Past Probability Seminars Spring 2020: Difference between revisions

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= Fall 2014 =
= Fall 2018 =


<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>


If you would like to sign up for the email list to receive seminar announcements then please send an email to
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]


Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.


<b>
If you would like to sign up for the email list to receive seminar announcements then please send an email to [[File:probsem.jpg]]
</b>


= =
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==


<!--
== Thursday, February 6, [http://people.mbi.ohio-state.edu/newby.23/ Jay Newby],  [http://mbi.osu.edu/ Mathematical Biosciences Institute]  ==


Title: Applications of large deviation theory in neuroscience
Title: '''The distribution of sandpile groups of random regular graphs'''


Abstract:
Abstract:
The membrane voltage of a neuron is modeled with a piecewise deterministic stochastic process. The membrane voltage changes deterministically while the population of open ion channels, which allow current to flow across the membrane, is constant.  Ion channels open and close randomly, and the transition rates depend on voltage, making the process nonlinear.  In the limit of infinite transition rates, the process becomes deterministic. The deterministic process is the well known Morris-Lecar model.  Under certain conditions, the deterministic process has one stable fixed point and is excitable. An excitable event, called an action potential, is a single large transient spike in voltage that eventually returns to the stable steady state.  I will discuss recent development of large deviation theory to study noise induced action potentials.
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.


== Thursday, February 13, [http://www.math.wisc.edu/~holcomb/ Diane Holcomb], [http://www.math.wisc.edu/ UW-Madison] ==
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.


Title: Large deviations for point process limits of random matrices.
<!-- ==September 13, TBA == -->


Abstract: The Gaussian Unitary ensemble (GUE) is one of the most studied Hermitian random matrix model. When appropriately rescaled the eigenvalues in the bulk of the spectrum converge to a translation invariant limiting point process called the Sine process. On large intervals one expects the Sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==


== Thursday, February 20, Philip Matchett Wood, UW-Madison ==
Title: '''Stochastic quantization of Yang-Mills'''


Title: The empirical spectral distribution (ESD) of a fixed matrix plus small random noise.
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.
Abstract: A fixed matrix has a distribution of eigenvalues in the complex
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].
plane. Small random noise can be formed by a random matrix with iid mean 0
 
variance 1 entries scaled by <math>n^{-\gamma -1/2}</math> for <math>\gamma > 0</math>, which
 
by itself has eigenvalues collapsing to the origin.  What happens to the
 
eigenvalues when you add a small random noise matrix to the fixed matrix?
==September 27, [https://www.math.wisc.edu/~seppalai/  Timo Seppäläinen] [https://www.math.wisc.edu/ UW-Madison] ==
There are interesting cases where the eigenvalue distribution is known to
change dramatically when small Gaussian random noise is added, and this talk
will focus on what happens when the noise is <i>not</i> Gaussian..


== Thursday, February 27, [http://mypage.iu.edu/~jthanson/ Jack Hanson], Indiana University Bloomington ==
Title:'''Random walk in random environment and the Kardar-Parisi-Zhang class'''
Abstract:This talk concerns a relationship between two much-studied classes of models  of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia)  discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior.  In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3.   Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).


Title: '''Subdiffusive Fluctuations in First-Passage Percolation'''
==October 4, [https://people.math.osu.edu/paquette.30/  Elliot Paquette], [https://math.osu.edu/ OSU] ==


Abstract: First-passage percolation is a model consisting of a random metric t(x,y) generated by random variables associated to edges of a graph. Many questions and conjectures in this model revolve around the fluctuating properties of this metric on the graph Z^d. In the early 1990s, Kesten showed an upper bound of Cn for the variance of t(0,nx); this was improved to Cn/log(n) by Benjamini-Kalai-Schramm and Benaim-Rossignol for particular choices of distribution. I will discuss recent work (with M. Damron and P. Sosoe) extending this upper bound to general classes of distributions.
Title: '''Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble'''


Abstract:
The characteristic polynomial of the Gaussian beta--ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two--by--two matrices.  For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane.  Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation.
We explain the origins of this decomposition, and we show partial progress in establishing part of it.


