Past Probability Seminars Spring 2020: Difference between revisions

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= Fall 2014 =
= Spring 2018 =


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


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


Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.
<!-- == Thursday, January 25, 2018, TBA== -->


<b>
== Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==
If you would like to sign up for the email list to receive seminar announcements then please send an email to [[File:probsem.jpg]]
</b>


= =
Title: '''A remark on long-range repulsion in spectrum'''
== Thursday, January 23, <span style="color:red"> CANCELED--NO SEMINAR </span>  ==


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.


<!--
== Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==
== Thursday, February 6, [http://people.mbi.ohio-state.edu/newby.23/ Jay Newby], [http://mbi.osu.edu/ Mathematical Biosciences Institute] ==


Title: Applications of large deviation theory in neuroscience
Title: '''Quantitative CLTs for random walks in random environments'''


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


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


Title: Large deviations for point process limits of random matrices.


Abstract: The Gaussian Unitary ensemble (GUE) is one of the most studied Hermitian random matrix model. When appropriately rescaled the eigenvalues in the bulk of the spectrum converge to a translation invariant limiting point process called the Sine process. On large intervals one expects the Sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.
<div style="width:400px;height:75px;border:5px solid black">
<b><span style="color:red"> This is a probability-related colloquium---Please note the unusual room, day, and time! </span></b>
</div>


== Thursday, February 20, Philip Matchett Wood, UW-Madison ==
Title: '''The fractal nature of the Abelian Sandpile'''


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


== Thursday, February 27, [http://mypage.iu.edu/~jthanson/ Jack Hanson], Indiana University Bloomington ==
== Thursday, February 15, 2018, Benedek Valkó, UW-Madison ==


Title: '''Subdiffusive Fluctuations in First-Passage Percolation'''
Title: '''Random matrices, operators and analytic functions'''


Abstract: First-passage percolation is a model consisting of a random metric t(x,y) generated by random variables associated to edges of a graph. Many questions and conjectures in this model revolve around the fluctuating properties of this metric on the graph Z^d. In the early 1990s, Kesten showed an upper bound of Cn for the variance of t(0,nx); this was improved to Cn/log(n) by Benjamini-Kalai-Schramm and Benaim-Rossignol for particular choices of distribution. I will discuss recent work (with M. Damron and P. Sosoe) extending this upper bound to general classes of distributions.
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.  


<!--== Thursday, March 6, TBA ==-->
Joint with B. Virág (Toronto/Budapest).


<!-- Thursday, March 13, TBA -->
== 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]==


== Thursday, March 20, No Seminar due to Spring Break ==
Title: '''Estimation of large-scale time series network models'''
 
== Thursday, March 27, [http://www.stat.wisc.edu/~ane/ Cécile Ané], UW-Madison Department of Statistics ==
 
Title: <b> Application of a birth-death process to model gene gains and losses on a phylogenetic tree </b>


Abstract:
Abstract:
Over time, genes can duplicate or be lost. The history of a gene family is a tree whose nodes represent duplications, speciations, or losses. A birth-and-death process is used to model this gene family tree, embedded within a species tree. I will present this phylogenetic version of the birth and death tree process, along with a probability model for whole-genome duplications. If there is interest and time, I will talk about learning birth and death rates and detecting ancient whole-genome duplications from genomic data.
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
<!--== Thursday, April 3, TBA ==-->
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, April 10, [https://www.math.ucdavis.edu/~romik/home/Dan_Romik_home.html Dan Romik] UC-Davis ==
== Thursday, March 8, 2018, [http://www.math.cmu.edu/~eemrah/ Elnur Emrah], [http://www.math.cmu.edu/index.php CMU] ==


Title: <b>Connectivity patterns in loop percolation and pipe percolation</b>
Title: '''Busemann limits for a corner growth model with deterministic inhomogeneity'''


Abstract:
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.


Loop percolation is a random collection of closed cycles in the square lattice Z^2, that is closely related to critical bond percolation. Its "connectivity pattern" is a random noncrossing matching associated with a loop percolation configuration that encodes information about connectivity of endpoints. The same probability measure on noncrossing matchings arises in several different and seemingly unrelated settings, for example in connection with alternating sign matrices, the quantum XXZ spin chain, and another type of percolation model called pipe percolation. In the talk I will describe some of these connections and discuss some results about the study of pipe percolation from the point of view of the theory of interacting particle systems. I will also mention the "rationality phenomenon" which causes the probabilities of certain natural connectivity events to be dyadic rational numbers such as 3/8, 97/512 and 59/1024. The reasons for this are not completely understood and are related to certain algebraic conjectures that I will discuss separately in Friday's talk in the Applied Algebra seminar.
== Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&T]==
 
<!--== Thursday, April 17, TBA ==-->


<!-- Thursday, April 24, TBA -->
Title: '''A random perturbation approach to some stochastic approximation algorithms in optimization'''


== Thursday, May 1, [http://math.uchicago.edu/~auffing/ Antonio Auffinger] U Chicago ==
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: '''Strict Convexity of the Parisi Functional'''
Title: On shocks in the TASEP


Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.
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.


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


Title: '''Modeling patterns of DNA sequence diversity with Cox Processes'''
== Thursday, April 12, 2018, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison]==


Abstract:
== Thursday, April 19, 2018, TBA==
Events in the evolutionary history of a population can leave subtle signals in the patterns of diversity of its DNA sequences. Identifying those signals from the DNA sequences of present-day populations and using them to make inferences about selection is a well-studied and challenging problem.
== Thursday, April 26, 2018, TBA==
== Thursday, May 3, 2018,TBA==
== Thursday, May 10, 2018, TBA==


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


The model treats the polymorphic sites along the genome as a realization of a Cox Process, a point process with a random intensity.  Within the context of this model, the underlying problem translates to making inferences about the distribution of the intensity function, given the sequence data.
We give a proof of principle, showing that even a simplistic application of the model can quantify differences in diversity between regions with varying recombination rates.  We also suggest a number of directions for applying and extending the model.
-->




== ==
== ==


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

Revision as of 16:24, 9 March 2018


Spring 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.


Thursday, February 1, 2018, Hoi Nguyen, OSU

Title: A remark on long-range repulsion in spectrum

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.

Thursday, February 8, 2018, Jon Peterson, Purdue

Title: Quantitative CLTs for random walks in random environments

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.

Friday, 4pm February 9, 2018, Van Vleck B239 Wes Pegden, CMU

This is a probability-related colloquium---Please note the unusual room, day, and time!

Title: The fractal nature of the Abelian Sandpile

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. 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

Title: Random matrices, operators and analytic functions

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.

Joint with B. Virág (Toronto/Budapest).

Thursday, February 22, 2018, Garvesh Raskutti UW-Madison Stats and WID

Title: Estimation of large-scale time series network models

Abstract: 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 8, 2018, Elnur Emrah, CMU

Title: Busemann limits for a corner growth model with deterministic inhomogeneity

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 Seppäläinen.

Thursday, March 15, 2018, Wenqing Hu Missouri S&T

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, Mustazee Rahman, 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 12, 2018, Sebastien Roch, UW-Madison

Thursday, April 19, 2018, TBA

Thursday, April 26, 2018, TBA

Thursday, May 3, 2018,TBA

Thursday, May 10, 2018, TBA

Past Seminars