Past Probability Seminars Spring 2020: Difference between revisions

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__NOTOC__
__NOTOC__


= Spring 2015 =
= Fall 2017 =


<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
<b>We  usually end for questions at 3:15 PM.</b>


<b>
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.
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.


<!-- [[File:probsem.jpg]] -->
</b>


= =
== Thursday, September 14, 2017, [https://math.temple.edu/~brider/ Brian Rider] [https://math.temple.edu/ Temple University] ==


== Thursday, January 15, [http://www.stat.berkeley.edu/~racz/ Miklos Racz], [http://statistics.berkeley.edu/ UC-Berkeley Stats] ==
'''A universality result for the random matrix hard edge'''


The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and “quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by Jose Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.


Title: Testing for high-dimensional geometry in random graphs
<!-- == Thursday, September 21, 2017, TBA==-->


Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.
<!-- == Thursday, September 28, 2017, TBA ==
== Thursday, October 5, 2017 ==
== Thursday, October 12, 2017 == -->
== Thursday, October 19, 2017  [https://sites.google.com/wisc.edu/vjog/ Varun Jog], [https://www.engr.wisc.edu/department/electrical-computer-engineering/ UW-Madison ECE] and [https://graingerinstitute.engr.wisc.edu/ Grainger Institute] ==


== Thursday, January 22, No Seminar  ==
Title: '''Teaching and learning in uncertainty'''


== Thursday, January 29, [http://www.math.umn.edu/~arnab/ Arnab Sen], [http://www.math.umn.edu/ University of Minnesota]  ==
Abstract:
We investigate a simple model for social learning with two characters: a teacher and a student. The teacher's goal is to teach the student the state of the world <math>\Theta</math>, however, the teacher herself is not certain about <math>\Theta</math> and needs to simultaneously learn it and teach it. We examine several natural strategies the teacher may employ to make the student learn as fast as possible. Our primary technical contribution is analyzing the exact learning rates for these strategies by studying the large deviation properties of the sign of a transient random walk on <math>\mathbb Z</math>.


Title: '''Double Roots of Random Littlewood Polynomials'''
== Thursday, October 26, 2017, [http://www.math.toronto.edu/matetski/ Konstantin Matetski]  [https://www.math.toronto.edu/ Toronto] ==


Abstract:
Title: '''The KPZ fixed point'''
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions. 
 
This is joint work with Ron Peled and Ofer Zeitouni.
 
== Thursday, February 5, No seminar this week  ==
 
== Thursday, February 12, No Seminar this week==
 
 
<!--
== Wednesday, <span style="color:red">February 11</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison]  ==
 
<span style="color:red">Please note the unusual time and room.
</span>
 
 
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
 
 
Abstract:
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.
--->
 
== Thursday, February 19, [http://www.math.purdue.edu/people/bio/guo297 Xiaoqin Guo], [http://www.math.purdue.edu/ Purdue]  ==
 
Title: Quenched invariance principle for random walks in time-dependent random environment
 
Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in <math>Z^d</math>. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.
 
== Thursday, February 26, [http://wwwf.imperial.ac.uk/~dcrisan/ Dan Crisan], [http://www.imperial.ac.uk/natural-sciences/departments/mathematics/ Imperial College London]  ==
 
Title: TBA
 
Abstract:
 
== Thursday, March 5, Kurt Helms,  Humboldt-Universität zu Berlin  ==
 
Title: TBA


Abstract:
Abstract:
The KPZ fixed point is the Markov process at the centre of the KPZ universality class. In the talk we describe the exact solution of the totally asymmetric simple exclusion process, which is one of the models in the KPZ universality class, and provide a description of the KPZ fixed point in the 1:2:3 scaling limit. This is a joint work with Jeremy Quastel and Daniel Remenik.


== Wednesday, <span style="color:red">March 11</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison]  ==
<!--== Thursday, November 2, 2017, TBA ==-->
 
<span style="color:red">Please note the unusual time and room.
</span>
 


Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
== Thursday, November 9, 2017, Chen Jia, University of Texas at Dallas  ==




Abstract:
'''Mathematical foundation of nonequilibrium fluctuation-dissipation theorems and a biological application'''
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.


