Difference between revisions of "Past Probability Seminars Spring 2020"

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__NOTOC__
 
__NOTOC__
  
== Fall 2010 ==
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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.
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<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.
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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.
 +
 
 +
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, [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'''
 +
 
 +
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'''
 +
 
 +
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 ==
 +
''' '''
  
==Friday, September 3, 4PM B239 [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen (UW Madison)]   ([[Colloquia|Math Colloquium]])==
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== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
:Title: '''Scaling exponents for a 1+1 dimensional directed polymer'''
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''' Large Deviation Principles via Spherical Integrals'''
  
:Abstract: Directed polymer in a random environment is a model from statistical physics that has been around for 25 years. It is a type of random walk that evolves in a random potential. This means that the walk lives in a random landscape, some parts of which are favorable and other parts unfavorable to the walk. The objective is to understand the behavior of the walk on large space and time scales.
+
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
:I will begin the talk with simple random walk straight from undergraduate probability and explain what diffusive behavior of random walk means and how Brownian motion figures into the picture. The recent result of the talk concerns a particular 1+1 dimensional polymer model: the order of magnitude of the fluctuations of the polymer path is described by the exponent 2/3, in contrast with the exponent 1/2 of diffusive paths. Finding a rigorous proof of this exponent has been an open problem since the introduction of the model.
 
  
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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, September 16, [http://www.math.wisc.edu/~moreno/ Gregorio Moreno Flores (UW - Madison)]==
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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: '''Asymmetric directed polymers in random environments.'''
 
  
:Abstract: It is well known that the asymmetric last passage percolation problem can be approximated by a Brownian percolation model, which is itself related to the GUE random matrices. This allows to transfer many results about random matrices to the setting of asymmetric last passage percolation.<br>
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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;
:In this talk, we will introduce two different schemes to treat asymmetric directed polymers in random environments.
 
  
==Thursday, September 30, [http://math.colorado.edu/~brider/ Brian Rider (University of Colorado at Boulder)]==
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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.
:Title: '''Solvable two-charge models'''
 
  
:Abstract: I'll describe recent progress on ensembles of random matrix type which can be viewed as having particles of two distinct "charges", subject to coulombic interaction.  The natural (and classic) example is Ginibre's  non-symmetric Gaussian matrix in which the particles (eigenvalues) live in the complex plane.  Taking this as a starting point and forcing the particles down to the line produces a family of ensembles which interpolate (though not in the way we might want) between the well studied Gaussian Orthogonal and Symplectic Ensembles.
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This is a joint work with Belinschi and Guionnet.
  
:Joint work with Christopher Sinclair and Yuan Xu (Univ. Oregon).
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== March 12, 2020, No seminar ==
 +
''' '''
  
==Thursday, October 7, [http://www.math.wisc.edu/~valko/ Benedek Valko (UW - Madison)]==
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== March 19, 2020, Spring break ==
:Title: '''Scaling limits of tridiagonal matrices'''
+
''' '''
  
:Abstract: I will describe the point process limits of the spectrum for a certain class of tridiagonal matrices. The limiting point process can be defined through a coupled system of stochastic differential equations. I will discuss various applications of this description, e.g. eigenvalue repulsion, probability of large gaps and central limit theorem for the number of points in an interval.
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== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
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== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
 +
''' '''
  
:Joint work with E. Kritchevski and B. Virag (Toronto).
+
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
==Thursday, October 14, [http://www.math.northwestern.edu/mwp/ MIDWEST PROBABILITY COLLOQUIUM], (no seminar)==
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== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
<br>
+
''' '''
  
==Thursday, October 21, [http://www.math.wisc.edu/~kuelbs/ Jim Kuelbs (UW - Madison)]==
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== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
:Title: '''An Empirical Process CLT for Time Dependent Data'''
 
