Difference between revisions of "Probability Seminar"

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(Wednesday, 3/29/2017, 1:00pm, Po-Ling Loh, UW-Madison)
(Wednesday, December 5 at 4pm in Van Vleck 911 Subhabrata Sen, MIT and Microsoft Research New England)
 
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__NOTOC__
 
__NOTOC__
  
= Spring 2017 =
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= Fall 2018 =
  
 
<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>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 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  
 +
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
  
<!-- == Monday, January 9, [http://www.stat.berkeley.edu/~racz/ Miklos Racz], Microsoft Research ==-->
 
  
  
== <span style="color:red"> Monday</span>, January 9, <span style="color:red"> 4pm, B233 Van Vleck </span> [http://www.stat.berkeley.edu/~racz/ Miklos Racz], Microsoft Research ==
+
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==
  
<div style="width:320px;height:50px;border:5px solid black">
 
<b><span style="color:red"> Please note the unusual day and time </span></b>
 
</div>
 
  
 +
Title: '''The distribution of sandpile groups of random regular graphs'''
  
Title: '''Statistical inference in networks and genomics'''
+
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.
  
Abstract:
+
<!-- ==September 13, TBA == -->
From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.
+
  
I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data.
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==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==
  
<!--
+
Title: '''Stochastic quantization of Yang-Mills'''
== Thursday, January 19, TBA  ==
+
-->
+
  
== Thursday, January 26, [http://mathematics.stanford.edu/people/name/erik-bates/ Erik Bates], [http://mathematics.stanford.edu/ Stanford]  ==
+
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.
Title: '''The endpoint distribution of directed polymers'''
+
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].
 
+
Abstract: On the d-dimensional integer lattice, directed polymers are paths of a random walk in random environment, except that the environment updates at each time step. The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment. Historically, the phase transitions have been best understood by whether or not the path’s endpoint localizes.  While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers.
+
This talk is based on joint work with Sourav Chatterjee.
+
  
  
<!--
 
== Thursday, 2/2/2017, TBA ==
 
-->
 
  
 +
==September 27, [https://www.math.wisc.edu/~seppalai/  Timo Seppäläinen] [https://www.math.wisc.edu/ UW-Madison] ==
  
<!--
+
Title:'''Random walk in random environment and the Kardar-Parisi-Zhang class'''
== Thursday, February 9, TBA ==
+
 
   
 
   
== Thursday, 2/16/2017, TBA ==
+
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).
-->
+
  
== Thursday, February 23, [http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault], [http://www.math.wisc.edu/ UW-Madison] ==
+
==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==
  
== Thursday, March 2, [http://people.maths.ox.ac.uk/woolley/ Thomas Wooley], [https://www.maths.ox.ac.uk/ Oxford] ==
+
Title: '''Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble'''
  
== Thursday, 3/9/2017, TBA ==
+
Abstract:
== Thursday, March 16, [http://www-users.math.umn.edu/~wkchen/ Wei-Kuo Chen], [http://math.umn.edu/ Minnesota] ==
+
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.
  
Title: '''Energy landscape of mean-field spin glasses'''
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Joint work with Diane Holcomb and Gaultier Lambert.
  
Abstract:  
+
==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==
  
The Sherrington-Kirkpatirck (SK) model is a mean-field spin glass introduced by theoretical physicists in order to explain the strange behavior of certain alloy, such as CuMn. Despite of its seemingly simple formulation, it was conjectured to possess a number of fruitful properties. This talk will be focused on the energy landscape of the SK model. First, we will present a formula for the maximal energy in Parisi’s formulation. Second, we will give a description of the energy landscape by showing that near any given energy level between zero and maximal energy, there exist exponentially many equidistant spin configurations. Based on joint works with Auffinger, Handschy, and Lerman.
 
  
== Thursday, March 23, Spring Break ==
+
Title: '''Busemann functions and Gibbs measures in directed polymer models on Z^2'''
  
== <span style="color:red"> Wednesday, 3/29/2017, 1:00pm, </span> [http://homepages.cae.wisc.edu/~loh/index.html Po-Ling Loh], [http://www.engr.wisc.edu/department/electrical-computer-engineering/ UW-Madison] ==
+
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.
  
