https://www.asc.ohio-state.edu/sivakoff.2// David Sivakoff] (Ohio State) ==</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=24456Graduate student reading seminar2023-02-13T20:59:19Z<p>Valko: </p>
<hr />
<div>(... in probability)<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list]<br />
<br />
== 2023 Spring ==<br />
We are reading parts of the textbook "Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction" by Friedli and Velenik (<nowiki>https://www.unige.ch/math/folks/velenik/smbook/</nowiki>).<br />
<br />
Feb 7: Benedek<br />
<br />
Feb 14 Jiaming<br />
<br />
Feb 21 Erik<br />
<br />
Feb 28 David<br />
<br />
Mar 7 Jiahong<br />
<br />
Mar 21 Jiahong<br />
<br />
Mar 28 Zhengwei<br />
<br />
April 4 Yahui<br />
<br />
April 11 Yahui<br />
<br />
April 18 Zhengwei<br />
<br />
April 25 Evan<br />
<br />
May 2 Evan<br />
<br />
== 2022 Fall ==<br />
We read the following lecture notes of Hugo Dominil-Copin on percolation: https://www.unige.ch/~duminil/publi/2017percolation.pdf<br />
<br />
== 2022 Spring ==<br />
We read the first couple of chapters of the lecture notes “Gaussian free field, Liouville quantum gravity and Gaussian multiplicative chaos” by Nathanaël Berestycki and Ellen Powell (https://www.ams.org/open-math-notes/omn-view-listing?listingId=111291). <br />
<br />
== 2021 Fall ==<br />
We discussed the lecture notes "Lectures on integrable probability" by Borodin and Gorin (https://arxiv.org/abs/1212.3351)<br />
<br />
==2020 Fall==<br />
<br />
The graduate probability seminar will be on Zoom this semester. Please sign up for the email list if you would like to receive notifications about the talks.<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/4, 2/11: Edwin<br />
<br />
2/18, 2/25: Chaojie<br />
<br />
3/3. 3/10: Yu Sun<br />
<br />
3/24, 3/31: Tony<br />
<br />
4/7, 4/14: Tung<br />
<br />
4/21, 4/28: Tung<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Colloquia/Spring2023&diff=23944Colloquia/Spring20232022-10-26T17:30:41Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 24, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Tushar Das] (University of Wisconsin - La Crosse) ==<br />
(hosts: Burkart, Stovall)<br />
<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
<br />
(host: Dymarz, Uyanik, WIMAW)<br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
(host: Benedek Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://math.wisc.edu TBA] (TBA) ==<br />
<br />
(Wasow lecture)<br />
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
<br />
<br />
== Past Colloquia ==<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Colloquia/Spring2022|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Colloquia&diff=23787Colloquia2022-09-30T19:14:04Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated. <br />
<br />
==September 9 , 2022, Friday at 4pm [https://math.ou.edu/~jing/ Jing Tao] (University of Oklahoma)==<br />
(host: Dymarz, Uyanik, WIMAW)<br />
<br />
'''On surface homeomorphisms'''<br />
<br />
In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.<br />
==September 23, 2022, Friday at 4pm [https://www.pabloshmerkin.org/ Pablo Shmerkin] (University of British Columbia) ==<br />
(host: Guo, Seeger)<br />
<br />
'''Incidences and line counting: from the discrete to the fractal setting'''<br />
<br />
How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.<br />
<br />
==September 30, 2022, Friday at 4pm [https://alejandraquintos.com/ Alejandra Quintos] (University of Wisconsin-Madison, Statistics) ==<br />
(host: Stovall)<br />
<br />
'''Dependent Stopping Times and an Application to Credit Risk Theory'''<br />
<br />
Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.<br />
<br />
In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.<br />
==October 7, 2022, Friday at 4pm [https://www.daniellitt.com/ Daniel Litt] (University of Toronto)==<br />
(host: Ananth Shankar)<br />
<br />
==October 14, 2022, Friday at 4pm [https://math.sciences.ncsu.edu/people/asagema/ Andrew Sageman-Furnas] (North Carolina State)==<br />
(host: Mari-Beffa)<br />
<br />
== October 20, 2022, Thursday at 4pm, VV911 [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==<br />
(host: Kurtz, Roch)<br />
<br />
''Note the unusual time and room!''<br />
<br />
'''An introduction to counts-of-counts data'''<br />
<br />
Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size<br />
<br />
''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>'' containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.<br />
<br />
In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.<br />
<br />
''References''<br />
<br />
[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943<br />
<br />
[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002<br />
<br />
[3] Ewens WJ. Theoret Popul Biol, 3, 1972<br />
<br />
[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)<br />
<br />
==October 21, 2022, Friday at 4pm [https://web.ma.utexas.edu/users/ntran/ Ngoc Mai Tran] (Texas)==<br />
(host: Rodriguez)<br />
== November 7-9, 2022, [https://ai.facebook.com/people/kristin-lauter/ Kristen Lauter] (Facebook) ==<br />
Distinguished lectures<br />
<br />
(host: Yang).<br />
<br />
== November 11, 2022, Friday at 4pm [http://users.cms.caltech.edu/~jtropp/ Joel Tropp] (Caltech)==<br />
This is the Annual LAA lecture. See [https://math.wisc.edu/laa-lecture/ this] for its history.<br />
<br />
(host: Qin, Jordan)<br />
==November 18, 2022, Friday at 4pm [TBD]==<br />
(reserved by HC. contact: Stechmann)<br />
==December 2, 2022, Friday at 4pm [TBD]==<br />
(reserved by HC. contact: Stechmann)<br />
==December 9, 2022, Friday at 4pm [TBD]==<br />
(reserved by HC. contact: Stechmann)<br />
== Future Colloquia ==<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Colloquia/Spring2023|Spring 2023]]<br />
<br />
== Past Colloquia ==<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=23786Probability Seminar2022-09-30T18:53:52Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
= Fall 2022 =<br />
<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> <br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
<br />
== September 22, 2022, in person: [https://sites.google.com/site/pierreyvesgl/home Pierre Yves Gaudreau Lamarre] (University of Chicago) ==<br />
<br />
'''Moments of the Parabolic Anderson Model with Asymptotically Singular Noise'''<br />
<br />
The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.<br />
<br />
One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.<br />
<br />
In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.<br />
<br />
This talk is based on a joint work with Promit Ghosal and Yuchen Liao.<br />
<br />
== September 29, 2022, in person: Christian Gorski (Northwestern University) ==<br />
<br />
'''Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees'''<br />
<br />
I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance. <br />
If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.<br />
<br />
== October 6, 2022, in person: [https://danielslonim.github.io/ Daniel Slonim] (University of Virginia) == <br />
<br />
'''Random Walks in (Dirichlet) Random Environments with Jumps on Z'''<br />
<br />
We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models. <br />
<br />
== October 13, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.univ-evry.fr/pages_perso/loukianova/ Dasha Loukianova] (Université d'Évry Val d'Essonne) ==<br />
<br />
== October 20, 2022, '''4pm, VV911''', in person: [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==<br />
''Note the unusual time and room!''<br />
<br />
'''An introduction to counts-of-counts data'''<br />
<br />
Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size<br />
<br />
''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>'' containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.<br />
<br />
In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.<br />
<br />
<br />
''References''<br />
<br />
[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943<br />
<br />
[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002<br />
<br />
[3] Ewens WJ. Theoret Popul Biol, 3, 1972<br />
<br />
[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online) <br />
<br />
<br />
<br />
== October 27, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~arnab/ Arnab Sen] (University of Minnesota, Twin Cities) == <br />
<br />
<br />
== November 3, 2022, in person: [https://www.ias.edu/scholars/sky-yang-cao Sky Cao] (Institute for Advanced Study) == <br />
<br />
<br />
== November 10, 2022, in person: TBD == <br />
<br />
<br />
== November 17, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/site/leandroprpimentel/ Leandro Pimentel] (Federal University of Rio de Janeiro) == <br />
<br />
<br />
== December 1, in person: [https://cims.nyu.edu/~ajd594/ Alex Dunlap] (Courant Institute) == <br />
<br />
<br />
== December 8, 2022, in person: [https://sites.northwestern.edu/juliagaudio/ Julia Gaudio] (Northwestern University) == <br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=23785Probability Seminar2022-09-30T18:53:28Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
[[Probability | Back to Probability Group]]<br />
<br />
= Fall 2022 =<br />
<br />
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> <br />
<br />
We usually end for questions at 3:20 PM.<br />
<br />
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
<br />
== September 22, 2022, in person: [https://sites.google.com/site/pierreyvesgl/home Pierre Yves Gaudreau Lamarre] (University of Chicago) ==<br />
<br />
'''Moments of the Parabolic Anderson Model with Asymptotically Singular Noise'''<br />
<br />
The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.<br />
<br />
One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.<br />
<br />
In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.<br />
<br />
This talk is based on a joint work with Promit Ghosal and Yuchen Liao.<br />
<br />
== September 29, 2022, in person: Christian Gorski (Northwestern University) ==<br />
<br />
'''Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees'''<br />
<br />
I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance. <br />
If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.<br />
<br />
== October 6, 2022, in person: [https://danielslonim.github.io/ Daniel Slonim] (University of Virginia) == <br />
<br />
'''Random Walks in (Dirichlet) Random Environments with Jumps on Z'''<br />
<br />
We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models. <br />
<br />
== October 13, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.univ-evry.fr/pages_perso/loukianova/ Dasha Loukianova] (Université d'Évry Val d'Essonne) ==<br />
<br />
== October 20, 2022, '''4pm, VV911''', in person: [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==<br />
Note the unusual time and room!<br />
<br />
'''An introduction to counts-of-counts data'''<br />
<br />
Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size<br />
<br />
''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>'' containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.<br />
<br />
In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.<br />
<br />
<br />
''References''<br />
<br />
[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943<br />
<br />
[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002<br />
<br />
[3] Ewens WJ. Theoret Popul Biol, 3, 1972<br />
<br />
[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online) <br />
<br />
<br />
<br />
== October 27, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~arnab/ Arnab Sen] (University of Minnesota, Twin Cities) == <br />
<br />
<br />
== November 3, 2022, in person: [https://www.ias.edu/scholars/sky-yang-cao Sky Cao] (Institute for Advanced Study) == <br />
<br />
<br />
== November 10, 2022, in person: TBD == <br />
<br />
<br />
== November 17, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/site/leandroprpimentel/ Leandro Pimentel] (Federal University of Rio de Janeiro) == <br />
<br />
<br />
== December 1, in person: [https://cims.nyu.edu/~ajd594/ Alex Dunlap] (Courant Institute) == <br />
<br />
<br />
