Difference between revisions of "Probability Seminar"

From UW-Math Wiki
Jump to: navigation, search
(Fall 2015)
(April 30, 2020, Will Perkins (University of Illinois at Chicago))
 
(474 intermediate revisions by 8 users not shown)
Line 1: Line 1:
 
__NOTOC__
 
__NOTOC__
  
= Spring 2015 =
+
= Spring 2020 =
  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
+
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
 +
<b>We  usually end for questions at 3:20 PM.</b>
  
<b>
+
If you would like to sign up for the email list to receive seminar announcements then please send an email to  
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.
+
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
</b>
 
  
= =
+
 
+
== January 23, 2020, [https://www.math.wisc.edu/~seppalai/ Timo Seppalainen] (UW Madison) ==
== Thursday, September 17, [http://www.math.ucla.edu/~nickcook/ Nicholas A. Cook], [http://www.math.ucla.edu/ UCLA], <span style="color:red"> 2:25pm Van Vleck B325</span> ==
+
'''Non-existence of bi-infinite geodesics in the exponential corner growth model
 
+
'''
<div style="width:430px;height:25px;border:5px solid black">
 
<b><span style="color:red"> Please note the unusual location, Van Vleck Hall B325 </span></b>
 
</div>
 
 
 
Title: '''Random regular digraphs: singularity and spectrum'''
 
 
 
We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices. Towards establishing the same result for the adjacency matrix without iid weights, we prove that it is invertible with high probability. Along the way we make use of Stein's method of exchangeable pairs to establish some graph discrepancy properties.
 
 
 
== Thursday, September 24, No seminar <!--[http://www.math.wisc.edu/~ogrosky/ Reed Ogrosky], [http://www.math.wisc.edu/ UW-Madison]--> ==
 
 
 
== Thursday, October 1 [http://www.math.wisc.edu/~roch Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison]  ==
 
  
Title: '''Mathematics of the Tree of Life--From Genomes to Phylogenetic Trees and Beyond'''
+
Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s.  A non-existence proof  in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in  November 2018. Their proof utilizes estimates from integrable probability.    This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).
  
Abstract:
+
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
The reconstruction of the Tree of Life is an old problem in evolutionary biology which has benefited from various branches of mathematics, including probability, combinatorics, algebra, and geometry. Modern DNA sequencing technologies are producing a deluge of new data on a vast array of organisms--transforming how we view the Tree of Life and how it is reconstructed. I will survey recent progress on some mathematical and computational questions that arise in this context. No biology background will be assumed. (This is a practice run for a plenary talk at an AMS meeting.)
+
'''Quasi-linear parabolic equations with singular forcing'''
  
== Thursday, October 8, No Seminar due to the [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
+
The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral.  By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noiseThe theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.
 
 
[http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium]
 
 
 
== Thursday, October 15, <!--TBA--> [http://math.wisc.edu/~louisfan Louis Fan], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
 
Title: '''Reflected diffusions with partial annihilations on a membrane (Part two)'''
 
 
 
Abstract:
 
Mathematicians and scientists use interacting particle models to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. In this talk, I will introduce an interacting particle system used to model the transport of positive and negative charges in solar cells. To connect the microscopic mechanisms with the macroscopic behaviors at two different scales, we show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. This is the second part of a previous talk given in the Applied and Computation math seminar. Our proofs are based on a correlation function technique (studying the BBGKY hierarchy) and its generalization. This is joint work with Zhen-Qing Chen.
 
