Difference between revisions of "Probability Seminar"

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(Thursday, April 12, 2018, Sebastien Roch, UW-Madison)
(November 14, 2019, Benjamin Landon, MIT)
 
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__NOTOC__
 
__NOTOC__
  
= Spring 2018 =
+
= Fall 2019 =
  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
+
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
<b>We  usually end for questions at 3:15 PM.</b>
+
<b>We  usually end for questions at 3:20 PM.</b>
  
If you would like to sign up for the email list to receive seminar announcements then please send an email to 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]
  
<!-- == Thursday, January 25, 2018, TBA== -->
+
 +
== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==
 +
'''Furstenberg theorem: now with a parameter!'''
  
== Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==
+
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter.
 +
It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.
 +
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.
  
Title: '''A remark on long-range repulsion in spectrum'''
+
== September 19, 2019, [http://math.columbia.edu/~xuanw  Xuan Wu], Columbia University==
  
Abstract: In this talk we will address a "long-range" type repulsion among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under  arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.
+
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''
  
== Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==
+
In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.
  
Title: '''Quantitative CLTs for random walks in random environments'''
+
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
  
Abstract:The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.
+
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==
  
== <span style="color:red"> Friday, 4pm </span> February 9, 2018, <span style="color:red">Van Vleck B239</span> [http://www.math.cmu.edu/~wes/ Wes Pegden], [http://www.math.cmu.edu/ CMU]==
+
''' Simplified dynamics for noisy systems with delays.'''
  
 +
Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed.  In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.
  
<div style="width:400px;height:75px;border:5px solid black">
+
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==
<b><span style="color:red"> This is a probability-related colloquium---Please note the unusual room, day, and time! </span></b>
 
</div>
 
  
Title: '''The fractal nature of the Abelian Sandpile'''
+
'''A general beta crossover ensemble'''
  
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.
+
I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known  "soft" and "hard"  edge point processes.  This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved  uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.
 
  
== Thursday, February 15, 2018, Benedek Valkó, UW-Madison ==
+
== October 31, 2019, Vadim Gorin, UW Madison==
  
Title: '''Random matrices, operators and analytic functions'''
+
'''Shift invariance for the six-vertex model and directed polymers.'''
  
Abstract: Many of the important results of random matrix theory deal with limits of the eigenvalues of certain random matrix ensembles. In this talk I review some recent results on limits of `higher level objects' related to random matrices: the limits of random matrices viewed as operators and also limits of the corresponding characteristic functions.  
+
I will explain a recently discovered mysterious property in a variety of stochastic systems ranging  from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.
  
Joint with B. Virág (Toronto/Budapest).
+
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==
 +
'''Domino tilings of the Aztec diamond with doubly periodic weightings​'''
  
== Thursday, February 22, 2018, [http://pages.cs.wisc.edu/~raskutti/ Garvesh Raskutti] [https://www.stat.wisc.edu/ UW-Madison Stats] and [https://wid.wisc.edu/people/garvesh-raskutti/ WID]==
+
This talk will be centered around domino tilings of the Aztec diamond with doubly periodic weightings. In particular asymptotic results of the $ 2 \times k $-periodic Aztec diamond will be discussed, both in the macroscopic and microscopic scale. The macroscopic picture is described using a close connection to a Riemann surface. For instance, the number of smooth regions (also called gas regions) is the same as the genus of the mentioned Riemann surface.  
 +
 +
The starting point of the asymptotic analysis is a non-intersecting path formulation and a double integral formula for the correlation kernel. The proof of this double integral formula is based on joint work with M. Duits, which will be discuss briefly if time permits.
  