== December 8, 2022, in person: [https://sites.northwestern.edu/juliagaudio/ Julia Gaudio] (Northwestern University) == <br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=23469Probability2022-08-22T19:29:59Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[https://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[https://hanbaeklyu.com/ Hanbaek Lyu] (Ohio State, 2018) discrete probability, dynamical systems, networks, optimization, machine learning <br />
<br />
[https://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied discrete probability, mathematical and computational biology, networks.<br />
<br />
[https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[https://math.wisc.edu/staff/shcherbyna-tatiana/ Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[https://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[https://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
== Postdocs ==<br />
<br />
[https://www.ewbates.com/ Erik Bates] (Stanford, 2019)<br />
<br />
David Keating (UC Berkeley, 2021)<br />
<br />
David Clancy (UWashington, 2022)<br />
<br />
== Graduate students ==<br />
<br />
<br />
<br />
Max Bacharach<br />
<br />
[https://sites.google.com/wisc.edu/evan-sorensen Evan Sorensen]<br />
<br />
Yu Sun<br />
<br />
Jiaming Xu<br />
<br />
Shuqi Yu<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2022 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math/ECE/Stat 888 Topics in Mathematical Data Science<br />
<br />
Math 717 Stochastic Computational Methods<br />
<br />
<br />
<br />
<br />
'''2023 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability: Stochastic Partial Differential Equations<br />
<br />
Math/ECE/Stat 888 Topics in Mathematical Data Science</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=23468Probability2022-08-22T19:26:09Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[https://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[https://hanbaeklyu.com/ Hanbaek Lyu] (Ohio State, 2018) discrete probability, dynamical systems, networks, optimization, machine learning <br />
<br />
[https://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied discrete probability, mathematical and computational biology, networks.<br />
<br />
[https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[https://math.wisc.edu/staff/shcherbyna-tatiana/ Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[https://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[https://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
== Postdocs ==<br />
<br />
[https://www.ewbates.com/ Erik Bates] (Stanford, 2019)<br />
<br />
David Keating (UC Berkeley, 2021)<br />
<br />
David Clancy (UWashington, 2022)<br />
<br />
== Graduate students ==<br />
<br />
<br />
<br />
Max Bacharach<br />
<br />
[https://sites.google.com/wisc.edu/evan-sorensen Evan Sorensen]<br />
<br />
Yu Sun<br />
<br />
Jiaming Xu<br />
<br />
Shuqi Yu<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability: Integrable probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Undergraduate_Linear_Algebra_Courses&diff=23092Undergraduate Linear Algebra Courses2022-04-07T18:21:35Z<p>Valko: </p>
<hr />
<div>'''Note: the links to the course webpages are broken. You can find a short description of all of our courses at the [https://guide.wisc.edu/courses/math/ UW-Guide].'''<br />
<br />
<br />
<br />
In order to complete the [https://www.math.wisc.edu/undergraduate/mathmajor major in mathematics] you must take a course in linear algebra. At [http://www.math.wisc.edu/ UW-Madison] we offer several versions of linear algebra. '''Note that in all versions of the major and certificate, only ONE of the following courses may be used to fulfill any of the requirements.''' The purpose of this page is to describe the essential differences between the courses. <br />
<br />
== [https://www.math.wisc.edu/320-Linear-Algebra-Differential-Equations Math 320] Linear Algebra and Differential Equations ==<br />
Math 320 covers both some linear algebra and some differential equation theory. As such, students who complete this course can consider themselves as also having some of the content of [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319] (introduction to differential equations). The difference between this course and taking both 319 and 340 is that one will be able to see how theory and applications unite in a meaningful way. This course also lends itself to the [http://www.math.wisc.edu/321-applied-mathematical-analysis 321]-[http://www.math.wisc.edu/322-applied-mathematical-analysis 322] applied analysis sequence.<br />
<br />
Students who have completed Math 320 will need to complete either 1) [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or 2) the applied analysis sequence [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322] or [https://www.math.wisc.edu/node/786 Math 467] before moving on to the 500 level.<br />
<br />
In summary math 320 is:<br />
* Useful for students interested in classical applications of mathematics (i.e., physics, engineering, continuous modeling, etc.)<br />
* Covers material in Math 319 and therefore '''credit for only one of Math 319 and 320 can be applied to the math major or certificate'''.<br />
* Not by itstelf sufficient for taking advanced math classes.<br />
* Good introduction to how theory and applications support each other.<br />
* '''Is offered with an honors(!) version. This version is suggested for potential math majors and those in the [https://www.math.wisc.edu/amep AMEP] program'''.<br />
* Suggested further courses are <br />
** The applied analysis sequence [https://www.math.wisc.edu/321-Applied-Mathematical-Analysis Math 321] - [https://www.math.wisc.edu/322-Applied-Mathematical-Analysis math 322] which covers more mathematics useful for traditional applications.<br />
** Dynamical systems [https://www.math.wisc.edu/415-Applied-Dynamical-Systems-Chaos-Modeling Math 415] which includes both continuous and discrete models of changing systems.<br />
** Theory of Calculus [https://www.math.wisc.edu/421-Theory-Single-Variable-Calculus Math 421] for an introduction to more formal mathematical arguments.<br />
<br />
== [https://www.math.wisc.edu/340-Elementary-Matrix-Linear-Algebra Math 340] Elementary Matrix and Linear Algebra ==<br />
Math 340 a basic linear algebra course which focuses on vectors as ordered sets of real numbers and linear operators as matrices. In this course the focus is typically on computational aspects of the subject with some lighter treatment of the more theoretical points.<br />
<br />
Students who complete this course and would also like exposure to differential equations should consider [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319].<br />
<br />
Students who have completed Math 320 will need to complete either 1) [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or 2) the applied analysis sequence [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322] or [https://www.math.wisc.edu/node/786 Math 467] before moving on to the 500 level.<br />
<br />
In summary math 340 is:<br />
* Ideal for students who need functional knowledge of basic matrix algebra in particular those looking for applications featuring discrete mathematics (i.e., computer science and possibly statistics).<br />
* Not by itself sufficient for enrollment in advance Math courses<br />
* Subsequent math courses:<br />
** [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] for those interested in advanced undergradaute math courses above the 500 level.<br />
** ODE [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319] for those interested in the applied analysis sequence.<br />
<br />
== [http://www.math.wisc.edu/341-linear-algebra Math 341] Linear Algebra ==<br />
Math 341 is a linear algebra course which is also meant to be an introduction to proofs and proofwriting. The linear algebra content of the course is more robust than any of the others listed on this page. Students who complete the course should be well prepared to move onto any upper level course, in particular [https://www.math.wisc.edu/521-Analysis-1 Math 521], [https://www.math.wisc.edu/522-Analysis-2 541], or [https://www.math.wisc.edu/551-Elementary-Topology 551].<br />
<br />
It is the recommended linear algebra course for majors interested in moving to advanced undergraduate courses quickly<br />
<br />
Students who complete this course and would also like exposure to differential equations should consider [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319].<br />
<br />
In summary math 341 is:<br />
* A good introduction to proofs and proofwriting.<br />
* Will give students access to advanced level math courses.<br />
* Subsequent courses:<br />
** [https://www.math.wisc.edu/421-Theory-Single-Variable-Calculus Math 421] for another exposure to formal mathematical arguments at the introductory level.<br />
** Any math course above the 500 level (possibly assuming other prereqs).<br />
<br />
== [https://www.math.wisc.edu/375-Multivariable-Calculus-Linear-Algebra Math 375] Topics in Multi-Variable Calculus and Linear Algebra==<br />
Math 375 is an Honors course which features the role that linear algebra has in multivariable calculus. Students who have completed [https://www.math.wisc.edu/234-Calculus-Functions-Several-Variables Math 234] (Calculus - Functions of Several Variables) may not take this course.<br />
<br />
Students who complete this course are also expected to complete the sequel course: [https://www.math.wisc.edu/376-Multivariable-Calculus-Differential-Equations Math 376].<br />
<br />
In summary Math 375 is:<br />
* '''Honors level.'''<br />
* Enrollment is by permission only.<br />
* Not open to students who have credit for Math 234.<br />
* A good introduction to proofs and proofwriting.<br />
* Students who complete Math 375 and not Math 376 are not considered to have completed the content of Math 234! So by enrolling in Math 375 in the Fall, you should be prepared to enroll in Math 376 in the spring, or Math 234 in order to complete multivariate calculus.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Undergraduate_Linear_Algebra_Courses&diff=23084Undergraduate Linear Algebra Courses2022-04-06T19:09:47Z<p>Valko: /* Math 341 Linear Algebra */</p>
<hr />
<div>In order to complete the [https://www.math.wisc.edu/undergraduate/mathmajor major in mathematics] you must take a course in linear algebra. At [http://www.math.wisc.edu/ UW-Madison] we offer several versions of linear algebra. '''Note that in all versions of the major and certificate, only ONE of the following courses may be used to fulfill any of the requirements.''' The purpose of this page is to describe the essential differences between the courses. <br />
<br />
== [https://www.math.wisc.edu/320-Linear-Algebra-Differential-Equations Math 320] Linear Algebra and Differential Equations ==<br />
Math 320 covers both some linear algebra and some differential equation theory. As such, students who complete this course can consider themselves as also having some of the content of [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319] (introduction to differential equations). The difference between this course and taking both 319 and 340 is that one will be able to see how theory and applications unite in a meaningful way. This course also lends itself to the [http://www.math.wisc.edu/321-applied-mathematical-analysis 321]-[http://www.math.wisc.edu/322-applied-mathematical-analysis 322] applied analysis sequence.<br />
<br />
Students who have completed Math 320 will need to complete either 1) [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or 2) the applied analysis sequence [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322] or [https://www.math.wisc.edu/node/786 Math 467] before moving on to the 500 level.<br />
<br />
In summary math 320 is:<br />
* Useful for students interested in classical applications of mathematics (i.e., physics, engineering, continuous modeling, etc.)<br />
* Covers material in Math 319 and therefore '''credit for only one of Math 319 and 320 can be applied to the math major or certificate'''.<br />
* Not by itstelf sufficient for taking advanced math classes.<br />
* Good introduction to how theory and applications support each other.<br />
* '''Is offered with an honors(!) version. This version is suggested for potential math majors and those in the [https://www.math.wisc.edu/amep AMEP] program'''.<br />