The fluctuation-dissipation theorem (FDT) for equilibrium states is one of the classical results in equilibrium statistical physics. In recent years, many efforts have been devoted to generalizing the classical FDT to systems far from equilibrium. This was considered as one of the most significant progress of nonequilibrium statistical physics over the past two decades. In this talk, I will introduce our recent work on the rigorous mathematical foundation of the nonequilibrium FDTs for inhomogeneous diffusion processes and inhomogeneous continuous-time Markov chains. I will also talk about the application of the nonequilibrium FDTs to a practical biological problem called sensory adaptation.


<!--
== Thursday, November 16, 2017, [http://louisfan.web.unc.edu/ Louis Fan], [http://www.math.wisc.edu/ UW-Madison]  ==
== Thursday, March 12, TBA  ==


Title: TBA
Title: '''Stochastic and deterministic spatial models for complex systems'''


Abstract:
Abstract:  
-->


== Thursday, March 19, [http://www.cmc.edu/pages/faculty/MHuber/ Mark Huber], [http://www.cmc.edu/math/ Claremont McKenna Math]  ==
Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.


Title: Understanding relative error in Monte Carlo simulations
In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.


AbstractThe problem of estimating the probability $p$ of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem.  In this talk, I'll consider a new twist: given an estimate $\hat p$, suppose we want to understand the behavior of the relative error $(\hat p - p)/p$. In classic estimators, the values that the relative error can take on depends on the value of $p$. I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of $p$. Moreover, this new estimate is very fast:  it takes a number of coin flips that is very close to the theoretical minimum. Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.
== <span style="color:red"> Friday,</span> November 17, 2017, <span style="color:red"> 1pm, Van Vleck B223, </span> [http://math.depaul.edu/kliechty/ Karl Leichty] [https://csh.depaul.edu/academics/mathematical-sciences/Pages/default.aspx DePaul University] ==


== Thursday, March 26, [http://mathsci.kaist.ac.kr/~jioon/ Ji Oon Lee], [http://www.kaist.edu/html/en/index.html KAIST]  ==


Title: TBA
<div style="width:400px;height:50px;border:5px solid black">
<b><span style="color:red"> Please note the unusual room, day, and time </span></b>
</div>


Title: '''Nonintersecting Brownian motions on the unit circle'''
Abstract:
Abstract:
Nonintersecting Brownian bridges on the unit circle form a determinantal point process whose kernel is expressed in terms of a system of discrete orthogonal polynomials which may be studied using Riemann--Hilbert techniques. If the Brownian motions have a drift, then the weight of the orthogonal polynomials becomes complex. I will discuss the tacnode and k-tacnode processes, which are related to the Painleve II function, as scaling limits of Nonintersecting Brownian motions on the unit circle and will discuss some of the features and difficulties of Riemann--Hilbert analysis of discrete orthogonal polynomials with varying complex weights.
This is joint work with Dong Wang and Robert Buckingham.


== Thursday, November 30, 2017, [https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo], [https://www.math.wisc.edu/ UW-Madison] ==


== Thursday, April 2, No Seminar, Spring Break  ==
Title: '''Harnack inequality, homogenization and random walks in a degenerate random environment'''


Abstract: Stochastic homogenization studies the effective equations or laws that characterize the large scale phenomena for systems with complicated random dynamics at microscopic levels. In this talk, we explore the relation between stochastic homogenization and a probabilistic model called random motion in a random medium. In particular we focus on dynamics on the integer lattice which is non-reversible in time and defined by a non-divergence form operator which is non-elliptic. A difficulty in studying this problem is that coefficients of the operator are allowed to be zero. Using random walks in random media, we present a Harnack inequality and a quantitative result for homogenization for this random operator. Joint work with N.Berger (TU-Munich), M.Cohen (Jerusalem) and J.-D. Deuschel (TU-Berlin).


<!--== Thursday, December 7, 2017,  TBA ==


== Thursday, December 14, 2017, TBA ==-->


== Thursday, April 9, [http://www.math.wisc.edu/~emrah/ Elnur Emrah], [http://www.math.wisc.edu/ UW-Madison]  ==