Title: '''Estimation of large-scale time series network models'''
+
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==
 +
'''Universality of extremal eigenvalue statistics of random matrices'''
  
Abstract:
+
The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory.  However, the behavior of certain ``extremal'' or ``critical'' observables is not fully understood.  Towards the former, we discuss progress  on the universality of the largest gap between consecutive eigenvalues. With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.
Estimating networks from multi-variate time series data
 
is an important problem that arises in many applications including
 
computational neuroscience, social network analysis, and many
 
others. Prior approaches either do not scale to multiple time series
 
or rely on very restrictive parametric assumptions in order to
 
guarantee mixing. In this talk, I present two approaches that provide
 
learning guarantees for large-scale multi-variate time series. The first
 
involves a parametric GLM framework where non-linear clipping and  
 
saturation effects that guarantee mixing. The second involves a
 
non-parametric sparse additive model framework where beta-mixing
 
conditions are considered. Learning guarantees are provided in both
 
cases and theoretical results are supported both by simulation results
 
and performance comparisons on various data examples.
 
<!-- == Thursday, March 1, 2018, TBA== -->
 
  
== Thursday, March 8, 2018, [http://www.math.cmu.edu/~eemrah/ Elnur Emrah], [http://www.math.cmu.edu/index.php CMU] ==
+
== November 21, 2019, Tung Nguyen, UW Madison ==
  
Title: '''Busemann limits for a corner growth model with deterministic inhomogeneity'''
+
== November 28, 2019, Thanksgiving (no seminar) ==
  
Abstract:
 
Busemann limits have become a useful tool in study of geodesics in percolation models. The
 
properties of these limits are closely related to the curvature of the limit shapes in the associated
 
growth models. In this talk, we will consider a corner growth model (CGM) with independent
 
exponential weights. The rates of the exponentials are deterministic and inhomogeneous across
 
columns and rows. (An equivalent model is the TASEP with step initial condition and with
 
particlewise and holewise deterministic disorder). In particular, the model lacks stationarity.
 
Under mild assumptions on the rates, the limit shape in our CGM exists, is concave and can
 
develop flat regions only near the axes. In contrast, flat regions can only occur away from the axes
 
in the CGM with general i.i.d. weights. This feature and stationarity have been instrumental in
 
proving the existence of the Busemann limits in past work. We will discuss how to adapt and
 
extend these arguments to establish the existence and main properties of the Busemann limits
 
in both flat and strictly concave regions for our CGM. The results we will present are from an
 
ongoing joint project with Chris Janjigian and Timo Sepp&auml;l&auml;inen.
 
  
== Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&T]==
+
==December 5, 2019 ==
 
 
Title: '''A random perturbation approach to some stochastic approximation algorithms in optimization'''
 
 
 
Abstract: Many large-scale learning problems in modern statistics and machine learning can be reduced to solving stochastic optimization problems, i.e., the search for (local) minimum points of the expectation of an objective random function (loss function). These optimization problems are usually solved by certain stochastic approximation algorithms, which are recursive update rules with random inputs in each iteration. In this talk, we will be considering various types of such stochastic approximation algorithms, including the stochastic gradient descent, the stochastic composite gradient descent, as well as the stochastic heavy-ball method. By introducing approximating diffusion processes to the discrete recursive schemes, we will analyze the convergence of the diffusion limits to these algorithms via delicate techniques in stochastic analysis and asymptotic methods, in particular random perturbations of dynamical systems.  This talk is based on a series of joint works with Chris Junchi Li (Princeton), Weijie Su (UPenn) and Haoyi Xiong (Missouri S&T).
 
 
 
== Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==
 
 
 
Title: On shocks in the TASEP
 
 
 
Abstract: The TASEP particle system, moving rightward, runs into traffic jams when the initial particle density to the left of the origin is smaller than the density to the right. The density function satisfies Burgers' equation and traffic jams correspond to its shocks. I will describe work with Jeremy Quastel on a specialization of the TASEP where we identify joint fluctuations of particles at the shock by using determinantal formulae for correlation functions of TASEP and its KPZ scaling limit. The limit process is expressed in terms of GOE Tracy-Widom laws.
 
 
 
This video shows the shock forming in Burgers' equation: https://www.youtube.com/watch?v=d49agpI0vu4
 
 
 
== Thursday, March 29, 2018, Spring Break ==
 
== Thursday, April 5, 2018, [http://www.math.wisc.edu/~qinli/ Qin Li], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
 
Title: '''PDE compression — asymptotic preserving, numerical homogenization and randomized solvers'''
 
 
 
Abstract:
 
All classical PDE numerical solvers are deterministic. Grids are sampled and basis functions are chosen a priori. The corresponding discrete operators are then inverted for the numerical solutions.
 
