Difference between revisions of "Probability Seminar"

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__NOTOC__
 
__NOTOC__
  
= Fall 2014 =
+
= Spring 2020 =
  
 +
<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>
  
 +
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]
  
Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.  
+
 +
== 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
 +
'''
  
<b>
+
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).
If you would like to sign up for the email list to receive seminar announcements then please send an email to [[File:probsem.jpg]]
 
</b>
 
  
= =
+
== 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, February 6, [http://people.mbi.ohio-state.edu/newby.23/ Jay Newby], [http://mbi.osu.edu/ Mathematical Biosciences Institute]  ==
 
  
Title: Applications of large deviation theory in neuroscience
+
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.
  
Abstract:
+
== February 6, 2020, [https://sites.google.com/site/cyleeken/ Cheuk-Yin Lee] (Michigan State) ==
The membrane voltage of a neuron is modeled with a piecewise deterministic stochastic process.  The membrane voltage changes deterministically while the population of open ion channels, which allow current to flow across the membrane, is constant.  Ion channels open and close randomly, and the transition rates depend on voltage, making the process nonlinear.  In the limit of infinite transition rates, the process becomes deterministic. The deterministic process is the well known Morris-Lecar model.  Under certain conditions, the deterministic process has one stable fixed point and is excitable.  An excitable event, called an action potential, is a single large transient spike in voltage that eventually returns to the stable steady state.  I will discuss recent development of large deviation theory to study noise induced action potentials.
+
'''Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points'''
  
== Thursday, February 13, [http://www.math.wisc.edu/~holcomb/ Diane Holcomb], [http://www.math.wisc.edu/ UW-Madison] ==
+
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.
  
Title: Large deviations for point process limits of random matrices.
+
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
'''Langevin Monte Carlo Without Smoothness'''
  
Abstract: The Gaussian Unitary ensemble (GUE) is one of the most studied Hermitian random matrix model. When appropriately rescaled the eigenvalues in the bulk of the spectrum converge to a translation invariant limiting point process called the Sine process. On large intervals one expects the Sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.
+
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.
  
== Thursday, February 20, Philip Matchett Wood, UW-Madison ==
+
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
'''A replacement principle for perturbations of non-normal matrices'''
  
Title: The empirical spectral distribution (ESD) of a fixed matrix plus small random noise.
+
There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is addedHowever, 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.
 
 
Abstract: A fixed matrix has a distribution of eigenvalues in the complex
 
planeSmall random noise can be formed by a random matrix with iid mean 0
 
variance 1 entries scaled by <math>n^{-\gamma -1/2}</math> for <math>\gamma > 0</math>, which
 
by itself has eigenvalues collapsing to the origin.  What happens to the
 
eigenvalues when you add a small random noise matrix to the fixed matrix?
 
There are interesting cases where the eigenvalue distribution is known to
 
change dramatically when small Gaussian random noise is added, and this talk
 
will focus on what happens when the noise is <i>not</i> Gaussian..
 
  
== Thursday, February 27, [http://mypage.iu.edu/~jthanson/ Jack Hanson], Indiana University Bloomington ==
+
== February 27, 2020, No seminar ==
 +
''' '''
  
Title: '''Subdiffusive Fluctuations in First-Passage Percolation'''
+
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
 +
''' Large Deviation Principles via Spherical Integrals'''
  
Abstract: First-passage percolation is a model consisting of a random metric t(x,y) generated by random variables associated to edges of a graph. Many questions and conjectures in this model revolve around the fluctuating properties of this metric on the graph Z^d. In the early 1990s, Kesten showed an upper bound of Cn for the variance of t(0,nx); this was improved to Cn/log(n) by Benjamini-Kalai-Schramm and Benaim-Rossignol for particular choices of distribution. I will discuss recent work (with M. Damron and P. Sosoe) extending this upper bound to general classes of distributions.
+
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 $U$;
  
== Thursday, March 20, No Seminar due to Spring Break ==
+
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;
  
== Thursday, March 27, [http://www.stat.wisc.edu/~ane/ Cécile Ané], UW-Madison Department of Statistics ==
+
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;
  
Title: <b> Application of a birth-death process to model gene gains and losses on a phylogenetic tree </b>
+
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.
  
Abstract:
+
This is a joint work with Belinschi and Guionnet.
Over time, genes can duplicate or be lost. The history of a gene family is a tree whose nodes represent duplications, speciations, or losses. A birth-and-death process is used to model this gene family tree, embedded within a species tree. I will present this phylogenetic version of the birth and death tree process, along with a probability model for whole-genome duplications. If there is interest and time, I will talk about learning birth and death rates and detecting ancient whole-genome duplications from genomic data.
 
  
 +
== March 12, 2020, No seminar ==
 +
''' '''
  
== Thursday, April 10, [https://www.math.ucdavis.edu/~romik/home/Dan_Romik_home.html Dan Romik] UC-Davis ==
+
== March 19, 2020, Spring break ==
 +
''' '''
  
Title: <b>Connectivity patterns in loop percolation and pipe percolation</b>
+
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
Abstract:
+
== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
 +
''' '''
  
Loop percolation is a random collection of closed cycles in the square lattice Z^2, that is closely related to critical bond percolation. Its "connectivity pattern" is a random noncrossing matching associated with a loop percolation configuration that encodes information about connectivity of endpoints. The same probability measure on noncrossing matchings arises in several different and seemingly unrelated settings, for example in connection with alternating sign matrices, the quantum XXZ spin chain, and another type of percolation model called pipe percolation. In the talk I will describe some of these connections and discuss some results about the study of pipe percolation from the point of view of the theory of interacting particle systems. I will also mention the "rationality phenomenon" which causes the probabilities of certain natural connectivity events to be dyadic rational numbers such as 3/8, 97/512 and 59/1024. The reasons for this are not completely understood and are related to certain algebraic conjectures that I will discuss separately in Friday's talk in the Applied Algebra seminar.
+
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
 +
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 +
''' '''
  
== Thursday, May 1, [http://math.uchicago.edu/~auffing/ Antonio Auffinger] U Chicago ==
+
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
  
 +
3-day event in Van Vleck 911
  
Title: '''Strict Convexity of the Parisi Functional'''
+
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
  
== Thursday, May 8, [http://wid.wisc.edu/profile/steve-goldstein/ Steve Goldstein], [http://wid.wisc.edu/ WID]==
+
== April 30, 2020, CANCELLED, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
  
Title: '''Modeling patterns of DNA sequence diversity with Cox Processes'''
 
  
Abstract:
 
Events in the evolutionary history of a population can leave subtle signals in the patterns of diversity of its DNA sequences. Identifying those signals from the DNA sequences of present-day populations and using them to make inferences about selection is a well-studied and challenging problem.
 
  
Next generation sequencing provides an opportunity for making inroads on that problem.  In this talk, I will present a novel model for the analysis of sequence diversity data and use the model to motivate analyses of whole-genome sequences from 11 strains of Drosophila pseudoobscura.
 
  
The model treats the polymorphic sites along the genome as a realization of a Cox Process, a point process with a random intensity.  Within the context of this model, the underlying problem translates to making inferences about the distribution of the intensity function, given the sequence data.
 
 
We give a proof of principle, showing that even a simplistic application of the model can quantify differences in diversity between regions with varying recombination rates.  We also suggest a number of directions for applying and extending the model.
 
-->
 
 
 
== ==
 
  
  
  
 
[[Past Seminars]]
 
[[Past Seminars]]

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