* Suggested further courses are <br />
** The applied analysis sequence [https://www.math.wisc.edu/321-Applied-Mathematical-Analysis Math 321] - [https://www.math.wisc.edu/322-Applied-Mathematical-Analysis math 322] which covers more mathematics useful for traditional applications.<br />
** Dynamical systems [https://www.math.wisc.edu/415-Applied-Dynamical-Systems-Chaos-Modeling Math 415] which includes both continuous and discrete models of changing systems.<br />
** Theory of Calculus [https://www.math.wisc.edu/421-Theory-Single-Variable-Calculus Math 421] for an introduction to more formal mathematical arguments.<br />
<br />
== [https://www.math.wisc.edu/340-Elementary-Matrix-Linear-Algebra Math 340] Elementary Matrix and Linear Algebra ==<br />
Math 340 a basic linear algebra course which focuses on vectors as ordered sets of real numbers and linear operators as matrices. In this course the focus is typically on computational aspects of the subject with some lighter treatment of the more theoretical points.<br />
<br />
Students who complete this course and would also like exposure to differential equations should consider [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319].<br />
<br />
Students who have completed Math 320 will need to complete either 1) [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or 2) the applied analysis sequence [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322] or [https://www.math.wisc.edu/node/786 Math 467] before moving on to the 500 level.<br />
<br />
In summary math 340 is:<br />
* Ideal for students who need functional knowledge of basic matrix algebra in particular those looking for applications featuring discrete mathematics (i.e., computer science and possibly statistics).<br />
* Not by itself sufficient for enrollment in advance Math courses<br />
* Subsequent math courses:<br />
** [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] for those interested in advanced undergradaute math courses above the 500 level.<br />
** ODE [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319] for those interested in the applied analysis sequence.<br />
<br />
== [http://www.math.wisc.edu/341-linear-algebra Math 341] Linear Algebra ==<br />
Math 341 is a linear algebra course which is also meant to be an introduction to proofs and proofwriting. The linear algebra content of the course is more robust than any of the others listed on this page. Students who complete the course should be well prepared to move onto any upper level course, in particular [https://www.math.wisc.edu/521-Analysis-1 Math 521], [https://www.math.wisc.edu/522-Analysis-2 541], or [https://www.math.wisc.edu/551-Elementary-Topology 551].<br />
<br />
It is the recommended linear algebra course for majors interested in moving to advanced undergraduate courses quickly<br />
<br />
Students who complete this course and would also like exposure to differential equations should consider [https://www.math.wisc.edu/319-Tech-Ordinary-Differential-Equations Math 319].<br />
<br />
In summary math 341 is:<br />
* A good introduction to proofs and proofwriting.<br />
* Will give students access to advanced level math courses.<br />
* Subsequent courses:<br />
** [https://www.math.wisc.edu/421-Theory-Single-Variable-Calculus Math 421] for another exposure to formal mathematical arguments at the introductory level.<br />
** Any math course above the 500 level (possibly assuming other prereqs).<br />
<br />
== [https://www.math.wisc.edu/375-Multivariable-Calculus-Linear-Algebra Math 375] Topics in Multi-Variable Calculus and Linear Algebra==<br />
Math 375 is an Honors course which features the role that linear algebra has in multivariable calculus. Students who have completed [https://www.math.wisc.edu/234-Calculus-Functions-Several-Variables Math 234] (Calculus - Functions of Several Variables) may not take this course.<br />
<br />
Students who complete this course are also expected to complete the sequel course: [https://www.math.wisc.edu/376-Multivariable-Calculus-Differential-Equations Math 376].<br />
<br />
In summary Math 375 is:<br />
* '''Honors level.'''<br />
* Enrollment is by permission only.<br />
* Not open to students who have credit for Math 234.<br />
* A good introduction to proofs and proofwriting.<br />
* Students who complete Math 375 and not Math 376 are not considered to have completed the content of Math 234! So by enrolling in Math 375 in the Fall, you should be prepared to enroll in Math 376 in the spring, or Math 234 in order to complete multivariate calculus.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Problem_Solver%27s_Toolbox&diff=21537Problem Solver's Toolbox2021-09-12T15:31:55Z<p>Valko: /* Angles in the circle */</p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or up to a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
<br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Problem_Solver%27s_Toolbox&diff=21536Problem Solver's Toolbox2021-09-12T15:31:21Z<p>Valko: /* Proof by contradiction */</p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or up to a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
<br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Problem_Solver%27s_Toolbox&diff=21535Problem Solver's Toolbox2021-09-12T15:31:06Z<p>Valko: /* Modular arithmetic */</p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or up to a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
<br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Problem_Solver%27s_Toolbox&diff=21534Problem Solver's Toolbox2021-09-12T15:30:20Z<p>Valko: /* Modular arithmetic */</p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or up to a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://talent.math.wisc.edu/wp-content/uploads/sites/1570/2021/01/Talent16-2a.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
<br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Option_2_packages&diff=21201Option 2 packages2021-05-09T16:43:57Z<p>Valko: </p>
<hr />
<div>'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see the named options section in the [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext Guide page of the major]). The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options. <br />
<br />
'''<br />
<br />
<br />
<br />
<br />
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample course collections in several popular areas.<br />
<br />
NOTES: <br />
<br />
1) '''These course collections do not include course prerequisites.''' For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.<br />
<br />
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.''' For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.<br />
<br />
== Economics and Business ==<br />
<br />
=== Actuarial Mathematics ===<br />
Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.<br />
<br />
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).<br />
<br />
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.<br />
<br />
'''Application Courses'''<br />
* Act Sci 303<br />
<br />
* Act. Sci 650 and 652<br />
<br />
* Act. Sci. 651 or 653<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. <br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
** Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Linear Programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
'''Additional Courses to Consider'''<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
<br />
'''Also:''' Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:<br />
<br />
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].<br />
<br />
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.<br />
<br />
=== Business ===<br />
Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.<br />
<br />
The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[http://www.history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[http://www.informs.org/]<br />
<br />
Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.<br />
<br />
'''Application Courses'''<br />
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Note that this course cannot also be used as a core math course.<br />
* Operations Research: OTM 410<br />
* At least two from the following: Gen Bus 306, Gen Bus 307; OTM 451, 411, 633, 654<br />
** Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]<br />
<br />
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastics: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics.<br />
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].<br />
<br />
'''Also:''' Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].<br />
<br />
=== Economics ===<br />
Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.<br />
<br />
Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[http://en.wikipedia.org/wiki/Economics_imperialism]<br />
<br />
Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis. <br />
<br />
'''Application Courses'''<br />
* Microeconomics: Econ 301 or 311.<br />
* Macroeconomics: Econ 302 or 312.<br />
* Economic Electives: At least two courses from Econ 410, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].<br />
<br />
=== Finance ===<br />
Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.<br />
<br />
'''Application Courses'''<br />
* Statistics: Econ 410 or Math/Stat 310.<br />
* Finance core: Finance 300, 320, 330.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.<br />
<br />
== Physical Sciences ==<br />
The physical sciences and mathematics have grown hand-in-hand since antiquity.<br />
Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.<br />
<br />
The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.<br />
<br />
=== Atmospheric & Oceanic Sciences ===<br />
Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.<br />
<br />
A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [http://www.nsf.gov/funding/]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.<br />
<br />
Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* ATM OCN 310, 311, and 330 [http://www.aos.wisc.edu/education/Syllabus/courses_majors.html]<br />
** 310 and 330 have Physics 208/248 as a prerequisite.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619]<br />
<br />
'''Also:''' Students who are interested in this area might consider <br />
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].<br />
* The [http://www.math.wisc.edu/amep AMEP] program.<br />
<br />
=== Chemistry ===<br />
The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].<br />
<br />
All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]<br />
** Prerequisite: Chem 104 or 109 <br />
* Physical Chemistry: Chem 561 and 562<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 320 recommended.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Math 513 or 514 suggested.<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
** Any of these courses are acceptable in lieu of the 500 level courses above.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate Department of Chemistry].<br />
<br />
=== Physics ===<br />
Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.<br />
<br />
Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.<br />
<br />
'''Application Courses'''<br />
* Mechanics, Electricity, and Magnetism: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 Physics 322]<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 421 is suggested to prepare students for math 521.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. See suggested courses below.<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].<br />
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].<br />
<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
=== Astronomy ===<br />
The Astronomy package has the same mathematics core, but different suggested application courses:<br />
<br />
'''Application Courses'''<br />
* Astronomy core: Choose two courses from Astron 310, 320, or 335.<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses above the 500 level.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
== Biological Sciences ==<br />
Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[http://www.ams.org/notices/199509/hoppensteadt.pdf]. The following lists some possibilities.<br />
<br />