 
 
We study if randomized solvers could be used to compute PDEs. More specifically, for PDEs that demonstrate multiple scales, we study if the macroscopic behavior in the solution could be quickly captured via random sampling. The framework we build is general and it compresses PDE solution spaces with no analytical PDE knowledge required. The concept, when applied onto kinetic equations and elliptic equations with porous media, is equivalent to asymptotic preserving and numerical homogenization respectively.
 
 
 
== Thursday, April 12, 2018, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison]==
 
 
 
 
 
Title: '''Circular Networks from Distorted Metrics'''
 
 
 
Abstract: Trees have long been used as a graphical representation of species relationships. However
 
complex evolutionary events, such as genetic reassortments or hybrid speciations which
 
occur commonly in viruses, bacteria and plants, do not fit into this elementary framework.
 
Alternatively, various network representations have been developed. Circular networks are a
 
natural generalization of leaf-labeled trees interpreted as split systems, that is, collections of
 
bipartitions over leaf labels corresponding to current species. Although such networks do not
 
explicitly model specific evolutionary events of interest, their straightforward visualization and
 
fast reconstruction have made them a popular exploratory tool to detect network-like evolution
 
in genetic datasets.
 
 
 
Standard reconstruction methods for circular networks rely on an
 
associated metric on the species set. Such a metric is first estimated from DNA sequences,
 
which leads to a key difficulty: distantly related sequences produce statistically unreliable
 
estimates. In the tree case, reconstruction methods have been developed using
 
the notion of a distorted metric, which captures the dependence of the error in the distance
 
through a radius of accuracy. I will present the first circular network reconstruction method
 
based on distorted metrics. This is joint work with Jason Wang.
 
 
 
== Thursday, April 19, 2018, TBA==
 
== Thursday, April 26, 2018, TBA==
 
== Thursday, May 3, 2018,TBA==
 
== Thursday, May 10, 2018, TBA==
 
 
 
 
 
 
 
 
 
== ==
 
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 16:10, 6 November 2019


Fall 2019

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


September 12, 2019, Victor Kleptsyn, CNRS and University of Rennes 1

Furstenberg theorem: now with a parameter!

The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes. Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.

September 19, 2019, Xuan Wu, Columbia University

A Gibbs resampling method for discrete log-gamma line ensemble.

In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.

October 10, 2019, NO SEMINAR - Midwest Probability Colloquium

October 17, 2019, Scott Hottovy, USNA

Simplified dynamics for noisy systems with delays.

Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

October 24, 2019, Brian Rider, Temple University

A general beta crossover ensemble

I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).

October 31, 2019, Vadim Gorin, UW Madison

Shift invariance for the six-vertex model and directed polymers.

I will explain a recently discovered mysterious property in a variety of stochastic systems ranging from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.

November 7, 2019, Tomas Berggren, KTH Stockholm

Domino tilings of the Aztec diamond with doubly periodic weightings​

This talk will be centered around domino tilings of the Aztec diamond with doubly periodic weightings. In particular asymptotic results of the $ 2 \times k $-periodic Aztec diamond will be discussed, both in the macroscopic and microscopic scale. The macroscopic picture is described using a close connection to a Riemann surface. For instance, the number of smooth regions (also called gas regions) is the same as the genus of the mentioned Riemann surface.

The starting point of the asymptotic analysis is a non-intersecting path formulation and a double integral formula for the correlation kernel. The proof of this double integral formula is based on joint work with M. Duits, which will be discuss briefly if time permits.

November 14, 2019, Benjamin Landon, MIT

Universality of extremal eigenvalue statistics of random matrices

The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory. However, the behavior of certain ``extremal or ``critical observables is not fully understood. Towards the former, we discuss progress on the universality of the largest gap between consecutive eigenvalues. With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.

November 21, 2019, Tung Nguyen, UW Madison

November 28, 2019, Thanksgiving (no seminar)

December 5, 2019

Past Seminars