Title: TBA


Abstract:
== Thursday, April 16, TBA  ==
Title: TBA
Abstract:
== Thursday, April 23, [http://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [http://math.osu.edu/ Ohio State University]  ==
Title: TBA
Abstract:
== Thursday, April 30, TBA  ==
Title: TBA
Abstract:
== Thursday, May 7, TBA  ==
Title: TBA
Abstract:
<!--
== Thursday, December 11, TBA  ==
Title: TBA
Abstract:
-->
<!--
== Thursday, September 11, <span style="color:red">Van Vleck B105,</span> [http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood], UW-Madison ==
Please note the non-standard room.
Title: '''The distribution of sandpile groups of random graphs'''
Abstract:<br>
The sandpile group is an abelian group associated to a graph, given as
the cokernel of the graph Laplacian.  An Erdős–Rényi random graph
then gives some distribution of random abelian groups.  We will give
an introduction to various models of random finite abelian groups
arising in number theory and the connections to the distribution
conjectured by Payne et. al. for sandpile groups.  We will talk about
the moments of random finite abelian groups, and how in practice these
are often more accessible than the distributions themselves, but
frustratingly are not a priori guaranteed to determine the
distribution.  In this case however, we have found the moments of the
sandpile groups of random graphs, and proved they determine the
measure, and have proven Payne's conjecture.
== Thursday, September 18, [http://www.math.purdue.edu/~peterson/ Jonathon Peterson], [http://www.math.purdue.edu/ Purdue University]  ==
Title: '''Hydrodynamic limits for directed traps and systems of independent RWRE'''
Abstract:
We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed <math>v_0 \neq 0)</math>. In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out'' and so the specific instance of the environment chosen actually matters.
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara.
== Thursday, September 25, [http://math.colorado.edu/~seor3821/ Sean O'Rourke],  [http://www.colorado.edu/math/ University of Colorado Boulder]  ==
Title: '''Singular values and vectors under random perturbation'''
Abstract:
Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors?
Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank.  This talk is based on joint work with Van Vu and Ke Wang.
== Thursday, October 2, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison]  ==
Title: '''Anisotropic local laws for random matrices'''
Abstract:
In this talk, we introduce a new method of deriving  local laws of random matrices.  As applications, we will show the local laws  and some universality results on general sample covariance matrices: TXX^*T^* (where $T$ is non-square deterministic matrix),  and deformed Wigner matrix: H+A (where A is deterministic symmetric matrix). Note: here $TT^*$ and $A$ could be full rank matrices.
== Thursday, October 9, No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium]  ==
No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
== Thursday, October 16, [http://www.math.utah.edu/~firas/ Firas Rassoul-Agha], [http://www.math.utah.edu/ University of Utah]==
Title: '''The growth model: Busemann functions, shape, geodesics, and other stories'''
Abstract:
We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface.  This is joint work with Nicos Georgiou and Timo Seppalainen.
== Thursday, November 6, Vadim Gorin, [http://www-math.mit.edu/people/profile.php?pid=1415 MIT]  ==
Title: '''Multilevel Dyson Brownian Motion and its edge limits.'''
Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of
interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of
random Hermitian matrices on the other side. In my talk I will explain some reasons for this
connection between two seemingly unrelated classes of stochastic systems, and how this relation can
be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion
will be the central object in the discussion.
(Based on joint papers with Misha Shkolnikov.)
==<span style="color:red"> Friday</span>, November 7, [http://tchumley.public.iastate.edu/ Tim Chumley], [http://www.math.iastate.edu/ Iowa State University] ==
<span style="color:darkgreen">Please note the unusual day.</span>
Title: '''Random billiards and diffusion'''
Abstract: We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system.  The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.
== Thursday, November 13, [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [http://www.math.wisc.edu/ UW-Madison]==
Title: '''Variational formulas for directed polymer and percolation models'''
Abstract:
Explicit formulas for subadditive limits of polymer and percolation models in probability and statistical mechanics have been difficult to find. We describe variational formulas for these limits and their connections with other features of the models such as Busemann functions and Kardar-Parisi-Zhang (KPZ) fluctuation exponents.
== <span style="color:red">Monday</span>, December 1,  [http://www.ma.utexas.edu/users/jneeman/index.html Joe Neeman], [http://www.ma.utexas.edu/ UT-Austin], <span style="color:red">4pm, Room B239 Van Vleck Hall</span>==
<span style="color:darkgreen">Please note the unusual time and room.</span>
Title: '''Some phase transitions in the stochastic block model'''
Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.
== Thursday, December 4, Arjun Krishnan, [http://www.fields.utoronto.ca/ Fields Institute] ==
Title: '''Variational formula for the time-constant of first-passage percolation'''
Abstract:
Consider first-passage percolation with positive, stationary-ergodic
weights on the square lattice in d-dimensions. Let <math>T(x)</math> be the
first-passage time from the origin to <math>x</math> in <math>Z^d</math>. The convergence of
<math>T([nx])/n</math> to the time constant as <math>n</math> tends to infinity is a consequence
of the subadditive ergodic theorem. This convergence can be viewed as
a problem of homogenization for a discrete Hamilton-Jacobi-Bellman
(HJB) equation. By borrowing several tools from the continuum theory
of stochastic homogenization for HJB equations, we derive an exact
variational formula (duality principle) for the time-constant. Under a
symmetry assumption, we will use the variational formula to construct
an explicit iteration that produces the limit shape.
-->


== ==
== ==


[[Past Seminars]]
[[Past Seminars]]

Revision as of 19:41, 30 November 2017


Fall 2017

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 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.