<div style="width:320px;height:50px;border:5px solid black">
+
Based on joint work with Firas Rassoul-Agha
<b><span style="color:red"> Please note the unusual day and time </span>
+
</b>
+
</div>
+
  
TBA
+
==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==
  
== Thursday, 4/6/2017, TBA ==
+
==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==
== Thursday, 4/13/2017, TBA ==
+
== Thursday, 4/20/2017, TBA ==
+
== Thursday, 4/27/2017, TBA ==
+
== Thursday, 5/4/2017, TBA ==
+
== Thursday, 5/11/2017, TBA ==
+
  
  
<!--
+
Title: '''Tails of the KPZ equation'''
== Thursday, September 8, Daniele Cappelletti, [http://www.math.wisc.edu UW-Madison] ==
+
     
Title: '''Reaction networks: comparison between deterministic and stochastic models'''
+
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.
  
Abstract: Mathematical models for chemical reaction networks are widely used in biochemistry, as well as in other fields. The original aim of the models is to predict the dynamics of a collection of reactants that undergo chemical transformations. There exist two standard modeling regimes: a deterministic and a stochastic one. These regimes are chosen case by case in accordance to what is believed to be more appropriate. It is natural to wonder whether the dynamics of the two different models are linked, and whether properties of one model can shed light on the behavior of the other one. Some connections between the two modelling regimes have been known for forty years, and new ones have been pointed out recently. However, many open questions remain, and the issue is still largely unexplored.
+
This talk will be based on a joint work with my advisor Prof. Ivan Corwin.
  
== <span style="color:red"> Friday</span>, September 16, <span style="color:red"> 11 am </span> [http://www.baruch.cuny.edu/math/elenak/ Elena Kosygina], [http://www.baruch.cuny.edu/ Baruch College] and the [http://www.gc.cuny.edu/Page-Elements/Academics-Research-Centers-Initiatives/Doctoral-Programs/Mathematics CUNY Graduate Center] ==
+
==November 1, [https://math.umn.edu/directory/james-melbourne James Melbourne], [https://math.umn.edu/ University of Minnesota] ==
  
<div style="width:320px;height:50px;border:5px solid black">
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Title: '''Upper bounds on the density of independent vectors under certain linear mappings'''
<b><span style="color:red"> Please note the unusual day and time </span></b>
+
</div>
+
 
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The talk will be in Van Vleck 910 as usual.
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Title: '''Homogenization of viscous Hamilton-Jacobi equations: a remark and an application.'''
+
 
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Abstract: It has been pointed out in the seminal work of P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan that for the first order
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Hamilton-Jacobi (HJ) equation, homogenization starting with affine initial data should imply homogenization for general uniformly
+
continuous initial data. The argument utilized the properties of the HJ semi-group, in particular, the finite speed of propagation. The
+
last property is lost for viscous HJ equations. We remark that the above mentioned implication holds under natural conditions for both
+
viscous and non-viscous Hamilton-Jacobi equations. As an application of our result, we show homogenization in a stationary ergodic  setting for a special class of viscous HJ equations with a non-convex Hamiltonian in one space dimension.
+
This is a joint work with Andrea Davini, Sapienza Università di Roma.
+
 
+
== Thursday, September 22,  [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [https://www.math.wisc.edu/ UW-Madison] ==
+
Title:  '''Low-degree factors of random polynomials'''
+
 
+
Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers.
+
It is known that certain models are very likely to produce random polynomials that are irreducible, and our project
+
can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random
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polynomial models. Interestingly, though the question comes from algebra and number theory, we primarily use tools
+
from combinatorics, including additive combinatorics, and probability theory. We prove for a variety of models that it
+
is very unlikely for a random polynomial with integer coefficients to have a low-degree factor—suggesting that this is, in
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fact, a universal behavior. For example, we show that the characteristic polynomial of random matrix with independent
+
+1 or −1 entries is very unlikely to have a factor of degree up to <math>n^{1/2-\epsilon}</math>.  Joint work with Sean O’Rourke.  The talk will also discuss joint work with UW-Madison
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undergraduates Christian Borst, Evan Boyd, Claire Brekken, and Samantha Solberg, who were supported
+
by NSF grant DMS-1301690 and co-supervised by Melanie Matchett Wood.
+
 
+
== Thursday, September 29, [http://www.artsci.uc.edu/departments/math/fac_staff.html?eid=najnudjh&thecomp=uceprof Joseph Najnudel],  [http://www.artsci.uc.edu/departments/math.html University of Cincinnati]==
+
Title:  '''On the maximum of the characteristic polynomial of the Circular Beta Ensemble'''
+
 
+
In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2.
+
 
+
== Thursday, October 6, No Seminar ==
+
 
+
== Thursday, October 13, No Seminar due to [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
+
For details, see [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
+
 
+
== Thursday, October 20, [http://www.math.harvard.edu/people/index.html Amol Aggarwal], [http://www.math.harvard.edu/ Harvard] ==
+
Title: Current Fluctuations of the Stationary ASEP and Six-Vertex Model
+
 
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Abstract: We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them.
+
 
+
== Thursday, October 27, [http://www.math.wisc.edu/~hung/ Hung Tran], [http://www.math.wisc.edu/ UW-Madison] ==
+
  
Title: '''Homogenization of non-convex Hamilton-Jacobi equations'''
+
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.
  