 
 
== Thursday, October 22, [http://www.math.wisc.edu/~kurtz/ Tom Kurtz], [http://www.math.wisc.edu UW-Madison] ==
 
 
 
 
 
Title: '''Strong and weak solutions for general stochastic models'''
 
 
Abstract:
 
Typically, a stochastic model relates stochastic “inputs” and, perhaps, controls to stochastic “outputs. A general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations will be given in this context.  A notion of “compatibility” between inputs and outputs is critical in relating the general result to its classical forebears.  Time-change equations for diffusion processes provide an interesting example.  Such equations arise naturally as limits of analogous equations for Markov chains.  For one-dimensional diffusions they also are essentially given in the now-famous notebook of Doeblin.  Although requiring nothing more than standard Brownian motions and the Riemann integral to define, the question of strong uniqueness remains unresolved.  To prove weak uniqueness, the notion of compatible solution is employed and the martingale properties of compatible solutions used to reduce the uniqueness question to the corresponding question for a martingale problem or an Ito equation.
 
  
== Thursday, October 29, [http://www.math.cornell.edu/m/People/EcaterinaSavaHuss Ecaterina Sava-Huss], [http://www.math.cornell.edu/m/ Cornell]  ==
+
In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE.  This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.
  
Title: '''Interpolating between rotor walk and random walk'''
+
== February 6, 2020, [https://sites.google.com/site/cyleeken/ Cheuk-Yin Lee] (Michigan State) ==
 +
'''Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points'''
  
Abstract: After a short introduction on deterministic random walks (called also rotor-router walks)  
+
In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.
and some related cluster growth models, I will introduce a family of stochastic processes on the integers, depending on a parameter p. These processes interpolate between the deterministic rotor walk (for p=0) and the simple random walk (for p=1/2), and they are not Markovian.  
 
For such processes,  I will prove that the scaling limit is a one-sided perturbed Brownian motion, which is a linear combination of a Brownian motion and its running maximum. This is based on joint work with Wilfried Huss and Lionel Levine.
 
  
== Thursday, November 5, No Seminar this week ==
+
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
'''Langevin Monte Carlo Without Smoothness'''
  
== Thursday, November 12, SEMINAR CANCELLED ==
+
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation.
 +
Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.
  
Thunderstorms in Chicago an 11/11 cancelled the speaker's flights; we will try to re-schedule.
+
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
'''A replacement principle for perturbations of non-normal matrices'''
  
<!--[http://www.math.illinois.edu/~lierl/ Janna Lierl], [http://www.math.illinois.edu/ UIUC] ==
+
There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.
  
Title: '''Parabolic Harnack inequality on fractal Dirichlet spaces'''
+
== February 27, 2020, No seminar ==
 +
''' '''
  
Abstract: I will present some recent results on extending the parabolic Moser iteration method to the setting of (fractal-type) metric measure Dirichlet spaces. Under appropriate geometric conditions, we obtain that local weak solutions to heat equation are locally bounded, H\"older continuous, and satisfy a strong parabolic Harnack inequality. If time permits, I will also discuss the case of time-dependent Dirichlet forms, or non-symmetric perturbations of the Dirichlet form.
+
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
Applications of the parabolic Harnack inequality include sharp upper and lower bounds for the associated heat kernel.
+
''' Large Deviation Principles via Spherical Integrals'''
-->
 
  
== Thursday, November 19, [http://orion.math.iastate.edu/dherzog/ David Herzog] [http://www.math.iastate.edu/ Iowa State] ==
+
In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain
  
Title: '''Stabilization by noise and the existence of optimal Lyapunov functions'''
+
1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;
  
Abstract: We discuss certain, explosive ODEs in the plane that become stable under the addition of noise. In each equation, the process by which stabilization occurs is intuitively clear: Noise diverts the solution away from any instabilities in the underlying ODE. However, in many cases, proving rigorously this phenomenon occurs has thus far been difficult and the current methods used to do so are rather ad hoc. Here we present a general, novel approach to showing stabilization by noise and apply it to these examples. We will see that the methods used streamline existing arguments as well as produce optimal results, in the sense that they allow us to understand well the asymptotic behavior of the equilibrium measure at infinity.
+
2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;
  
== Thursday, November 26, No Seminar, Thanksgiving Break ==
+
3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;
== Thursday, December 3, [http://www.math.illinois.edu/~lierl/ Janna Lierl], [http://www.math.illinois.edu/ UIUC] ==
 
  
Title: '''Parabolic Harnack inequality on fractal Dirichlet spaces'''
+
4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.
  