== Thursday, March 20, No Seminar due to Spring Break ==
Joint work with Diane Holcomb and Gaultier Lambert.


== Thursday, March 27, [http://www.stat.wisc.edu/~ane/ Cécile Ané], UW-Madison Department of Statistics ==
==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==


Title: <b> Application of a birth-death process to model gene gains and losses on a phylogenetic tree </b>


Abstract:
Title: '''Busemann functions and Gibbs measures in directed polymer models on Z^2'''
Over time, genes can duplicate or be lost. The history of a gene family is a tree whose nodes represent duplications, speciations, or losses. A birth-and-death process is used to model this gene family tree, embedded within a species tree. I will present this phylogenetic version of the birth and death tree process, along with a probability model for whole-genome duplications. If there is interest and time, I will talk about learning birth and death rates and detecting ancient whole-genome duplications from genomic data.
 
Abstract: We consider the model of a nearest-neighbor random walk on the planar square lattice in a general iid space-time potential, which is also known as a directed polymer in a random environment. We prove results on existence, uniqueness (and non-uniqueness), and the law of large numbers for semi-infinite path measures. Our main tools are the Busemann functions, which are families of stochastic processes obtained through limits of ratios of partition functions.
 
Based on joint work with Firas Rassoul-Agha
 
==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==
 
==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==
 
 
Title: '''Tails of the KPZ equation'''
     
Abstract: The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the solution of the KPZ equation. For instance, we investigate the probability of the solution being smaller or larger than the expected value. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.
 
This talk will be based on a joint work with my advisor Prof. Ivan Corwin.
 
==November 1, [https://math.umn.edu/directory/james-melbourne James Melbourne], [https://math.umn.edu/ University of Minnesota] ==
 
Title: '''Upper bounds on the density of independent vectors under certain linear mappings'''


Abstract:  Using functional analytic techniques and rearrangement, we prove anti-concentration results for the linear images of independent random variables, in the form of density upper bounds.  For continuous variables the results unify and sharpen Bobkov-Chistyakov's for independent sums of vectors and Rudelson-Vershynin's bounds on projections of independent coordinates.  For integer valued variables the techniques reduce finding the maximum of the probability mass function of a sum of independent variables, to the case that each variable is uniform on a contiguous interval.  This problem is approached through analysis of characteristic functions and new $L^p$ bounds on the Dirichlet and Fejer Kernel are obtained and used to derive a discrete analog of Bobkov-Chistyakov.


== Thursday, April 10, [https://www.math.ucdavis.edu/~romik/home/Dan_Romik_home.html Dan Romik] UC-Davis ==
==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], [https://cims.nyu.edu/ NYU] ==


Title: <b>Connectivity patterns in loop percolation and pipe percolation</b>
Title: '''The Sine-beta process: DLR equations and applications'''


Abstract:
Abstract:
One-dimensional log-gases, or Beta-ensembles, are statistical physics models finding an incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE's. In a joint work with D. Dereudre, A. Hardy, and M. Maïda, we give a new description of Sine-beta as an "infinite volume Gibbs measure", using the Dobrushin-Lanford-Ruelle (DLR) formalism, and we use it to prove the rigidity of the process, in the sense of Ghosh-Peres. Another application is a CLT for fluctuations of linear statistics.
<!-- ==November 15, TBA == -->