Abstract: I will describe why it is hard to do homogenization for non-convex Hamilton-Jacobi equations and explain some recent results in this direction. I will also make a very brief connection to first passage percolation and address some challenging questions which appear in both directions. This is based on joint work with Qian and Yu.
+
==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], [https://cims.nyu.edu/ NYU] ==
  
== Thursday, November 3, Alejandro deAcosta, [http://math.case.edu/ Case-Western Reserve] ==
+
Title: '''The Sine-beta process: DLR equations and applications'''
Title: '''Large deviations for irreducible Markov chains with general state space'''
+
  
 
Abstract:
 
Abstract:
We study the large deviation principle for the empirical measure of general irreducible Markov chains in the tao topology for a broad class of initial distributions. The roles of several rate functions, including the rate function based on the convergence parameter of the transform kernel and the Donsker-Varadhan rate function, are clarified.
+
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.
  
== Thursday, November 10, [https://sites.google.com/a/wisc.edu/louisfan/home Louis Fan], [https://www.math.wisc.edu/ UW-Madison] ==
+
<!-- ==November 15, TBA == -->
  
Title: '''Particle representations for (stochastic) reaction-diffusion equations'''
+
==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==
  
 +
==  <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]  ==
  
Abstract:
 
  
Reaction diffusion equations (RDE) is a popular tool to model complex spatial-temporal patterns including Turing patterns, traveling waves and periodic switching.
+
<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>
  
These models, however, ignore the stochasticity and individuality of many complex systems in nature. Recognizing this drawback, scientists are developing individual-based models for model selection purposes. The latter models are sometimes studied under the framework of interacting particle systems (IPS) by mathematicians, who prove scaling limit theorems to connect various IPS with RDE across scales.
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Title: '''Macroscopic fluctuations through Schur generating functions'''
 
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In this talk, I will present some new limiting objects including SPDE on metric graphs and coupled SPDE. These SPDE reduce to RDE when the noise parameter tends to zero, therefore interpolates between IPS and RDE and identifies the source of stochasticity. Scaling limit theorems and novel duality formulas are obtained for these SPDE, which not only connect phenomena across scales, but also offer insights about the genealogies and time-asymptotic properties of certain population dynamics.  In particular, I will present rigorous results about the lineage dynamics for of a biased voter model introduced by Hallatschek and Nelson (2007).
+
 
+
== Thursday, November 24, No Seminar due to Thanksgiving ==
+
 
+
== Thursday, December 1, [http://math.columbia.edu/~hshen/ Hao Shen], [http://math.columbia.edu/~hshen/ Columbia] ==
+
Title: '''On scaling limits of Open ASEP and Glauber dynamics of ferromagnetic models'''
+
  
 
Abstract:
 
Abstract:
We discuss two scaling limit results for discrete models converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP.  We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. A common challenge in the proofs is how to identify the limiting process as the solution to the SPDE, and we will discuss how to overcome the difficulties in the two cases.(Based on joint works with Ivan Corwin and Hendrik Weber.)
+
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.
  
== '''Colloquium''' Friday, December 2, [http://math.columbia.edu/~hshen/ Hao Shen], [http://math.columbia.edu/~hshen/ Columbia] ==
+
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.
  
4pm, Van Vleck 9th floor
+
<!-- ==November 29, TBA == -->
  
Title: '''Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?'''
+
== <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] ==
  
Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
 
  
 +
<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>
  
<!--
 
  
== Thursday, January 28, [http://faculty.virginia.edu/petrov/ Leonid Petrov], [http://www.math.virginia.edu/ University of Virginia] ==
 
  
Title: '''The quantum integrable particle system on the line'''
+
Title: '''Random graphs, Optimization, and Spin glasses'''
 
+
I will discuss the higher spin six vertex model - an interacting  particle
+
system on the discrete 1d line in the Kardar--Parisi--Zhang universality
+
class. Observables of this system admit explicit contour integral expressions
+
which degenerate  to many known formulas of such type for other integrable
+
systems on the line in the KPZ class, including stochastic six vertex model,
+
ASEP, various <math>q</math>-TASEPs, and associated zero range processes. The structure
+
of the higher spin six vertex model (leading to contour integral formulas for
+
observables) is based on Cauchy summation identities for certain symmetric
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rational functions, which in turn can be traced back to the sl2 Yang--Baxter
+
equation. This framework allows to also include space and spin inhomogeneities
+
into the picture, which leads to new particle systems with unusual phase
+
transitions.
+
 
