Department of

# Mathematics

Seminar Calendar
for Probability Seminar events the next 12 months of Sunday, January 1, 2012.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 31, 2012

2:00 pm in 347 Altgeld Hall,Tuesday, January 31, 2012

#### Strong Central Limit Theorems in Elliptic and Parabolic PDE with random coefficients

###### Joseph Conlon   [email] (U Michigan Math)

Abstract: In this talk I will discuss some recent work of myself in collaboration with Tom Spencer and Arash Fahim. The basic object of study is uniformly elliptic and parabolic PDE in divergence form with random coefficients. It has been known since the 1980's that under suitable scaling the solutions of these equations converge in distribution to the solution of a constant coefficient PDE, the so called homogenized equation. In the talk I shall describe our results on the rate of convergence in homogenization when the random environment is a uniformly elliptic Euclidean Field Theory. The main tools needed to prove these theorems are the Poincare inequality and the Calderon-Zygmund theorem on boundedness in L^p of Fourier multipliers.

Tuesday, February 7, 2012

2:00 pm in Altgeld Hall 347,Tuesday, February 7, 2012

#### A modified diffusion Monte Carlo and other ensemble sampling methods

###### Jonathan Weare   [email] (U Chicago Math)

Abstract: This talk will survey my efforts with coworkers to develop and analyze Monte Carlo sampling algorithms for complex (usually high dimensional) probability distributions. These sampling problems are typically difficult because they have multiple high probability regions separated by low probability regions and/or they are badly scaled in the sense that there are strong unknown relationships between variables. I'll begin the talk by discussing a simple modification of the standard diffusion Monte Carlo algorithm that results in a more efficient and much more flexible tool for use, for example, in rare event simulation. If time permits I'll discuss a few other ensemble based sampling tools designed to directly address energy barriers and scaling issues.

Tuesday, February 28, 2012

2:00 pm in 347 Altgeld Hall,Tuesday, February 28, 2012

#### Random homogenization of Hamilton-Jacobi equations

###### Scott Armstrong   [email] (U Wisconsin Math)

Abstract: I will describe some recent work with Souganidis on the stochastic homogenization of Hamilton-Jacobi equations (both "viscous" as well as first-order equations). The homogenization of special cases of these equations has a direct connection to the work of Sznitman on the quenched large deviations of Brownian motion in the presence of Poissonian obstacles. It also benefits from some strong analogies to certain aspects of the theory of first-passage percolation.

Tuesday, March 6, 2012

2:00 pm in Altgeld Hall 347,Tuesday, March 6, 2012

#### Large deviations and slowdown asymptotics for excited random walks

###### Jonathon Peterson   [email] (Purdue)

Abstract: Excited random walks (also called cookie random walks) are self-interacting random walks where the transition probabilities depend on the number of previous visits to the current location. Although the models are quite different, many of the known results for one-dimensional excited random walks have turned out to be remarkably similar to the corresponding results for random walks in random environments. For instance, one can have transience with sub-linear speed and limiting distributions that are non-Gaussian. In this talk I will prove a large deviation principle for excited random walks. The main tool used will be what is known as the "backwards branching process" associated with the excited random walk, thus reducing the problem to proving a large deviation principle for the empirical mean of a Markov chain (a much simpler task). While we do not obtain an explicit formula for the large deviation rate function, we will be able to give a good qualitative description of the rate function. While many features of the rate function are similar to the corresponding rate function for RWRE, there are some interesting differences that highlight the major difference between RWRE and excited random walks.

Tuesday, March 27, 2012

2:00 pm in Altgeld Hall,Tuesday, March 27, 2012

#### Survey of the Circular Law

###### Philip Matchett Wood (U Wisconsin-Madison)

Abstract: What do the eigenvalues of a random matrix look like? This talk will focus on large square matrices where the entries are independent, identically distributed random variables. In the most basic case, the distribution of the eigenvalues in the complex plane (suitably scaled) approaches the uniform distribution on the unit disk, which is called the circular law. We will discuss some of the methods that have been used to prove the circular law, including recent work that has extended the circular law to the most general situation, and we will also discuss generalizations to situations where the eigenvalue distributions are stable, but non-circular.

