Difference between revisions of "Probability Seminar"

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(Thursday, May 3, Samuel Isaacson, Boston University)
(April 23, 2020, Martin Hairer (Imperial College))
 
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__NOTOC__
 
__NOTOC__
  
== Spring 2012 ==
+
= 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>
  
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).
 
 
  
== Thursday, January 26, Timo Seppäläinen, University of Wisconsin - Madison ==
+
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
 +
'''Quasi-linear parabolic equations with singular forcing'''
  
Title: The exactly solvable log-gamma polymer
+
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.
  
Abstract:  Among 1+1 dimensional directed lattice polymerslog-gamma distributed weights are a special case that is amenable to various useful exact calculations (an ''exactly solvable'' case).   This talk discusses various aspects of the log-gamma model, in particular an approach to analyzing the model through a geometric or "tropical" version of the Robinson-Schensted-Knuth correspondence.
+
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.
  
== Thursday, February 9, Arnab Sen, Cambridge ==
+
== 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'''
  
Title: Random Toeplitz matrices
+
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.
  
Abstract: Random Toeplitz matrices belong to the exciting area that lies at the intersection of the usual Wigner random matrices and random Schrodinger operators. In this talk I will describe two recent results on random Toeplitz matrices. First, the maximum eigenvalue, suitably normalized, converges to the 2-4 operator norm of the well-known Sine kernel. Second, the limiting eigenvalue distribution is absolutely continuous, which partially settles a conjecture made by Bryc, Dembo and Jiang (2006). I will also present several open questions and conjectures.
+
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
'''Langevin Monte Carlo Without Smoothness'''
  
This is a joint work with Balint Virag (Toronto).  
+
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, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
'''A replacement principle for perturbations of non-normal matrices'''
  
== Thursday, February 16, Benedek Valko, University of Wisconsin - Madison ==
+
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.
  
Title: Point processes and carousels
+
== February 27, 2020, No seminar ==
 +
''' '''
  
Abstract: For several classical matrix models the joint density of the eigenvalues can be written as an expression involving a Vandermonde determinant raised to the power of 1, 2 or 4. Most of these examples have beta-generalizations where this exponent is replaced by a parameter beta>0.
+
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
In recent years the point process limits of various beta ensembles have been derived. The limiting processes are usually described as the spectrum of certain stochastic operators or with the help of a coupled system of SDEs.
+
''' Large Deviation Principles via Spherical Integrals'''
In the bulk beta Hermite case (which is the generalization of GUE) there is a nice geometric construction of the point process involving a Brownian motion in the hyperbolic plane, this is the Brownian carousel. Surprisingly, there are a number of other limit processes that have carousel like representation. We will discuss a couple of examples and some applications of these new representations.
 
  
Joint with Balint Virag.
+
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, February 23, Tom Kurtz, University of Wisconsin - Madison ==
+
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;
  
Title: Particle representations for SPDEs and strict positivity of solutions
+
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;
  
Abstract: Stochastic partial differential equations arise naturally as limits of finite systems of interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. The support properties of the measure-valued solution can be studied using Girsanov change of measure techniques. The ideas will be illustrated by a model of asset prices set by an infinite system of competing traders. These latter results are joint work with Dan Crisan and Yoonjung Lee.
+
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.
  
== Wednesday, February 29, 2:30pm, Scott Armstrong, University of Wisconsin - Madison ==
+
== March 12, 2020, No seminar ==
 +
''' '''
  
VV B309
+
== March 19, 2020, Spring break ==
 +
''' '''
  
Title: PDE methods for diffusions in random environments
+
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
Abstract: I will summarize some recent work with Souganidis on the stochastic
+
== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
homogenization of (viscous) Hamilton-Jacobi equations. The
+
''' '''
homogenization of (special cases of) these equations can be shown to
 
be equivalent to some well-known results of Sznitman in the 90s on the
 
quenched large deviations of Brownian motion in the presence of
 
Poissonian obstacles. I will explain the PDE point of view and
 
speculate on some further connections that can be made with
 
probability.
 
