Difference between revisions of "Probability Seminar"

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(Friday, April 26, Colloquium, B 239 from 4pm to 5pm, Kavita Ramanan, Brown)
(February 6, 2020, Cheuk-Yin Lee (Michigan State))
 
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__NOTOC__
 
__NOTOC__
  
= Spring 2019 =
+
= Spring 2020 =
  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
+
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
<b>We  usually end for questions at 3:15 PM.</b>
+
<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  
 
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]
 
[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
 +
'''
  
 +
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 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==
+
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
 +
'''Quasi-linear parabolic equations with singular forcing'''
  
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''
+
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: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.
+
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.
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.
 
  
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==
+
== 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: '''When particle systems meet PDEs'''
+
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: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..
+
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
''' '''
  
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==
+
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
''' '''
  
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''
+
== February 27, 2020, TBA ==
 +
''' '''
  
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.
+
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
 +
''' '''
  
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==
+
== March 12, 2020, TBA ==
 +
''' '''
  
Title: '''Geometry of the corner growth model'''
+
== March 26, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).
+
== April 2, 2020, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
 +
''' '''
  
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==
+
== April 9, 2020, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
 +
== April 16, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 +
''' '''
  
Title: '''On the centered maximum of the Sine beta process'''
+
== April 22-24, 2020, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
  
 +
3-day event in Van Vleck 911
  
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.
+
== April 23, 2020, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
  
Title: Quantitative homogenization in a balanced random environment
+
== April 30, 2020, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).
 
  
==  <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==
 
  
  
<div style="width:520px;height:50px;border:5px solid black">
 
<b><span style="color:red">&emsp; Please note the unusual day and time.
 
&emsp; </span></b>
 
</div>
 
  
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''
 
  
Abstract:
 
  
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.
 
 
<!-- == March 7, TBA == -->
 
 
<!-- == March 14, TBA == -->
 
 
== March 21, Spring Break, No seminar ==
 
 
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==
 
 
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''
 
 
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the  ''character ratio'':
 
 
$$
 
\text{trace}(\rho(g))/\text{dim}(\rho),
 
$$
 
 
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant  ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).
 
 
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
Title: '''Outliers in the spectrum for products of independent random matrices'''
 
 
Abstract:  For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.  Joint work with Natalie Coston and Sean O'Rourke.
 
 
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==
 
 
'''Title:  Stabilization of Diffusion Limited Aggregation in a Wedge.'''
 
 
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.
 
 
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==
 
 
 
Title: '''Large Deviations Theory for Chemical Reaction Networks'''
 
 
Abstract:
 
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a  straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.
 
 
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==
 
 
Title:  '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''
 
 
Abstract:  Many applications can be modeled as a large system of homogeneous interacting  particle systems on a graph in which the infinitesimal evolution of each particle depends on  its own state and the empirical distribution of the states of neighboring particles.  When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit.  In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle.  This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.
 
 
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==
 
 
Title:  '''Tales of Random Projections'''
 
 
Abstract:  The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures.  In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics.  Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections.  A particular question of interest is to identify what properties of the high-dimensional measure are captured by  its lower-dimensional projections.  While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.
 
 
== May 7, '''Tuesday''', Duncan Dauvergne (Toronto) ==
 
 
 
<!--
 
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==
 
 
 
Title: '''The distribution of sandpile groups of random regular graphs'''
 
 
Abstract:
 
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.
 
 
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.
 
 
 
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==
 
 
Title: '''Stochastic quantization of Yang-Mills'''
 
 
Abstract:
 
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.
 
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].
 
 
-->
 
 
== ==
 
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 09:50, 23 January 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)

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

February 27, 2020, TBA

March 5, 2020, Jiaoyang Huang (IAS)

March 12, 2020, TBA

March 26, 2020, Philippe Sosoe (Cornell)

April 2, 2020, Tianyu Liu (UW Madison)

April 9, 2020, Alexander Dunlap (Stanford)

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

April 22-24, 2020, FRG Integrable Probability meeting

3-day event in Van Vleck 911

April 23, 2020, 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