Difference between revisions of "Probability Seminar"

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(Thursday, December 1, Hao Shen, Columbia)
(Thursday, April 19, 2018, Timo Seppäläinen, UW-Madison)
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__NOTOC__
 
__NOTOC__
  
= Fall 2016 =
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= Spring 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.  
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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 join-probsem@lists.wisc.edu.
  
 +
<!-- == Thursday, January 25, 2018, TBA== -->
  
 +
== Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==
  
== Thursday, September 8, Daniele Cappelletti, [http://www.math.wisc.edu UW-Madison] ==
+
Title: '''A remark on long-range repulsion in spectrum'''
Title: '''Reaction networks: comparison between deterministic and stochastic models'''
+
  
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.
+
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.
  
== <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] ==
+
== Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==
  
<div style="width:320px;height:50px;border:5px solid black">
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Title: '''Quantitative CLTs for random walks in random environments'''
<b><span style="color:red"> Please note the unusual day and time </span></b>
+
</div>
+
  
The talk will be in Van Vleck 910 as usual.
+
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.
  
Title: '''Homogenization of viscous Hamilton-Jacobi equations: a remark and an application.'''
+
== <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]==
  
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
 
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] ==
+
<div style="width:400px;height:75px;border:5px solid black">
Title:  '''Low-degree factors of random polynomials'''
+
<b><span style="color:red"> This is a probability-related colloquium---Please note the unusual room, day, and time! </span></b>
 +
</div>
  
Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers.
+
Title: '''The fractal nature of the Abelian Sandpile'''
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
+
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
+
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
+
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]==
+
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.
Title:  '''On the maximum of the characteristic polynomial of the Circular Beta Ensemble'''
+
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.
  
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, February 15, 2018, Benedek Valkó, UW-Madison ==
  
== Thursday, October 6, No Seminar ==
+
Title: '''Random matrices, operators and analytic functions'''
  
== Thursday, October 13, No Seminar due to [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
+
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.  
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] ==
+
Joint with B. Virág (Toronto/Budapest).
Title: Current Fluctuations of the Stationary ASEP and Six-Vertex Model
+
  
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, 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, October 27, [http://www.math.wisc.edu/~hung/ Hung Tran], [http://www.math.wisc.edu/ UW-Madison] ==
+
Title: '''Estimation of large-scale time series network models'''
 
+
Title: '''Homogenization of non-convex Hamilton-Jacobi equations'''
+
 
+
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.
+
 
+
== Thursday, November 3, Alejandro deAcosta, [http://math.case.edu/ Case-Western Reserve] ==
+
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.
+
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, November 10, [https://sites.google.com/a/wisc.edu/louisfan/home Louis Fan], [https://www.math.wisc.edu/ UW-Madison] ==
+
== Thursday, March 8, 2018, [http://www.math.cmu.edu/~eemrah/ Elnur Emrah], [http://www.math.cmu.edu/index.php CMU] ==
 
+
Title: '''Particle representations for (stochastic) reaction-diffusion equations'''
+
  
 +
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.
  
Reaction diffusion equations (RDE) is a popular tool to model complex spatial-temporal patterns including Turing patterns, traveling waves and periodic switching.
+
== Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&T]==
  
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.
+
Title: '''A random perturbation approach to some stochastic approximation algorithms in optimization'''
  
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 dynamicsIn particular, I will present rigorous results about the lineage dynamics for of a biased voter model introduced by Hallatschek and Nelson (2007).
+
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 systemsThis talk is based on a series of joint works with Chris Junchi Li (Princeton), Weijie Su (UPenn) and Haoyi Xiong (Missouri S&T).
  
== Thursday, November 24, No Seminar due to Thanksgiving ==
+
== Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==
  
== Thursday, December 1, [http://math.columbia.edu/~hshen/ Hao Shen], [http://math.columbia.edu/~hshen/ Columbia] ==
+
Title: On shocks in the TASEP
Title: On scaling limits of Open ASEP and Glauber dynamics of ferromagnetic models
+
  
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.)
+
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.
  
== '''Colloquium''' Friday, December 2, [http://math.columbia.edu/~hshen/ Hao Shen], [http://math.columbia.edu/~hshen/ Columbia] ==
+
This video shows the shock forming in Burgers' equation: https://www.youtube.com/watch?v=d49agpI0vu4
  
4pm, Van Vleck 9th floor
+
== 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:  TBA
+
Title: '''PDE compression — asymptotic preserving, numerical homogenization and randomized solvers'''
 
+
== Thursday, December 8, TBA, TBA ==
+
Title:  TBA
+
 
+
== Thursday, December 15, TBA, TBA ==
+
Title:  TBA
+
 
+
 
+
 
+
 
+
 
+
<!--
+
 
+
== 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'''
+
 
+
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
+
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.
+
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.
  