  
:Abstract: For stochastic processes <math>\{X_t: t \in E\}</math>, we establish sufficient conditions for the empirical process based on <math>\{ I_{X_t \le y} - P(X_t \le y): t \in E, y \in \mathbb{R}\}</math> to satisfy the CLT uniformly in <math> t \in E, y  \in \mathbb{R}</math>. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and <math>E= [0,1]</math>.
+
3-day event in Van Vleck 911
  
:Joint work with Tom Kurtz and Joel Zinn.
+
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
 +
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
  
==Thursday, October 28, [http://www.math.toronto.edu/alberts/ Tom Alberts (University of Toronto)]==
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== April 30, 2020, CANCELLED, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
:Title: '''Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer'''
+
''' '''
  
:Abstract: The 1+1 dimensional directed polymer model is a Gibbs measure on simple random walk paths of a prescribed length. The weights for the measure are determined by a random environment occupying space-time lattice sites, and the measure favors paths to which the environment gives high energy. For each inverse temperature <math>\beta</math> the polymer is said to be in the weak disorder regime if the environment has little effect on it, and the strong disorder regime otherwise. In dimension 1+1 it turns out that all positive <math>\beta</math> are in the strong disorder regime. I will introduce a new regime called intermediate disorder, which is accessed by scaling the inverse temperature to zero with the length <math>n</math> of the polymer. The precise scaling is <math>\beta n^{-1/4}</math>. The most interesting result is that under this scaling the polymer has diffusive fluctuations, but the fluctuations themselves are not Gaussian. Instead they are still coupled to the random environment, and their distribution is intimately related to the Tracy-Widom distribution for the largest eigenvalue of a random matrix from the GUE. More recent work also indicates that we can take a scaling limit of the entire intermediate disorder regime to construct a continuous random path under the effect of a continuum random environment. We call the scaling limit the continuum random polymer. I will discuss a few properties of the continuum random polymer and its intimate connection to the stochastic heat equation in one dimension.
 
  
:Joint work with Kostya Khanin and Jeremy Quastel.
 
  
==<span style="color:#009000"> Wednesday, November 17, 2:30pm</span>, [http://math.stanford.edu/~pmwood/ Philip Matchett Wood (Stanford)]==
 
<span style="color:#FF0000">'''NOTE THE UNUSUAL TIME!'''<span style="color:#009000">
 
:Title: '''Random tridiagonal doubly stochastic matrices'''
 
  
:Let <math>T_n</math> be the compact convex set of tridiagonal doubly stochastic matrices.  These arise naturally as birth and death chains with a uniform stationary distribution.  One can think of a ‘typical’ matrix <math>T_n</math> as one chosen uniformly at random, and this talk will present a simple algorithm to sample uniformly in <math>T_n</math>.  Once we have our hands on a 'typical' element of <math>T_n</math>, there are many natural questions to ask:  What are the eigenvalues? What is the mixing time?  What is the distribution of the entries?  This talk will explore these and other questions, with a focus on whether a random element of <math>T_n</math> exhibits a cutoff in its approach to stationarity.  Joint work with Persi Diaconis.
 
  
==Thursday, November 18, [http://www.math.wisc.edu/~jmiller/ Joseph S. Miller (UW - Madison)]==
 
:Title: '''TBA'''
 
  
  
==Thursday, December 2, [http://www.math.wisc.edu/~lin/ Hao Lin (UW - Madison)]==
 
:Title: '''Properties of the limit shape for some last passage growth models in random
 
environments'''
 
  
:We study directed last passage percolation on the first quadrant of the planar square lattice whose weights have general distributions, or equivalently, ./G/1 queues in series. The service time distributions of the servers vary randomly which constitutes a random environment for the model. Equivalently, each row of the last passage model has its own randomly chosen weight distribution. We investigate the limiting time constant close to the boundary of the quadrant. Close to the y-axis, where the number of random distributions averaged over stays large, the limiting time constant takes the same universal form as in the homogeneous model. But close to the x-axis we see the effect of the tail of the distribution of the random means attached to the rows.
+
[[Past Seminars]]

Latest revision as of 17:18, 12 August 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