Difference between revisions of "Probability Seminar"

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__NOTOC__
 
__NOTOC__
  
== Fall 2011 ==
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= Spring 2020 =
  
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<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
 +
<b>We  usually end for questions at 3:20 PM.</b>
  
Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit [https://www-old.cae.wisc.edu/mailman/listinfo/apseminar this page] to sign up for the email list.
+
If you would like to sign up for the email list to receive seminar announcements then please send an email to
 +
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
  
 +
 +
== January 23, 2020, [https://www.math.wisc.edu/~seppalai/ Timo Seppalainen] (UW Madison) ==
 +
'''Non-existence of bi-infinite geodesics in the exponential corner growth model
 +
'''
  
[[Past Seminars]]
+
Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s.  A non-existence proof  in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in  November 2018. Their proof utilizes estimates from integrable probability.    This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).
  
 +
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
 +
'''Quasi-linear parabolic equations with singular forcing'''
  
 +
The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral.  By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise.  In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise.  The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.
  
== Thursday, September 15, Jun Yin, University of Wisconsin - Madison ==
+
In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution.  These are known as quasi-linear equations.  Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization.  Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE.  This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.
'''Some recent results on random matrices with almost independent entries.'''
 
  
In this talk, we are going to introduce some recent work on a large class of random matrices, whose entries are (almost) independent. For example, the Wigner matrix, generalized Wigner matrix, Band random matrix, Covariance matrix and Sparse random matrix. We mainly focus on the local statistics of the  eigenvalues and eigenvectors of these random matrix ensembles. We will also introduce some applications of these results and some long-standing open questions.
+
== February 6, 2020, [https://sites.google.com/site/cyleeken/ Cheuk-Yin Lee] (Michigan State) ==
 +
'''Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points'''
  
 +
In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.
  
 +
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
'''Langevin Monte Carlo Without Smoothness'''
  
== Thursday, September 22, Philip Matchett Wood, University of Wisconsin - Madison ==
+
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation.
 +
Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.
  
'''Survey of the Circular Law'''
+
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
'''A replacement principle for perturbations of non-normal matrices'''
  
What do the eigenvalues of a random matrix look like?  This talk will focus on large square matrices where the entries are independent, identically distributed random variablesIn the most basic case, the distribution of the eigenvalues in the complex plane (suitably scaled) approaches the uniform distribution on the unit disk, which is called the circular lawWe will discuss some of the methods that have been used to prove the circular law, including recent work that has extended the circular law to the most general situation, and we will also discuss generalizations to situations where the eigenvalue distributions are stable, but non-circular.
+
There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added.  However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are addedMuch of the work is this situation has focused on iid random gaussian perturbationsIn this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure.  Interestingly, this even allows for deterministic perturbations to be considered.  Joint work with Sean O'Rourke.
  
== Thursday, September 29, Antonio Auffinger, University of Chicago ==
+
== February 27, 2020, No seminar ==
 +
''' '''
  
'''A simplified proof of the relation between scaling exponents in first passage percolation'''
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== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
 +
''' Large Deviation Principles via Spherical Integrals'''
  
In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved a version of this conjecture using a strong definition of the exponents. I will discuss work I just completed with Michael Damron, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the scaling relation. One advantage of our argument is that it does not require a non-trivial technical assumption of Chatterjee on the weight distribution.
+
In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain
  
== Tuesday, October 4, 2:30 PM, VV901, Gregorio Moreno Flores, University of Wisconsin - Madison ==
+
1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;
<span style="color:#FF0000">'''UNUSUAL TIME'''<span style="color:#009000">
 
  
'''Airy process and the polymer end point distribution'''
+
2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;
  
We give an explicit formula for the joint density of the max and argmax of the Airy process minus a parabola.  The argmax has a universal distribution which governs the rescaled endpoint of directed polymers in 1+1 dimensions.
+
3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;
  
 +
4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.
  
== Thursday, October 20, Kay Kirkpatrick, University of Illinois at Urbana-Champaign ==
+
This is a joint work with Belinschi and Guionnet.
  
'''Bose-Einstein condensation and a phase transition for the nonlinear Schrodinger equation'''
+
== March 12, 2020, No seminar ==
 +
''' '''
  
Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss work with Sourav Chatterjee about a phase transition for invariant measures of the discrete focusing NLS. Using techniques from probability theory, we show that the thermodynamics of the NLS are exactly solvable in dimensions three and higher. There are a number of consequences of this result, including a prediction for experimentalists to look for a new spatially localized phase of BEC.
+
== March 19, 2020, Spring break ==
 +
''' '''
  
== Monday, October 31, 2:30pm, VV B341, Ankit Gupta, Ecole Polytechnique, Centre de Mathematiques Appliqees ==
+
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
<span style="color:#FF0000">'''UNUSUAL TIME AND PLACE'''<span style="color:#009000">
+
''' '''
  
'''Modeling adaptive dynamics for structured populations with functional traits'''
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== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
 +
''' '''
  
We develop the framework of adaptive dynamics for populations that are structured by age and functional traits. The functional trait of an individual may express itself differently during the life of an individual according to her age and a random parameter that is chosen at birth to capture the environmental stochasticity. The population evolves through birth, death and selection mechanisms. At each birth, the new individual may be a clone of its parent or a mutant. Starting from an individual based model we use averaging techniques to take the large population and rare mutation limit under a well-chosen time-scale separation. This gives us the Trait Substitution Sequence process that describes the adaptive dynamics in our setting. Assuming small mutation steps we also derive the Canonical Equation which expresses the evolution of advantageous traits as a function-valued ordinary differential equation.
+
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
This is joint work with J.A.J Metz (Leiden University) and V.C. Tran (University of Lille).
+
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 +
''' '''
  