+
== Thursday, February 4, [http://homepages.math.uic.edu/~nenciu/Site/Contact.html Inina Nenciu], [http://www.math.uic.edu/ UIC], Joint Probability and Analysis Seminar ==
+
 
+
Title: '''On some concrete criteria for quantum and stochastic confinement'''
+
 
+
Abstract: In this talk we will present several recent results on criteria ensuring the confinement of a quantum or a stochastic particle to a bounded domain in <math>\mathbb{R}^n</math>. These criteria are given in terms of explicit growth and/or decay rates for the diffusion matrix and the drift potential close to the boundary of the domain. As an application of the general method, we will discuss several cases, including some where the background Riemannian manifold (induced by the diffusion matrix) is geodesically incomplete. These results are part of an ongoing joint project with G. Nenciu (IMAR, Bucharest, Romania).
+
 
+
== <span style="color:green">Friday, February 5</span>, [http://www.math.ku.dk/~d.cappelletti/index.html Daniele Cappelletti], [http://www.math.ku.dk/ Copenhagen University], speaks in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], <span style="color:green">2:25pm in Room 901 </span>==
+
 
+
'''Note:''' Daniele Cappelletti is speaking in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], but his research on stochastic reaction networks uses probability theory and is related to work of our own [http://www.math.wisc.edu/~anderson/ David Anderson].
+
 
+
Title: '''Deterministic and Stochastic Reaction Networks'''
+
 
+
Abstract:  Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.
+
 
+
 
+
== Thursday, February 25, [http://www.princeton.edu/~rvan/ Ramon van Handel], [http://orfe.princeton.edu/ ORFE] and [http://www.pacm.princeton.edu/ PACM, Princeton] ==
+
 
+
Title: '''The norm of structured random matrices'''
+
 
+
Abstract: Understanding the spectral norm of random matrices is a problem
+
of basic interest in several areas of pure and applied mathematics. While
+
the spectral norm of classical random matrix models is well understood,
+
existing methods almost always fail to be sharp in the presence of
+
nontrivial structure. In this talk, I will discuss new bounds on the norm
+
of random matrices with independent entries that are sharp under mild
+
conditions. These bounds shed significant light on the nature of the
+
problem, and make it possible to easily address otherwise nontrivial
+
phenomena such as the phase transition of the spectral edge of random band
+
matrices. I will also discuss some conjectures whose resolution would
+
complete our understanding of the underlying probabilistic mechanisms.
+
 
+
== Thursday,  March 3, [http://www.math.wisc.edu/~janjigia/ Chris Janjigian], [http://www.math.wisc.edu/ UW-Madison] ==
+
 
+
Title: '''Large deviations for certain inhomogeneous corner growth models'''
+
  
 
Abstract:
 
Abstract:
The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.
+
Combinatorial optimization problems are ubiquitous in diverse mathematical
 
+
applications. The desire to understand their “typical” behavior motivates
This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in this model. (Based on joint work with Elnur Emrah.)
+
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
== Thursday,  March 10, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison] ==
+
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
Title: '''Delocalization and Universality of band matrices.'''
+
questions using the non-rigorous “replica” and “cavity” methods, and predict
 
+
complex, intriguing features. In this talk, I will describe some recent progress
Abstract: in this talk we introduce our new work on band matrices, whose eigenvectors and eigenvalues are  widely believed  to have the same asymptotic behaviors as those of Wigner matrices.
+
in our understanding of their typical properties on random graphs, obtained via
We proved that this conjecture is true as long as the bandwidth is wide enough.
+
connections to the theory of mean-field spin glasses. The new techniques are
 
+
broadly applicable, and lead to novel algorithmic and statistical consequences.
== Thursday, March 17, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison] ==
+
 
+
 
+
Title: '''Recovering the Treelike Trend of Evolution Despite Extensive Lateral Genetic Transfer'''
+
 
+
Abstract
+
Reconstructing the tree of life from molecular sequences is a fundamental problem in computational
+
biology. Modern data sets often contain large numbers of genes. That can complicate the reconstruction because different genes often undergo different evolutionary histories. This is the case in particular in the presence of lateral genetic transfer (LGT), where a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from (but related to) the species phylogeny. In this talk I will sketch recent results showing that, under a natural stochastic model of LGT, the species phylogeny can be reconstructed from gene trees despite surprisingly high rates of LGT.
+
 