=== Bio-Informatics ===<br />
Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [http://www.genome.gov/12011238], it's no wonder that many mathematicians find compelling problems in the area to devote their time.<br />
<br />
Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [http://www.bioinformatics.org/jobs/]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [http://ils.unc.edu/informatics_programs/doc/Bioinformatics_2006.html].<br />
<br />
Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.<br />
<br />
'''Application Courses'''<br />
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).<br />
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]<br />
* Genetics: Gen 466<br />
** Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison [http://www.genetics.wisc.edu/UndergraduateProgram.htm genetics] program for more information.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: At least three of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
<br />
'''Also''' <br />
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].<br />
<br />
=== Bio-Statistics ===<br />
Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.<br />
<br />
Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[http://www.amstat.org/careers/biostatistics.cfm]. Students interested in this area should find excellent job prospects.<br />
<br />
Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.<br />
<br />
'''Application Courses'''<br />
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]<br />
** Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* More courses in computational mathematics listed above.<br />
* [http://www.math.wisc.edu/635-introduction-brownian-motion-and-stochastic-calculus Math 635]<br />
<br />
'''Also'''<br />
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].<br />
* Compare this major program to requirements for Medical School.<br />
<br />
=== Ecology, Forestry, Wildlife Ecology ===<br />
Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.<br />
<br />
'''Application Courses'''<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].<br />
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
=== Genetics ===<br />
Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:<br />
<br />
'''Application Courses'''<br />
* Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended for non-honors students.<br />
<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
<br />
=== Structural Biology ===<br />
Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.<br />
<br />
In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.<br />
<br />
'''Application Courses'''<br />
* Analytical Methods in Chemistry: Chem 327 or 329<br />
* Physical Chemistry: Chem 561 and 562<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
=== Systems Biology ===<br />
Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation predator-prey] model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.<br />
<br />
In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.<br />
<br />
'''Application Courses'''<br />
* Organic Chemistry: Chem 341 or 343<br />
* Introductory Biochemistry: Biochem 501<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/609-mathematical-methods-systems-biology Math 609]<br />
* One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
<br />
'''Additional Courses to Consider'''<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
== Engineering ==<br />
Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.<br />
<br />
In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the [http://www.engr.wisc.edu/current/undergrad.html College of Engineering]. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].<br />
<br />
So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.<br />
<br />
=== Chemical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.<br />
<br />
'''Application Courses'''<br />
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]<br />
** Note: All of these course are required for the undergraduate program in chemical engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
=== Civil Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Fluid Mechanics and Structural Analysis: [http://courses.engr.wisc.edu/cee/ CIV ENG 310, 311, 340]<br />
** Note: All of these course are required for the undergraduate program in civil engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]); and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
=== Electrical and Computer Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core ECE: [http://courses.engr.wisc.edu/ece/ ECE 220, 230, 352]<br />
** Note: All of these course are required for the undergraduate program in electrical and computer engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective: ECE 435, 525, or 533.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
** ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* At least two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-error-correcting-codes Math 641]<br />
<br />
===Engineering Mechanics and Astronautics===<br />
The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Engineering Mechanics: [http://courses.engr.wisc.edu/ema/ EMA 201, 202, 303]<br />
* One elective: EMA 521, 542, 545, or 563<br />
** All of the above courses may be used to satisfy the EMA program requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
<br />
=== Industrial Engineering ===<br />
Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.<br />
<br />
'''Application Courses[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''<br />
* Core Industrial engineering: I SY E 315, 320, and 323.<br />
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.<br />
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Both 309 and 431 are preferred over stat 311.<br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].<br />
<br />
'''Also:'''<br />
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html Industrial Engineering] offered by the College of Engineering.<br />
<br />
=== Materials Science ===<br />
The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Materials Courses: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]<br />
* One Engineering Elective: CBE 255, CS 300, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].<br />
** All of the above classes may be used to satisfy the program requirements for MS&E BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Mechanical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Mechanical Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]<br />
** All of the above courses are required by the Mechanical Engineering program.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Nuclear Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Nuclear Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf NE 305, 405, and 408]<br />
* One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.<br />
** All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
* Additional courses above the 500 level listed above.<br />
<br />
== Computer Science ==<br />
Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.<br />
<br />
Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.<br />
<br />
Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.<br />
<br />
=== Computational Methods ===<br />
Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level. <br />
<br />
The mathematical coursework focuses on combinatorics, analysis, and numerical methods. <br />
<br />
'''Application Courses'''<br />
* Any four courses from: CS 352, 367, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.<br />
** Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics above.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].<br />
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].<br />
<br />
'''Also:'''<br />
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science].<br />
<br />
=== Theoretical Computer Science ===<br />
If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:<br />
* Include both CS 520 and CS 577 into your core applied courses.<br />
* Replace the two computational methods courses with Math 567 and Math 571.<br />
<br />
=== Cryptography ===<br />
Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.<br />
<br />
The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.<br />
<br />
'''Application Courses'''<br />
* Programming: CS 300 and CS 400 (or CS 302 and 367).<br />
* One of the following two pairs:<br />
** The CS track: Operating systems (CS 537) and Security (CS 642)<br />
** The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435]<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]<br />
<br />
'''Additional Courses to Consider'''<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Also:'''<br />
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].<br />
<br />
== Secondary Education ==<br />
The so called STEM fields continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments. <br />
<br />
The following program was designed for a math major who is interested in becoming an educator at the secondary level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. <br />
<br />
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''<br />
<br />
'''Application Courses'''<br />
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].<br />
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]<br />
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.<br />
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 (or Math 375) suggested.<br />
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. <br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] <br />
** Math 431 and 309 are equivalent. <br />
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)<br />
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].<br />
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].<br />
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school<br />
<br />
'''Additional Courses to Consider'''<br />
* Math 421 can be a useful course to take before the 500 level coursework.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Additional courses at the 500 level in mathematics.<br />
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.<br />
<br />
== Statistics ==<br />
Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.<br />
<br />
Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.<br />
<br />
'''Application Courses'''<br />
* Core Statistics: Stat 333 and Stat 424<br />
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]<br />
** Math 431 may be used for Math 309.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].<br />
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].<br />
<br />
'''Also:'''<br />
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:<br />
* Stat 302 and 327.<br />
* A course in programming (e.g. CS 300).<br />
* At least one more course from the statistics electives above.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Option_2_packages&diff=21200Option 2 packages2021-05-09T16:43:22Z<p>Valko: </p>
<hr />
<div>'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext the Guide page of the major]). The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options. <br />
<br />
'''<br />
<br />
<br />
<br />
<br />
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample course collections in several popular areas.<br />
<br />
NOTES: <br />
<br />
1) '''These course collections do not include course prerequisites.''' For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.<br />
<br />
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.''' For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.<br />
<br />
== Economics and Business ==<br />
<br />
=== Actuarial Mathematics ===<br />
Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.<br />
<br />
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).<br />
<br />
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.<br />
<br />
'''Application Courses'''<br />
* Act Sci 303<br />
<br />
* Act. Sci 650 and 652<br />
<br />
* Act. Sci. 651 or 653<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. <br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
** Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Linear Programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
'''Additional Courses to Consider'''<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