Loop percolation is a random collection of closed cycles in the square lattice Z^2, that is closely related to critical bond percolation. Its "connectivity pattern" is a random noncrossing matching associated with a loop percolation configuration that encodes information about connectivity of endpoints. The same probability measure on noncrossing matchings arises in several different and seemingly unrelated settings, for example in connection with alternating sign matrices, the quantum XXZ spin chain, and another type of percolation model called pipe percolation. In the talk I will describe some of these connections and discuss some results about the study of pipe percolation from the point of view of the theory of interacting particle systems. I will also mention the "rationality phenomenon" which causes the probabilities of certain natural connectivity events to be dyadic rational numbers such as 3/8, 97/512 and 59/1024. The reasons for this are not completely understood and are related to certain algebraic conjectures that I will discuss separately in Friday's talk in the Applied Algebra seminar.
==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==


<!--== Thursday, April 17, TBA ==-->
==  <span style="color:red">  Monday, November 26, 4pm, Van Vleck 911</span>  [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin], [http://math.mit.edu/index.php MIT]  ==


<!-- Thursday, April 24, TBA -->


== Thursday, May 1, [http://math.uchicago.edu/~auffing/ Antonio Auffinger] U Chicago ==
<div style="width:320px;height:50px;border:5px solid black">
<b><span style="color:red">&emsp; Please note the unusual day, time, <br>
&emsp; and room.</span></b>
</div>


Title: '''Macroscopic fluctuations through Schur generating functions'''


Title: '''Strict Convexity of the Parisi Functional'''
Abstract:
I will talk about a special class of large-dimensional stochastic systems with
strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices,
and measures governing decompositions of group representations into irreducible components.


Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.
It is believed that macroscopic fluctuations in such systems are universally
described by log-correlated Gaussian fields. I will present an approach to
handle this question based on the notion of the Schur generating function of a probability
distribution, and explain how it leads to a rigorous confirmation of this belief in
a variety of situations.


== Thursday, May 8, [http://wid.wisc.edu/profile/steve-goldstein/ Steve Goldstein], [http://wid.wisc.edu/ WID]==
<!-- ==November 29, TBA == -->


== <span style="color:red">  Wednesday, December 5 at 4pm in Van Vleck 911</span> [http://www.mit.edu/~ssen90/ Subhabrata Sen], [https://math.mit.edu/ MIT] and [https://www.microsoft.com/en-us/research/lab/microsoft-research-new-england/ Microsoft Research New England] ==


Title: '''Modeling patterns of DNA sequence diversity with Cox Processes'''


Abstract:
<div style="width:320px;height:50px;border:5px solid black">
Events in the evolutionary history of a population can leave subtle signals in the patterns of diversity of its DNA sequences. Identifying those signals from the DNA sequences of present-day populations and using them to make inferences about selection is a well-studied and challenging problem.
<b><span style="color:red">&emsp; Please note the unusual day, time, <br>&emsp; and room. </span></b>
</div>


Next generation sequencing provides an opportunity for making inroads on that problem.  In this talk, I will present a novel model for the analysis of sequence diversity data and use the model to motivate analyses of whole-genome sequences from 11 strains of Drosophila pseudoobscura.


The model treats the polymorphic sites along the genome as a realization of a Cox Process, a point process with a random intensity.  Within the context of this model, the underlying problem translates to making inferences about the distribution of the intensity function, given the sequence data.


We give a proof of principle, showing that even a simplistic application of the model can quantify differences in diversity between regions with varying recombination rates.  We also suggest a number of directions for applying and extending the model.
Title: '''Random graphs, Optimization, and Spin glasses'''
-->


Abstract:
Combinatorial optimization problems are ubiquitous in diverse mathematical
applications. The desire to understand their “typical” behavior motivates
a study of these problems on random instances. In spite of a long and rich
history, many natural questions in this domain are still intractable to rigorous
mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection
are canonical examples in this class. On the other hand, physicists study these
questions using the non-rigorous “replica” and “cavity” methods, and predict
complex, intriguing features. In this talk, I will describe some recent progress
in our understanding of their typical properties on random graphs, obtained via
connections to the theory of mean-field spin glasses. The new techniques are
broadly applicable, and lead to novel algorithmic and statistical consequences.