Thursday, September 14, 2017, Brian Rider Temple University

A universality result for the random matrix hard edge

The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and “quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by Jose Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.


Thursday, October 19, 2017 Varun Jog, UW-Madison ECE and Grainger Institute

Title: Teaching and learning in uncertainty

Abstract: We investigate a simple model for social learning with two characters: a teacher and a student. The teacher's goal is to teach the student the state of the world [math]\displaystyle{ \Theta }[/math], however, the teacher herself is not certain about [math]\displaystyle{ \Theta }[/math] and needs to simultaneously learn it and teach it. We examine several natural strategies the teacher may employ to make the student learn as fast as possible. Our primary technical contribution is analyzing the exact learning rates for these strategies by studying the large deviation properties of the sign of a transient random walk on [math]\displaystyle{ \mathbb Z }[/math].

Thursday, October 26, 2017, Konstantin Matetski Toronto

Title: The KPZ fixed point

Abstract: The KPZ fixed point is the Markov process at the centre of the KPZ universality class. In the talk we describe the exact solution of the totally asymmetric simple exclusion process, which is one of the models in the KPZ universality class, and provide a description of the KPZ fixed point in the 1:2:3 scaling limit. This is a joint work with Jeremy Quastel and Daniel Remenik.


Thursday, November 9, 2017, Chen Jia, University of Texas at Dallas

Mathematical foundation of nonequilibrium fluctuation-dissipation theorems and a biological application

The fluctuation-dissipation theorem (FDT) for equilibrium states is one of the classical results in equilibrium statistical physics. In recent years, many efforts have been devoted to generalizing the classical FDT to systems far from equilibrium. This was considered as one of the most significant progress of nonequilibrium statistical physics over the past two decades. In this talk, I will introduce our recent work on the rigorous mathematical foundation of the nonequilibrium FDTs for inhomogeneous diffusion processes and inhomogeneous continuous-time Markov chains. I will also talk about the application of the nonequilibrium FDTs to a practical biological problem called sensory adaptation.

Thursday, November 16, 2017, Louis Fan, UW-Madison

Title: Stochastic and deterministic spatial models for complex systems

Abstract:

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.

In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.

Friday, November 17, 2017, 1pm, Van Vleck B223, Karl Leichty DePaul University

Please note the unusual room, day, and time

Title: Nonintersecting Brownian motions on the unit circle

Abstract:

Nonintersecting Brownian bridges on the unit circle form a determinantal point process whose kernel is expressed in terms of a system of discrete orthogonal polynomials which may be studied using Riemann--Hilbert techniques. If the Brownian motions have a drift, then the weight of the orthogonal polynomials becomes complex. I will discuss the tacnode and k-tacnode processes, which are related to the Painleve II function, as scaling limits of Nonintersecting Brownian motions on the unit circle and will discuss some of the features and difficulties of Riemann--Hilbert analysis of discrete orthogonal polynomials with varying complex weights.

This is joint work with Dong Wang and Robert Buckingham.

Thursday, November 30, 2017, Xiaoqin Guo, UW-Madison

Title: Harnack inequality, homogenization and random walks in a degenerate random environment

Abstract: Stochastic homogenization studies the effective equations or laws that characterize the large scale phenomena for systems with complicated random dynamics at microscopic levels. In this talk, we explore the relation between stochastic homogenization and a probabilistic model called random motion in a random medium. In particular we focus on dynamics on the integer lattice which is non-reversible in time and defined by a non-divergence form operator which is non-elliptic. A difficulty in studying this problem is that coefficients of the operator are allowed to be zero. Using random walks in random media, we present a Harnack inequality and a quantitative result for homogenization for this random operator. Joint work with N.Berger (TU-Munich), M.Cohen (Jerusalem) and J.-D. Deuschel (TU-Berlin).



Past Seminars