<br />
'''Also:''' Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:<br />
<br />
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].<br />
<br />
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.<br />
<br />
=== Business ===<br />
Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.<br />
<br />
The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[http://www.history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[http://www.informs.org/]<br />
<br />
Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.<br />
<br />
'''Application Courses'''<br />
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Note that this course cannot also be used as a core math course.<br />
* Operations Research: OTM 410<br />
* At least two from the following: Gen Bus 306, Gen Bus 307; OTM 451, 411, 633, 654<br />
** Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]<br />
<br />
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastics: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics.<br />
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].<br />
<br />
'''Also:''' Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].<br />
<br />
=== Economics ===<br />
Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.<br />
<br />
Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[http://en.wikipedia.org/wiki/Economics_imperialism]<br />
<br />
Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis. <br />
<br />
'''Application Courses'''<br />
* Microeconomics: Econ 301 or 311.<br />
* Macroeconomics: Econ 302 or 312.<br />
* Economic Electives: At least two courses from Econ 410, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].<br />
<br />
=== Finance ===<br />
Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.<br />
<br />
'''Application Courses'''<br />
* Statistics: Econ 410 or Math/Stat 310.<br />
* Finance core: Finance 300, 320, 330.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.<br />
<br />
== Physical Sciences ==<br />
The physical sciences and mathematics have grown hand-in-hand since antiquity.<br />
Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.<br />
<br />
The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.<br />
<br />
=== Atmospheric & Oceanic Sciences ===<br />
Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.<br />
<br />
A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [http://www.nsf.gov/funding/]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.<br />
<br />
Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* ATM OCN 310, 311, and 330 [http://www.aos.wisc.edu/education/Syllabus/courses_majors.html]<br />
** 310 and 330 have Physics 208/248 as a prerequisite.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619]<br />
<br />
'''Also:''' Students who are interested in this area might consider <br />
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].<br />
* The [http://www.math.wisc.edu/amep AMEP] program.<br />
<br />
=== Chemistry ===<br />
The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].<br />
<br />
All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]<br />
** Prerequisite: Chem 104 or 109 <br />
* Physical Chemistry: Chem 561 and 562<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 320 recommended.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Math 513 or 514 suggested.<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
** Any of these courses are acceptable in lieu of the 500 level courses above.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate Department of Chemistry].<br />
<br />
=== Physics ===<br />
Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.<br />
<br />
Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.<br />
<br />
'''Application Courses'''<br />
* Mechanics, Electricity, and Magnetism: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 Physics 322]<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 421 is suggested to prepare students for math 521.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. See suggested courses below.<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].<br />
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].<br />
<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
=== Astronomy ===<br />
The Astronomy package has the same mathematics core, but different suggested application courses:<br />
<br />
'''Application Courses'''<br />
* Astronomy core: Choose two courses from Astron 310, 320, or 335.<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses above the 500 level.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
== Biological Sciences ==<br />
Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[http://www.ams.org/notices/199509/hoppensteadt.pdf]. The following lists some possibilities.<br />
<br />
=== Bio-Informatics ===<br />
Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [http://www.genome.gov/12011238], it's no wonder that many mathematicians find compelling problems in the area to devote their time.<br />
<br />
Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [http://www.bioinformatics.org/jobs/]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [http://ils.unc.edu/informatics_programs/doc/Bioinformatics_2006.html].<br />
<br />
Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.<br />
<br />
'''Application Courses'''<br />
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).<br />
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]<br />
* Genetics: Gen 466<br />
** Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison [http://www.genetics.wisc.edu/UndergraduateProgram.htm genetics] program for more information.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: At least three of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
<br />
'''Also''' <br />
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].<br />
<br />
=== Bio-Statistics ===<br />
Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.<br />
<br />
Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[http://www.amstat.org/careers/biostatistics.cfm]. Students interested in this area should find excellent job prospects.<br />
<br />
Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.<br />
<br />
'''Application Courses'''<br />
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]<br />
** Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* More courses in computational mathematics listed above.<br />
* [http://www.math.wisc.edu/635-introduction-brownian-motion-and-stochastic-calculus Math 635]<br />
<br />
'''Also'''<br />
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].<br />
* Compare this major program to requirements for Medical School.<br />
<br />
=== Ecology, Forestry, Wildlife Ecology ===<br />
Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.<br />
<br />
'''Application Courses'''<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].<br />
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
=== Genetics ===<br />
Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:<br />
<br />
'''Application Courses'''<br />
* Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended for non-honors students.<br />
<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
<br />
=== Structural Biology ===<br />
Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.<br />
<br />
In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.<br />
<br />
'''Application Courses'''<br />
* Analytical Methods in Chemistry: Chem 327 or 329<br />
* Physical Chemistry: Chem 561 and 562<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
=== Systems Biology ===<br />
Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation predator-prey] model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.<br />
<br />
In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.<br />
<br />
'''Application Courses'''<br />
* Organic Chemistry: Chem 341 or 343<br />
* Introductory Biochemistry: Biochem 501<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/609-mathematical-methods-systems-biology Math 609]<br />
* One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
<br />
'''Additional Courses to Consider'''<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
== Engineering ==<br />
Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.<br />
<br />
In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the [http://www.engr.wisc.edu/current/undergrad.html College of Engineering]. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].<br />
<br />
So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.<br />
<br />
=== Chemical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.<br />
<br />
'''Application Courses'''<br />
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]<br />
** Note: All of these course are required for the undergraduate program in chemical engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
=== Civil Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Fluid Mechanics and Structural Analysis: [http://courses.engr.wisc.edu/cee/ CIV ENG 310, 311, 340]<br />
** Note: All of these course are required for the undergraduate program in civil engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]); and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
=== Electrical and Computer Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core ECE: [http://courses.engr.wisc.edu/ece/ ECE 220, 230, 352]<br />
** Note: All of these course are required for the undergraduate program in electrical and computer engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective: ECE 435, 525, or 533.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
** ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* At least two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-error-correcting-codes Math 641]<br />
<br />
===Engineering Mechanics and Astronautics===<br />
The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Engineering Mechanics: [http://courses.engr.wisc.edu/ema/ EMA 201, 202, 303]<br />
* One elective: EMA 521, 542, 545, or 563<br />
** All of the above courses may be used to satisfy the EMA program requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
<br />
=== Industrial Engineering ===<br />
Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.<br />
<br />
'''Application Courses[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''<br />
* Core Industrial engineering: I SY E 315, 320, and 323.<br />
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.<br />
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Both 309 and 431 are preferred over stat 311.<br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].<br />
<br />
'''Also:'''<br />
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html Industrial Engineering] offered by the College of Engineering.<br />
<br />
=== Materials Science ===<br />
The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Materials Courses: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]<br />
* One Engineering Elective: CBE 255, CS 300, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].<br />
** All of the above classes may be used to satisfy the program requirements for MS&E BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Mechanical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Mechanical Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]<br />
** All of the above courses are required by the Mechanical Engineering program.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Nuclear Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Nuclear Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf NE 305, 405, and 408]<br />
* One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.<br />
** All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
* Additional courses above the 500 level listed above.<br />
<br />
== Computer Science ==<br />
Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.<br />
<br />
Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.<br />
<br />
Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.<br />
<br />
=== Computational Methods ===<br />
Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level. <br />
<br />