+
== Thursday,  March 24, No Seminar, Spring Break ==
+
 
+
== Thursday,  March 31, [http://www.ssc.wisc.edu/~whs/ Bill Sandholm], [http://www.econ.wisc.edu/ Economics, UW-Madison] ==
+
 
+
Title: '''A Sample Path Large Deviation Principle for a Class of Population Processes'''
+
 
+
Abstract:  We establish a sample path large deviation principle for sequences of Markov chains arising in game theory and other applications. As the state spaces of these Markov chains are discrete grids in the simplex, our analysis must account for the fact that the processes run on a set with a boundary. A key step in the analysis establishes joint continuity properties of the state-dependent Cramer transform L(·,·), the running cost appearing in the large deviation principle rate function.
+
 
+
[http://www.ssc.wisc.edu/~whs/research/ldp.pdf paper preprint]
+
 
+
== Thursday,  April 7, No Seminar ==
+
 
+
== Thursday,  April 14, [https://www.math.wisc.edu/~jessica/ Jessica Lin], [https://www.math.wisc.edu/~jessica/ UW-Madison], Joint with [https://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE Geometric Analysis seminar] ==
+
 
+
Title: '''Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form'''
+
 
+
Abstract: I will present optimal quantitative error estimates in the
+
stochastic homogenization for uniformly elliptic equations in
+
nondivergence form. From the point of view of probability theory,
+
stochastic homogenization is equivalent to identifying a quenched
+
invariance principle for random walks in a balanced random
+
environment. Under strong independence assumptions on the environment,
+
the main argument relies on establishing an exponential version of the
+
Efron-Stein inequality. As an artifact of the optimal error estimates,
+
we obtain a regularity theory down to microscopic scale, which implies
+
estimates on the local integrability of the invariant measure
+
associated to the process. This talk is based on joint work with Scott
+
Armstrong.
+
 
+
== Thursday,  April 21, [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade], [https://www.cims.nyu.edu/ Courant Institute, NYU] ==
+
 
+
Title: '''Freezing and extremes of random unitary matrices'''
+
 
+
Abstract: A conjecture of Fyodorov, Hiary & Keating states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the log-correlated Gaussian fields. We will outline the proof of the conjecture for the leading order of the maximum, and a freezing of the free energy related to the matrix model. This talk is based on a joint work with Louis-Pierre Arguin and David Belius.
+
 
+
== Thursday, April 28, [http://www.ime.unicamp.br/~nancy/ Nancy Garcia], [http://www.ime.unicamp.br/conteudo/departamento-estatistica Statistics], [http://www.ime.unicamp.br/ IMECC], [http://www.unicamp.br/unicamp/ UNICAMP, Brazil] ==
+
 
+
Title: '''Rumor processes on <math>\mathbb{N}</math> and discrete renewal processe'''
+
 
+
Abstract: We study two rumor processes on the positive integers, the dynamics of which are related to an SI epidemic model with long range transmission. Start with one spreader at site <math>0</math> and ignorants situated at some other sites of <math>\mathbb{N}</math>. The spreaders transmit the information within a random distance on their right.  Depending on the initial distribution of the ignorants, we obtain  probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our approach is to relate this model to the house-of-cards.
+
 
+
== Thursday,  May 5, [http://math.arizona.edu/~dianeholcomb/ Diane Holcomb], [http://math.arizona.edu/ University of Arizona]  ==
+
 
+
 
+
Title: '''Local limits of Dyson's Brownian Motion at multiple times'''
+
  
Abstract: Dyson's Brownian Motion may be thought of as a generalization of  Brownian Motion to the matrix setting. We  can study the eigenvalues of a Dyson's Brownian motion at multiple times. The resulting object has different "color" points corresponding to the eigenvalues at different times. Similar to a single time, the correlation functions of the process may be described in terms of determinantal formulas. We study the local behavior of the eigenvalues as we take the dimension of the associated matrix to infinity. The resulting limiting process in the bulk is again determinantal and is described with an "extended sine kernel." This work aims to give an alternate description of the limiting process in terms of the counting function. In this seminar I will go over the the description and methods for finding such a limit. This is work in progress and is joint with Elliot Paquette (Weizmann Institute).
 
  
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+
<!-- ==December 6, TBA ==-->
  
 
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== ==
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 09:51, 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 d-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the p-Sylow subgroup of the sandpile group is a given p-group P, is proportional to |\operatorname{Aut}(P)|^{-1}. 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 d-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