Abstract: I will present some recent results on extending the parabolic Moser iteration method to the setting of (fractal-type) metric measure Dirichlet spaces. Under appropriate geometric conditions, we obtain that local weak solutions to heat equation are locally bounded, H\"older continuous, and satisfy a strong parabolic Harnack inequality. If time permits, I will also discuss the case of time-dependent Dirichlet forms, or non-symmetric perturbations of the Dirichlet form.
+
This is a joint work with Belinschi and Guionnet.
Applications of the parabolic Harnack inequality include sharp upper and lower bounds for the associated heat kernel.
 
  
== Thursday, December 10, [http://www.case.edu/artsci/math/esmeckes/ Elizabeth Meckes], [http://www.case.edu/artsci/math/ Case Western Reserve University] ==
+
== March 12, 2020, No seminar ==
 +
''' '''
  
Title: '''Patterns in Eigenvalues: Random matrices from the compact classical groups.'''
+
== March 19, 2020, Spring break ==
 +
''' '''
  
Abstract: There are many striking features of the eigenvalues of random orthogonal and unitary matrices. In this talk, I'll describe Haar measure on those groups and the resulting distributions of the eigenvalues. I will give a survey of now-classical asymptotic results, and then describe a result of E. Rains and a recent result of mine (joint with M. Meckes), which demonstrate some intriguing  self-similarities of the eigenvalue processes.  Prerequisites will be kept to a minimum.
+
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
 +
== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
 +
''' '''
  
<!--
+
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
== 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.
+
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
</span>
+
''' '''
  
 +
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
  
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
+
3-day event in Van Vleck 911
  
 +
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
Abstract:
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
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.
 
--->
 
  
== ==  
+
== April 30, 2020, CANCELLED, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
  
== Spring Semester ==
 
A selection of the spring semester schedule starts below
 
  
== ==
 
  
== Thursday, January 28, [http://faculty.virginia.edu/petrov/ Leonid Petrov], [http://www.math.virginia.edu/ University of Virginia] ==
 
  
== ==
 
  
  
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 13:59, 12 April 2020


Spring 2020

Thursdays in 901 Van Vleck Hall at 2:30 PM, unless otherwise noted. We usually end for questions at 3:20 PM.

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


January 23, 2020, Timo Seppalainen (UW Madison)

Non-existence of bi-infinite geodesics in the exponential corner growth model

Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s. A non-existence proof in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in November 2018. Their proof utilizes estimates from integrable probability. This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).

January 30, 2020, Scott Smith (UW Madison)

Quasi-linear parabolic equations with singular forcing

The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral. By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise. The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.

In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

February 6, 2020, Cheuk-Yin Lee (Michigan State)

Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points

In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.

February 13, 2020, Jelena Diakonikolas (UW Madison)

Langevin Monte Carlo Without Smoothness

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation. Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.

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

A replacement principle for perturbations of non-normal matrices

There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.

February 27, 2020, No seminar

March 5, 2020, Jiaoyang Huang (IAS)

Large Deviation Principles via Spherical Integrals

In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain

1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;

2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;

3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;

4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.

This is a joint work with Belinschi and Guionnet.

March 12, 2020, No seminar

March 19, 2020, Spring break

March 26, 2020, CANCELLED, Philippe Sosoe (Cornell)

April 2, 2020, CANCELLED, Tianyu Liu (UW Madison)

April 9, 2020, CANCELLED, Alexander Dunlap (Stanford)

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

April 22-24, 2020, CANCELLED, FRG Integrable Probability meeting

3-day event in Van Vleck 911

April 23, 2020, CANCELLED, Martin Hairer (Imperial College)

Wolfgang Wasow Lecture at 4pm in Van Vleck 911

April 30, 2020, CANCELLED, Will Perkins (University of Illinois at Chicago)





Past Seminars