The mathematical coursework focuses on combinatorics, analysis, and numerical methods. <br />
<br />
'''Application Courses'''<br />
* Any four courses from: CS 352, 367, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.<br />
** Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics above.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].<br />
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].<br />
<br />
'''Also:'''<br />
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science].<br />
<br />
=== Theoretical Computer Science ===<br />
If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:<br />
* Include both CS 520 and CS 577 into your core applied courses.<br />
* Replace the two computational methods courses with Math 567 and Math 571.<br />
<br />
=== Cryptography ===<br />
Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.<br />
<br />
The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.<br />
<br />
'''Application Courses'''<br />
* Programming: CS 300 and CS 400 (or CS 302 and 367).<br />
* One of the following two pairs:<br />
** The CS track: Operating systems (CS 537) and Security (CS 642)<br />
** The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435]<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]<br />
<br />
'''Additional Courses to Consider'''<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Also:'''<br />
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].<br />
<br />
== Secondary Education ==<br />
The so called STEM fields continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments. <br />
<br />
The following program was designed for a math major who is interested in becoming an educator at the secondary level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. <br />
<br />
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''<br />
<br />
'''Application Courses'''<br />
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].<br />
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]<br />
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.<br />
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 (or Math 375) suggested.<br />
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. <br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] <br />
** Math 431 and 309 are equivalent. <br />
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)<br />
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].<br />
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].<br />
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school<br />
<br />
'''Additional Courses to Consider'''<br />
* Math 421 can be a useful course to take before the 500 level coursework.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Additional courses at the 500 level in mathematics.<br />
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.<br />
<br />
== Statistics ==<br />
Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.<br />
<br />
Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.<br />
<br />
'''Application Courses'''<br />
* Core Statistics: Stat 333 and Stat 424<br />
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]<br />
** Math 431 may be used for Math 309.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].<br />
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].<br />
<br />
'''Also:'''<br />
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:<br />
* Stat 302 and 327.<br />
* A course in programming (e.g. CS 300).<br />
* At least one more course from the statistics electives above.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Option_2_packages&diff=21199Option 2 packages2021-05-09T16:43:10Z<p>Valko: </p>
<hr />
<div>'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext the Guide page of the major]. The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options. <br />
<br />
'''<br />
<br />
<br />
<br />
<br />
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample course collections in several popular areas.<br />
<br />
NOTES: <br />
<br />
1) '''These course collections do not include course prerequisites.''' For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.<br />
<br />
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.''' For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.<br />
<br />
== Economics and Business ==<br />
<br />
=== Actuarial Mathematics ===<br />
Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.<br />
<br />
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).<br />
<br />
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.<br />
<br />
'''Application Courses'''<br />
* Act Sci 303<br />
<br />
* Act. Sci 650 and 652<br />
<br />
* Act. Sci. 651 or 653<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. <br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
** Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Linear Programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
'''Additional Courses to Consider'''<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
<br />
'''Also:''' Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:<br />
<br />
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].<br />
<br />
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.<br />
<br />
=== Business ===<br />
Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.<br />
<br />
The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[http://www.history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[http://www.informs.org/]<br />
<br />
Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.<br />
<br />
'''Application Courses'''<br />
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Note that this course cannot also be used as a core math course.<br />
* Operations Research: OTM 410<br />
* At least two from the following: Gen Bus 306, Gen Bus 307; OTM 451, 411, 633, 654<br />
** Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]<br />
<br />
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastics: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics.<br />
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].<br />
<br />
'''Also:''' Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].<br />
<br />
=== Economics ===<br />
Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.<br />
<br />
Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[http://en.wikipedia.org/wiki/Economics_imperialism]<br />
<br />
Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis. <br />
<br />
'''Application Courses'''<br />
* Microeconomics: Econ 301 or 311.<br />
* Macroeconomics: Econ 302 or 312.<br />
* Economic Electives: At least two courses from Econ 410, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].<br />
<br />
=== Finance ===<br />
Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.<br />
<br />
'''Application Courses'''<br />
* Statistics: Econ 410 or Math/Stat 310.<br />
* Finance core: Finance 300, 320, 330.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.<br />
<br />
== Physical Sciences ==<br />
The physical sciences and mathematics have grown hand-in-hand since antiquity.<br />
Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.<br />
<br />
The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.<br />
<br />
=== Atmospheric & Oceanic Sciences ===<br />
Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.<br />
<br />
A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [http://www.nsf.gov/funding/]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.<br />
<br />
Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* ATM OCN 310, 311, and 330 [http://www.aos.wisc.edu/education/Syllabus/courses_majors.html]<br />
** 310 and 330 have Physics 208/248 as a prerequisite.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619]<br />
<br />
'''Also:''' Students who are interested in this area might consider <br />
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].<br />
* The [http://www.math.wisc.edu/amep AMEP] program.<br />
<br />
=== Chemistry ===<br />
The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].<br />
<br />
All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]<br />
** Prerequisite: Chem 104 or 109 <br />
* Physical Chemistry: Chem 561 and 562<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 320 recommended.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Math 513 or 514 suggested.<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
** Any of these courses are acceptable in lieu of the 500 level courses above.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate Department of Chemistry].<br />
<br />
=== Physics ===<br />
Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.<br />
<br />
Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.<br />
<br />
'''Application Courses'''<br />
* Mechanics, Electricity, and Magnetism: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 Physics 322]<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 421 is suggested to prepare students for math 521.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. See suggested courses below.<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].<br />
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].<br />
<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
=== Astronomy ===<br />
The Astronomy package has the same mathematics core, but different suggested application courses:<br />
<br />
'''Application Courses'''<br />
* Astronomy core: Choose two courses from Astron 310, 320, or 335.<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses above the 500 level.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
== Biological Sciences ==<br />
Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[http://www.ams.org/notices/199509/hoppensteadt.pdf]. The following lists some possibilities.<br />
<br />
=== Bio-Informatics ===<br />
Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [http://www.genome.gov/12011238], it's no wonder that many mathematicians find compelling problems in the area to devote their time.<br />
<br />
Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [http://www.bioinformatics.org/jobs/]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [http://ils.unc.edu/informatics_programs/doc/Bioinformatics_2006.html].<br />
<br />
Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.<br />
<br />
'''Application Courses'''<br />
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).<br />
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]<br />
* Genetics: Gen 466<br />
** Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison [http://www.genetics.wisc.edu/UndergraduateProgram.htm genetics] program for more information.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: At least three of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
<br />
'''Also''' <br />
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].<br />
<br />
=== Bio-Statistics ===<br />
Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.<br />
<br />
Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[http://www.amstat.org/careers/biostatistics.cfm]. Students interested in this area should find excellent job prospects.<br />
<br />
Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.<br />
<br />
'''Application Courses'''<br />
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]<br />
** Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* More courses in computational mathematics listed above.<br />
* [http://www.math.wisc.edu/635-introduction-brownian-motion-and-stochastic-calculus Math 635]<br />
<br />
'''Also'''<br />
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].<br />
* Compare this major program to requirements for Medical School.<br />
<br />
=== Ecology, Forestry, Wildlife Ecology ===<br />
Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.<br />
<br />
'''Application Courses'''<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].<br />
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
=== Genetics ===<br />
Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:<br />
<br />
'''Application Courses'''<br />
* Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended for non-honors students.<br />
<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
<br />
=== Structural Biology ===<br />
Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.<br />