== ==


<!-- ==December 6, TBA ==-->


== ==


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

Revision as of 15:51, 29 November 2018


Fall 2018

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


Friday, August 10, 10am, B239 Van Vleck 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]\displaystyle{ d }[/math]-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the [math]\displaystyle{ p }[/math]-Sylow subgroup of the sandpile group is a given [math]\displaystyle{ p }[/math]-group [math]\displaystyle{ P }[/math], is proportional to [math]\displaystyle{ |\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]\displaystyle{ d }[/math]-regular directed graphs are invertible with high probability to the undirected case.


September 20, Hao Shen, 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].


September 27, Timo Seppäläinen UW-Madison

Title:Random walk in random environment and the Kardar-Parisi-Zhang class

Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).

October 4, Elliot Paquette, OSU

Title: Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble

Abstract: The characteristic polynomial of the Gaussian beta--ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two--by--two matrices. For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation. We explain the origins of this decomposition, and we show partial progress in establishing part of it.

Joint work with Diane Holcomb and Gaultier Lambert.

October 11, Chris Janjigian, University of Utah

Title: Busemann functions and Gibbs measures in directed polymer models on Z^2

Abstract: We consider the model of a nearest-neighbor random walk on the planar square lattice in a general iid space-time potential, which is also known as a directed polymer in a random environment. We prove results on existence, uniqueness (and non-uniqueness), and the law of large numbers for semi-infinite path measures. Our main tools are the Busemann functions, which are families of stochastic processes obtained through limits of ratios of partition functions.

Based on joint work with Firas Rassoul-Agha

October 18-20, Midwest Probability Colloquium, No Seminar

October 25, Promit Ghosal, Columbia

Title: Tails of the KPZ equation

Abstract: The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the solution of the KPZ equation. For instance, we investigate the probability of the solution being smaller or larger than the expected value. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.

This talk will be based on a joint work with my advisor Prof. Ivan Corwin.

November 1, James Melbourne, University of Minnesota

Title: Upper bounds on the density of independent vectors under certain linear mappings

Abstract: Using functional analytic techniques and rearrangement, we prove anti-concentration results for the linear images of independent random variables, in the form of density upper bounds. For continuous variables the results unify and sharpen Bobkov-Chistyakov's for independent sums of vectors and Rudelson-Vershynin's bounds on projections of independent coordinates. For integer valued variables the techniques reduce finding the maximum of the probability mass function of a sum of independent variables, to the case that each variable is uniform on a contiguous interval. This problem is approached through analysis of characteristic functions and new $L^p$ bounds on the Dirichlet and Fejer Kernel are obtained and used to derive a discrete analog of Bobkov-Chistyakov.

November 8, Thomas Leblé, NYU

Title: The Sine-beta process: DLR equations and applications

Abstract: One-dimensional log-gases, or Beta-ensembles, are statistical physics models finding an incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE's. In a joint work with D. Dereudre, A. Hardy, and M. Maïda, we give a new description of Sine-beta as an "infinite volume Gibbs measure", using the Dobrushin-Lanford-Ruelle (DLR) formalism, and we use it to prove the rigidity of the process, in the sense of Ghosh-Peres. Another application is a CLT for fluctuations of linear statistics.


November 22, Thanksgiving Break, No Seminar

Monday, November 26, 4pm, Van Vleck 911 Vadim Gorin, MIT

  Please note the unusual day, time,
  and room.

Title: Macroscopic fluctuations through Schur generating functions

Abstract: I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.

It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.


Wednesday, December 5 at 4pm in Van Vleck 911 Subhabrata Sen, MIT and Microsoft Research New England

  Please note the unusual day, time,
  and room.


Title: Random graphs, Optimization, and Spin glasses

Abstract: Combinatorial optimization problems are ubiquitous in diverse mathematical applications. The desire to understand their “typical” behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous “replica” and “cavity” methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.


Past Seminars