  
== Wednesday, March 7, 2:30pm, Paul Bourgade, Harvard ==
+
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
VV B309
+
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 +
''' '''
  
Title: Universality for beta ensembles.
+
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
  
Abstract: Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis is for ensembles of large but finite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erdos and H.-T. Yau, which yields local universality for the log-gases at arbitrary temperature, for analytic external potential. The involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.
+
3-day event in Van Vleck 911
  
== Thursday, March 8, William Stanton, UC Boulder ==
+
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
Title: An Improved Right-Tail Upper Bound for a KPZ "Crossover Distribution"
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
  
Abstract:  In the last decade, there has been an explosion of research on KPZ universality: the notion that the Kardar-Parisi-Zhang stochastic PDE from statistical physics describes the fluctuations of a large class of models. In 2010, Amir, Corwin, and Quastel proved that the KPZ equation arises as a scaling limit of the Weakly Asymmetric Simple Exclusion Process (WASEP).  In this sense, the KPZ equation interpolates between the KPZ universality class and the Edwards-Wilkinson class.  Thus, the distributions of height functions of the KPZ equation are called the "crossover distributions."  In this talk, I will introduce the notion of KPZ universality and the crossover distributions and present a new result giving an improved upper bound for the right-tail of a particular crossover distribution.
+
== April 30, 2020, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
== Friday, April  13, 2:30pm,  Gregory Shinault, UC Davis ==
 
VV B305
 
  
Title: Inhomogeneous Tilings of the Aztec Diamond
 
  
Random domino tilings of the Aztec diamond have been a
 
subject of much interest in the past 20 years. The main result of the
 
subject, the Arctic Circle theorem, is a gem of modern mathematics
 
which gives the limiting shape of the tiling. When we examine
 
fluctuations around the limiting shape, we do not see a Gaussian
 
distribution as one might expect in classical probability. Instead we
 
see the Tracy-Widom GUE distribution of random matrix theory. These
 
theorems were originally proven for a random tiling chosen by a
 
uniform distribution. In this talk we examine the effects of choosing
 
the tiling via a distribution in an inhomogeneous environment (and
 
we'll explain what we mean by this!).
 
  
== Thursday, April  19, Nancy Garcia, Universidade Estadual de Campinas ==
 
Title: Perfect simulation for chains and processes with infinite range
 
  
Abstract: In this talk we discuss how to perform Kalikow-type decompositions
 
for discontinuous chains of infinite memory and for interacting particle
 
systems
 
with interactions of infinite range. Then, we will show how this
 
decomposition can be used
 
to generate samples from these systems.
 
  
== Thursday, April  26, Jim Kuelbs, University of Wisconsin - Madison ==
 
  
  
A CLT for Empirical Processes and Empirical Quantile Processes  Involving Time Dependent Data
+
[[Past Seminars]]
 
 
We establish empirical quantile process CLT's based on <math>n</math> independent copies of a stochastic process <math>\{X_t: t  \in E\}</math> that are uniform in <math>t \in E</math> and quantile levels <math>\alpha \in I</math>, where <math>I</math> is a closed sub-interval of <math>(0,1)</math>. Typically <math>E=[0,T]</math>, or a finite product of such intervals. Also included are CLT's  for the empirical process based on <math>\{I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R \}</math> that are uniform in <math>t \in E, y \in R</math>. The process <math>\{X_t: t \in E\}</math> may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.
 
 
 
== Monday, April 30, 2:30pm, Lukas Szpruch, Oxford ==
 
VV B337
 
 
 
Title: Antithetic multilevel Monte Carlo method.
 
 
 
 
We introduce a new multilevel Monte Carlo
 
(MLMC) estimator for multidimensional SDEs driven by Brownian motion.
 
Giles has previously shown that if we combine a numerical approximation
 
with strong order of convergence $O(\D t)$ with MLMC we can reduce
 
the computational complexity to estimate expected values of
 
functionals of SDE solutions with a root-mean-square error of $\eps$
 
from $O(\eps^{-3})$ to $O(\eps^{-2})$. However, in general, to obtain
 
a rate of strong convergence higher than  $O(\D t^{1/2})$ requires
 
simulation, or approximation, of \Levy areas.  Through
 
the construction of a suitable antithetic multilevel correction estimator,
 
we are able to avoid the simulation of \Levy areas and still achieve an
 
$O(\D t2)$ variance for smooth payoffs, and almost an $O(\D t^{3/2})$ variance for
 
piecewise  smooth payoffs,
 
even though there is only $O(\D t^{1/2})$ strong convergence. This
 
results in an $O(\eps^{-2})$ complexity for estimating the
 
value of European and Asian put and call options.
 
We also comment on the extension of the  antithetic
 
approach to pricing Asian and barrier options.
 
 
 
== Wednesday, May 2, Wenbo Li, University of Delaware  ==
 
Title: Probabilities of all real zeros for random polynomials
 
 
 
Abstract: There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables.
 
We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions.
 
The talk is accessible to undergraduate and graduate students in any areas of mathematics.
 
 
 
== Thursday, May 3, Samuel Isaacson, Boston University ==
 
 
 
VV B337
 
 
 
Title: Relationships between several particle-based stochastic reaction-diffusion models.
 
 
 
Abstract:
 
Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.
 
 
 
As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.
 

Latest revision as of 13:51, 24 March 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, Will Perkins (University of Illinois at Chicago)





Past Seminars