<br />
In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.<br />
<br />
'''Application Courses'''<br />
* Analytical Methods in Chemistry: Chem 327 or 329<br />
* Physical Chemistry: Chem 561 and 562<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
=== Systems Biology ===<br />
Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation predator-prey] model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.<br />
<br />
In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.<br />
<br />
'''Application Courses'''<br />
* Organic Chemistry: Chem 341 or 343<br />
* Introductory Biochemistry: Biochem 501<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/609-mathematical-methods-systems-biology Math 609]<br />
* One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
<br />
'''Additional Courses to Consider'''<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
== Engineering ==<br />
Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.<br />
<br />
In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the [http://www.engr.wisc.edu/current/undergrad.html College of Engineering]. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].<br />
<br />
So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.<br />
<br />
=== Chemical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.<br />
<br />
'''Application Courses'''<br />
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]<br />
** Note: All of these course are required for the undergraduate program in chemical engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
=== Civil Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Fluid Mechanics and Structural Analysis: [http://courses.engr.wisc.edu/cee/ CIV ENG 310, 311, 340]<br />
** Note: All of these course are required for the undergraduate program in civil engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]); and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
=== Electrical and Computer Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core ECE: [http://courses.engr.wisc.edu/ece/ ECE 220, 230, 352]<br />
** Note: All of these course are required for the undergraduate program in electrical and computer engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective: ECE 435, 525, or 533.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
** ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* At least two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-error-correcting-codes Math 641]<br />
<br />
===Engineering Mechanics and Astronautics===<br />
The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Engineering Mechanics: [http://courses.engr.wisc.edu/ema/ EMA 201, 202, 303]<br />
* One elective: EMA 521, 542, 545, or 563<br />
** All of the above courses may be used to satisfy the EMA program requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
<br />
=== Industrial Engineering ===<br />
Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.<br />
<br />
'''Application Courses[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''<br />
* Core Industrial engineering: I SY E 315, 320, and 323.<br />
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.<br />
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Both 309 and 431 are preferred over stat 311.<br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].<br />
<br />
'''Also:'''<br />
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html Industrial Engineering] offered by the College of Engineering.<br />
<br />
=== Materials Science ===<br />
The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Materials Courses: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]<br />
* One Engineering Elective: CBE 255, CS 300, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].<br />
** All of the above classes may be used to satisfy the program requirements for MS&E BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Mechanical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Mechanical Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]<br />
** All of the above courses are required by the Mechanical Engineering program.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Nuclear Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Nuclear Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf NE 305, 405, and 408]<br />
* One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.<br />
** All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
* Additional courses above the 500 level listed above.<br />
<br />
== Computer Science ==<br />
Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.<br />
<br />
Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.<br />
<br />
Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.<br />
<br />
=== Computational Methods ===<br />
Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level. <br />
<br />
The mathematical coursework focuses on combinatorics, analysis, and numerical methods. <br />
<br />
'''Application Courses'''<br />
* Any four courses from: CS 352, 367, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.<br />
** Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics above.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].<br />
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].<br />
<br />
'''Also:'''<br />
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science].<br />
<br />
=== Theoretical Computer Science ===<br />
If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:<br />
* Include both CS 520 and CS 577 into your core applied courses.<br />
* Replace the two computational methods courses with Math 567 and Math 571.<br />
<br />
=== Cryptography ===<br />
Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.<br />
<br />
The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.<br />
<br />
'''Application Courses'''<br />
* Programming: CS 300 and CS 400 (or CS 302 and 367).<br />
* One of the following two pairs:<br />
** The CS track: Operating systems (CS 537) and Security (CS 642)<br />
** The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435]<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]<br />
<br />
'''Additional Courses to Consider'''<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Also:'''<br />
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].<br />
<br />
== Secondary Education ==<br />
The so called STEM fields continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments. <br />
<br />
The following program was designed for a math major who is interested in becoming an educator at the secondary level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. <br />
<br />
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''<br />
<br />
'''Application Courses'''<br />
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].<br />
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]<br />
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.<br />
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 (or Math 375) suggested.<br />
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. <br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] <br />
** Math 431 and 309 are equivalent. <br />
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)<br />
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].<br />
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].<br />
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school<br />
<br />
'''Additional Courses to Consider'''<br />
* Math 421 can be a useful course to take before the 500 level coursework.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Additional courses at the 500 level in mathematics.<br />
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.<br />
<br />
== Statistics ==<br />
Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.<br />
<br />
Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.<br />
<br />
'''Application Courses'''<br />
* Core Statistics: Stat 333 and Stat 424<br />
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]<br />
** Math 431 may be used for Math 309.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].<br />
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].<br />
<br />
'''Also:'''<br />
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:<br />
* Stat 302 and 327.<br />
* A course in programming (e.g. CS 300).<br />
* At least one more course from the statistics electives above.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=20963Probability Seminar2021-03-09T13:26:45Z<p>Valko: /* March 18, 2021, Theo Assiotis (Edinburgh) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2021 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== January 28, 2021, no seminar ==<br />
<br />
== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==<br />
<br />
'''Dynamic polymers: invariant measures and ordering by noise'''<br />
<br />
We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.<br />
<br />
== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==<br />
<br />
'''Non-stationary fluctuations for some non-integrable models'''<br />
<br />
We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.<br />
<br />
== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==<br />
<br />
'''Signature moments to characterize laws of stochastic processes'''<br />
<br />
The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.<br />
<br />
== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==<br />
<br />
'''Random matrices, random groups, singular values, and symmetric functions'''<br />
<br />
Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.<br />
<br />
== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==<br />
<br />
'''The Coleman correspondence at the free fermion point'''<br />
<br />
Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. <br />
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. <br />
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.<br />
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent. <br />
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$. <br />
This is joint work with C. Webb (arXiv:2010.07096).<br />
<br />
== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester) ==<br />
<br />
'''The limit shape of the Leaky Abelian Sandpile Model'''<br />
<br />
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.<br />
<br />
We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.<br />
<br />
We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.<br />
<br />
== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==<br />
<br />
'''On the joint moments of characteristic polynomials of random unitary matrices'''<br />
<br />
I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.<br />
<br />
== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==<br />
'''Fluctuations of particle density for open ASEP'''<br />
<br />
I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.<br />
<br />
The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.<br />
<br />
== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University) ==<br />
<br />
<br />
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==<br />
<br />
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==<br />
<br />
== April 16, 2021, [http://www.mathjunge.com/ Matthew Junge] (CUNY) <span style="color:red">FRIDAY at 2:25pm, joint with</span> [https://www.math.wisc.edu/wiki/index.php/Applied/ACMS ACMS] ==<br />
<br />
== April 22, 2021, TBA ==<br />
<br />
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=20962Probability Seminar2021-03-09T13:26:24Z<p>Valko: /* March 18, 2021, Theo Assiotis (Edinburgh) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2021 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== January 28, 2021, no seminar ==<br />
<br />
== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==<br />
<br />
'''Dynamic polymers: invariant measures and ordering by noise'''<br />
<br />
We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.<br />
<br />
== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==<br />
<br />
'''Non-stationary fluctuations for some non-integrable models'''<br />
<br />
We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.<br />
<br />
== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==<br />
<br />
'''Signature moments to characterize laws of stochastic processes'''<br />