== Friday, November 4, 2:30pm, VV B341, Michael Cranston, University of California, Irvine ==
+
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
<span style="color:#FF0000">'''UNUSUAL TIME AND PLACE'''<span style="color:#009000">
 
  
'''Overlaps and Pathwise Localization in the Anderson Polymer Model'''
+
3-day event in Van Vleck 911
  
We consider large time behavior of typical paths under the Anderson poly-
+
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
mer measure. We establish that the polymer measure gives a macroscopic mass to a  typical path and this mass grows to 1 in the limit as a particular parameter tends to infinity.  This gives a rigourous approach to the polymer localization. The localization is measured by considering the overlap between two independent samples drawn from the polymer measure.
 
  
== Thursday, November 10, David Anderson, University of Wisconsin - Madison ==
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
  
 +
== April 30, 2020, CANCELLED, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
'''Computational methods for stochastic models in biology'''
 
  
I will focus on computational methods for continuous time Markov chains, which includes the large class of stochastically modeled biochemical reaction networks and population processes.    I will show how different computational methods can be understood and analyzed by using different representations for the processes.  Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.
 
  
== Wednesday, December 7,  2:30pm, B313, Alexander Fish, University of Wisconsin - Madison ==
 
<span style="color:#FF0000">'''UNUSUAL TIME AND PLACE'''<span style="color:#009000">
 
  
'''Fast Matched Filter algorithm and applications'''
 
  
We will explain the mathematical model of a wireless communication.
 
One of the main problems is finding (time,frequency) shift of a signal in a noisy environment which is caused by time asynchronization of a sender with a receiver and by a non-zero speed of a sender w.r.t a receiver.
 
A classical solution (Matched Filter Algorithm) of a discrete analog of the problem uses a pseudo-random waveform S(t) of the length p and gives rise to the complexity p^2 log(p) operations (using fast Fourier transform).
 
We will explain how to use techniques from group representation theory to construct waveforms S(t) which enable us to introduce a fast matched filter  algorithm, called the "flag algorithm", which solves (time,frequency) shift problem in O(p*log (p)) operations.
 
We will discuss applications to radars, GPS system, and Mobile Communication.
 
  
This is a joint work with S. Gurevich (Mathematics, UW Madison), R. Hadani
 
(Mathematics, UT Austin), A. Sayeed (Electrical Engineering, UW Madison),
 
and O. Schwartz (Computer Science, UC Berkeley).
 
  
  
 
+
[[Past Seminars]]
== Spring 2012 ==
 
 
 
 
 
== Thursday, January 26, Timo Seppalainen, University of Wisconsin - Madison ==
 
 
 
== Thursday, February 9, Scott Armstrong, University of Wisconsin - Madison ==
 
 
 
== Thursday, February 22, Tom Kurtz, University of Wisconsin - Madison ==
 
 
 
== Thursday, March 7, Paul Bourgade, Harvard ==
 
 
 
== Thursday, April  19, Nancy Garcia, Universidade Estadual de Campinas ==
 
 
 
== Thursday, April  26, Jim Kuelbs, University of Wisconsin - Madison ==
 

Latest revision as of 13:59, 12 April 2020


Spring 2020

Thursdays in 901 Van Vleck Hall at 2:30 PM, unless otherwise noted. We usually end for questions at 3:20 PM.

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu


January 23, 2020, Timo Seppalainen (UW Madison)

Non-existence of bi-infinite geodesics in the exponential corner growth model

Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s. A non-existence proof in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in November 2018. Their proof utilizes estimates from integrable probability. This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).

January 30, 2020, Scott Smith (UW Madison)

Quasi-linear parabolic equations with singular forcing

The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral. By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise. The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.

In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

February 6, 2020, Cheuk-Yin Lee (Michigan State)

Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points

In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.

February 13, 2020, Jelena Diakonikolas (UW Madison)

Langevin Monte Carlo Without Smoothness

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation. Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.

February 20, 2020, Philip Matchett Wood (UC Berkeley)

A replacement principle for perturbations of non-normal matrices

There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.

February 27, 2020, No seminar

March 5, 2020, Jiaoyang Huang (IAS)

Large Deviation Principles via Spherical Integrals

In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain

1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;

2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;

3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;

4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.

This is a joint work with Belinschi and Guionnet.

March 12, 2020, No seminar

March 19, 2020, Spring break

March 26, 2020, CANCELLED, Philippe Sosoe (Cornell)

April 2, 2020, CANCELLED, Tianyu Liu (UW Madison)

April 9, 2020, CANCELLED, Alexander Dunlap (Stanford)

April 16, 2020, CANCELLED, Jian Ding (University of Pennsylvania)

April 22-24, 2020, CANCELLED, FRG Integrable Probability meeting

3-day event in Van Vleck 911

April 23, 2020, CANCELLED, Martin Hairer (Imperial College)

Wolfgang Wasow Lecture at 4pm in Van Vleck 911

April 30, 2020, CANCELLED, Will Perkins (University of Illinois at Chicago)





Past Seminars