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.)
+
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, March 10, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison] ==
+
== Thursday, April 12, 2018, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison]==
  
Title: '''Delocalization and Universality of band matrices.'''
 
  
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.
+
Title: '''Circular Networks from Distorted Metrics'''
We proved that this conjecture is true as long as the bandwidth is wide enough.
+
  
== Thursday, March 17, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison] ==
+
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.
  
Title: '''Recovering the Treelike Trend of Evolution Despite Extensive Lateral Genetic Transfer'''
+
== Thursday, April 19, 2018, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [https://www.math.wisc.edu UW-Madison]==
  
Abstract
+
Title: '''Shifted weights and  restricted path length in first-passage percolation'''
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 ==
+
First-passage percolation has remained a challenging field of study since its introduction in 1965 by Hammersley and Welsh. There are many outstanding open problems.  Among these are properties of the limit shape and the Euclidean length of geodesics.  This talk describes a convex duality between a shift of the edge weights and the length of the geodesic, together with related results on the regularity of the limit shape as a function of the shift.  The talk is based on joint work  with Arjun Krishnan (Rochester) and Firas Rassoul-Agha (Utah).
  
== Thursday, March 31, [http://www.ssc.wisc.edu/~whs/ Bill Sandholm], [http://www.econ.wisc.edu/ Economics, UW-Madison] ==
+
== Thursday, April 26, 2018, [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==
  
Title: '''A Sample Path Large Deviation Principle for a Class of Population Processes'''
+
Title: '''Limiting eigenvalue distribution for the non-backtracking matrix of an Erdos-Renyi random graph'''
  
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.
+
Abstract: A non-backtracking random walk on a graph is a directed walk with the constraint that the last edge crossed may not be immediately crossed again in the opposite direction. This talk will give a precise description of the eigenvalues of the transition matrix for the non-backtracking random walk when the underlying graph is an Erdos-Renyi random graph on n vertices, where edges present independently with probability p. We allow p to be constant or decreasing with n, so long as p*sqrt(n) tends to infinity.  The key ideas in the proof are partial derandomization, applying the Tao-Vu Replacement Principle in a novel context, and showing that partial derandomization may be interpreted as a perturbation, allowing one to apply the Bauer-Fike Theorem.  Joint work with Ke Wang at HKUST (Hong Kong University of Science and Technology).
  
[http://www.ssc.wisc.edu/~whs/research/ldp.pdf paper preprint]
+
== Thursday, May 3, 2018,TBA==
 +
== Thursday, May 10, 2018, TBA==
  
== 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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== ==
 
== ==
 
 
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Revision as of 13:45, 13 April 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 5, 2018, Qin Li, 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, Sebastien Roch, 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, Timo Seppäläinen, UW-Madison

Title: Shifted weights and restricted path length in first-passage percolation

First-passage percolation has remained a challenging field of study since its introduction in 1965 by Hammersley and Welsh. There are many outstanding open problems. Among these are properties of the limit shape and the Euclidean length of geodesics. This talk describes a convex duality between a shift of the edge weights and the length of the geodesic, together with related results on the regularity of the limit shape as a function of the shift. The talk is based on joint work with Arjun Krishnan (Rochester) and Firas Rassoul-Agha (Utah).

Thursday, April 26, 2018, Philip Matchett Wood, UW-Madison

Title: Limiting eigenvalue distribution for the non-backtracking matrix of an Erdos-Renyi random graph

Abstract: A non-backtracking random walk on a graph is a directed walk with the constraint that the last edge crossed may not be immediately crossed again in the opposite direction. This talk will give a precise description of the eigenvalues of the transition matrix for the non-backtracking random walk when the underlying graph is an Erdos-Renyi random graph on n vertices, where edges present independently with probability p. We allow p to be constant or decreasing with n, so long as p*sqrt(n) tends to infinity. The key ideas in the proof are partial derandomization, applying the Tao-Vu Replacement Principle in a novel context, and showing that partial derandomization may be interpreted as a perturbation, allowing one to apply the Bauer-Fike Theorem. Joint work with Ke Wang at HKUST (Hong Kong University of Science and Technology).

Thursday, May 3, 2018,TBA

Thursday, May 10, 2018, TBA

Past Seminars