<br />
The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.<br />
<br />
== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==<br />
<br />
'''Random matrices, random groups, singular values, and symmetric functions'''<br />
<br />
Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.<br />
<br />
== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==<br />
<br />
'''The Coleman correspondence at the free fermion point'''<br />
<br />
Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. <br />
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. <br />
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.<br />
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent. <br />
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$. <br />
This is joint work with C. Webb (arXiv:2010.07096).<br />
<br />
== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester) ==<br />
<br />
'''The limit shape of the Leaky Abelian Sandpile Model'''<br />
<br />
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.<br />
<br />
We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.<br />
<br />
We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.<br />
<br />
== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==<br />
<br />
Title: On the joint moments of characteristic polynomials of random unitary matrices.<br />
<br />
Abstract: I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.<br />
<br />
== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==<br />
'''Fluctuations of particle density for open ASEP'''<br />
<br />
I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.<br />
<br />
The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.<br />
<br />
== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University) ==<br />
<br />
<br />
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==<br />
<br />
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==<br />
<br />
== April 16, 2021, [http://www.mathjunge.com/ Matthew Junge] (CUNY) <span style="color:red">FRIDAY at 2:25pm, joint with</span> [https://www.math.wisc.edu/wiki/index.php/Applied/ACMS ACMS] ==<br />
<br />
== April 22, 2021, TBA ==<br />
<br />
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=20161Graduate student reading seminar2020-10-18T15:41:42Z<p>Valko: /* 2020 Fall */</p>
<hr />
<div>(... in probability)<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
==2020 Fall==<br />
<br />
The graduate probability seminar will be on Zoom this semester. Please sign up for the email list if you would like to receive notifications about the talks.<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/4, 2/11: Edwin<br />
<br />
2/18, 2/25: Chaojie<br />
<br />
3/3. 3/10: Yu Sun<br />
<br />
3/24, 3/31: Tony<br />
<br />
4/7, 4/14: Tung<br />
<br />
4/21, 4/28: Tung<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=20113Probability Seminar2020-10-12T18:43:04Z<p>Valko: /* October 22, 2020, Balint Virag (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2020 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==<br />
<br />
'''Pre-Talk: (1:00pm)'''<br />
<br />
'''Neural Networks for Probabilists''' <br />
<br />
Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.<br />
<br />
'''Talk: (2:30pm)'''<br />
<br />
'''Effective Theory of Deep Neural Networks''' <br />
<br />
Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.<br />
<br />
== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==<br />
<br />
'''Some new perspectives on moments of random matrices'''<br />
<br />
The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.<br />
<br />
== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==<br />
<br />
'''Roots of random polynomials near the unit circle'''<br />
<br />
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.<br />
<br />
== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==<br />
<br />
'''Large deviations for dense random graphs: beyond mean-field'''<br />
<br />
In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.<br />
<br />
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model<br />
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.<br />
<br />
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.<br />
<br />
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==<br />
<br />
Title: '''Concentration in integrable polymer models'''<br />
<br />
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the<br />
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed<br />
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain<br />
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}). <br />
<br />
Joint work with Christian Noack.<br />
<br />
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==<br />
<br />
Title: '''The heat and the landscape'''<br />
<br />
Abstract: The directed landscape is the conjectured universal scaling limit of the<br />
most common random planar metrics. Examples are planar first passage<br />
percolation, directed last passage percolation, distances in percolation<br />
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.<br />
<br />
We show that the KPZ fixed point is characterized by the Baik Ben-Arous<br />
Peche statistics well-known from random matrix theory.<br />
<br />
This provides a general and elementary method for showing convergence to<br />
the KPZ fixed point. We apply this method to two models related to<br />
random heat flow: the O'Connell-Yor polymer and the KPZ equation.<br />
<br />
Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.<br />
<br />
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_Seminar&diff=20112Probability Seminar2020-10-12T18:41:34Z<p>Valko: /* October 22, 2020, Balint Virag (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2020 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==<br />
<br />
'''Pre-Talk: (1:00pm)'''<br />
<br />
'''Neural Networks for Probabilists''' <br />
<br />
Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.<br />
<br />
'''Talk: (2:30pm)'''<br />
<br />
'''Effective Theory of Deep Neural Networks''' <br />
<br />
Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.<br />
<br />
== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==<br />
<br />
'''Some new perspectives on moments of random matrices'''<br />
<br />
The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.<br />
<br />
== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==<br />
<br />
'''Roots of random polynomials near the unit circle'''<br />
<br />
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.<br />
<br />
== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==<br />
<br />
'''Large deviations for dense random graphs: beyond mean-field'''<br />
<br />
In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.<br />
<br />
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model<br />
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.<br />
<br />
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.<br />
<br />
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==<br />
<br />
Title: '''Concentration in integrable polymer models'''<br />
<br />
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the<br />
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed<br />
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain<br />
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}). <br />
<br />
Joint work with Christian Noack.<br />
<br />
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==<br />
<br />
Title: '''The heat and the landscape'''<br />
<br />
Abstract: The directed landscape is the conjectured universal scaling limit of the<br />
most common random planar metrics. Examples are planar first passage<br />
percolation, directed last passage percolation, distances in percolation<br />
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.<br />
<br />
We show that the KPZ fixed point is characterized by the Baik Ben-Arous<br />
Peche statistics well-known from random matrix theory.<br />
<br />
This provides a general and elementary method for showing convergence to<br />
the KPZ fixed point. We apply this method to two models related to<br />
random heat flow: the O'Connell-Yor polymer and the KPZ equation.<br />
<br />
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability_group_timetable&diff=19627Probability group timetable2020-09-01T20:33:59Z<p>Valko: </p>
<hr />
<div>2020 Fall<br />
<br />
<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| || || || || <br />
|- <br />
| 10-11|| || || || || <br />
|-<br />
| 11-12|| || || || ||<br />
|-<br />
| 12-1|| || || || || <br />
|-<br />
| 1-2|| || || || ||<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || || probability seminar (2:25) || <br />
|-<br />
| 3-4|| || || || || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
<br />
<br />
<!-- <br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br />
|-<br />
| 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br />
|-<br />
| 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br />
|-<br />
| 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br />
|-<br />
| 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br />
|-<br />
| 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br />
|-<br />
| 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--><br />
<br />
<!--<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br />
|-<br />
| 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br />
|-<br />
| 11-12|| || Hans 820 || || Hans 820 ||<br />
|-<br />
| 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br />
|-<br />
| 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br />
|-<br />
| 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br />
|-<br />
| 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--></div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19626Probability2020-09-01T20:33:44Z<p>Valko: /* Graduate Courses in Probability */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability: Integrable probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19625Probability2020-09-01T20:33:26Z<p>Valko: /* Graduate students */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19520Probability2020-08-04T21:49:25Z<p>Valko: /* Graduate Courses in Probability */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19519Probability2020-08-04T21:49:16Z<p>Valko: /* Graduate Courses in Probability */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
Math/Stat 735 Stochastic Analysis<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19518Probability2020-08-04T21:47:32Z<p>Valko: /* Postdocs */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19517Probability2020-08-04T21:47:22Z<p>Valko: /* Postdocs */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19516Probability2020-08-04T21:46:43Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19515Probability2020-08-04T21:46:26Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19514Probability2020-08-04T21:46:14Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[[http://www.math.wisc.edu/??? Tatyana Shcherbyna]] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Probability&diff=19513Probability2020-08-04T21:45:56Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[[???? Tatyana Shcherbyna]] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Transition_to_proof_courses&diff=18950Transition to proof courses2020-02-07T15:03:48Z<p>Valko: Created page with "Under construction Math 341 Math 375 Math 421 Math 467"</p>
<hr />
<div>Under construction<br />
<br />
<br />
Math 341<br />
Math 375<br />
Math 421<br />
Math 467</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=18747Graduate student reading seminar2020-01-22T22:57:27Z<p>Valko: /* 2020 Spring */</p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/4, 2/11: Edwin<br />
<br />
2/18, 2/25: Chaojie<br />
<br />
3/3. 3/10: Yu Sun<br />
<br />
3/24, 3/31: Tony<br />
<br />
4/7, 4/14: Tung<br />
<br />
4/21, 4/28: Tung<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=18621Graduate student reading seminar2020-01-13T23:16:14Z<p>Valko: </p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=18620Graduate student reading seminar2020-01-13T23:13:55Z<p>Valko: /* 2018 Fall */</p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valko