Tuesday, April 3, 2012

2:00 pm in Altgeld Hall 347,Tuesday, April 3, 2012

#### Eigenvalue and Eigenvector distributions of Random matrices

Abstract: In the current study of the random matrix theory, many long time open problems have been solved in the past three years. Right now, the study of the distribution of individual eigenvalues, even eigenvectors has become possible. In some works, we even obtained some brand new formulas which were not predicted before. And our methods have been successfully applied on many different matrix ensembles, like (generalized) Wigner matrix, covariance matrix, band matrix, Erdoes-renyi Graph, correlation matrix, etc. In this talk, besides the recent process on random matrix theory, we will also introduce the main open questions in this field.

Tuesday, April 10, 2012

2:00 pm in Altgeld Hall 347,Tuesday, April 10, 2012

#### A construction of bivariate distributions with arbitrary marginals and specified correlation

###### Nevena Maric   [email] (U Missouri - St Louis)

Abstract: We propose a simple and efficient algorithm for exact generation of bivariate samples from two arbitrary (possibly different) marginal distributions and with any attainable correlation coefficient (positive or negative). Our algorithm is related to the ideas of trivariate reduction (introduced by Arnold in 1967) in a sense that we use three independent uniforms in order to obtain a pair of correlated variables with desired marginals. This way we construct a bivariate distribution that is a mixture of Fr\'echet bounds and marginal products. The method allows for fast simulation, and does not have any theoretical limitation in terms of types of distributions and ranges of correlations. [joint work with Vanja Dukić]

Tuesday, April 17, 2012

2:00 pm in 347 Altgeld Hall,Tuesday, April 17, 2012

#### Noisy Perturbations of Kuramoto Oscillators

###### Lee DeVille   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: We consider the Kuramoto model of coupled oscillators with various couplings and additive white noise. Synchronous solutions which are stable without the addition of noise become metastable and there are transitions amongst synchronous solutions on long timescales. We compute these timescales and, moreover, compute the most likely path in phase space that transitions will follow. We compute these timescales for various families of connection graphs and show that for several families, these timescales do not increase as the system size increases (i.e the action governing the large deviation timescale does not depend significantly on the number of oscillators).

Thursday, April 19, 2012

2:00 pm in 347 Altgeld Hall,Thursday, April 19, 2012

#### Lake Wobegon Dice

###### David Stork (Distinguished Research Scientist and Research Director, Rambus Labs, Computational Sensing and Imaging Initiative)

Abstract: We present sets of n non-standard dice-Lake Wobegon Dice-having the following paradoxical property: On every (random) roll of a set, each die is more likely to roll greater than the set average than less than the set average; in a speci c statistical sense, then, each die is "better than the set average." We defi ne the Lake Wobegon Dominance of a die in a set as the probability the die rolls greater than the set average minus the probability the die rolls less than the set average. We further defi ne the Lake Wobegon Dominance of the set to be the dominance of the set's least dominant die and prove that such paradoxical dominance is bounded above by $(n-2)/n$ regardless of the number of sides s on each die and the maximum number of pips p on each side. A set achieving this bound is called Lake Wobegon Optimal. We give a constructive proof that Lake Wobegon Optimal sets exist for all $n \geq 3$ if one is free to choose $s$ and $p$. We also show how to construct minimal optimal sets, that is, that set that requires the smallest range in the number of pips on the faces. We determine the frequency of such Lake Wobegon sets in the $n = 3$ case through exhaustive computer search and nd the unique optimal $n = 3$ set having minimal $s$ and $p$. We investigate symmetry properties of such sets, and present equivalence classes having identical paradoxical dominance. We construct inverse sets, in which on any roll each die is more likely to roll less than the set average than greater than the set average, and thus each die is "worse than the set average." We show the unique extreme "worst" case, the Lake Wobegon Pessimal set. We speculate on the application of such paradoxical Lake Wobegon Dominance for collusion among agents, such as economic agents, each seeking to be "above average" as often as possible. [Joint work with Jorge Moraleda]

Tuesday, April 24, 2012

2:00 pm in 347 Altgeld Hall,Tuesday, April 24, 2012

#### A Large Deviations Theory for Heavy-tailed Processes via Martingale Arguments and its Connections to Monte Carlo

###### Jose Blanchet (Columbia Engineering)

Abstract: Modern large deviations theory centers around the work of Donsker and Varadhan in the setting of processes whose marginal distributions have a finite moment generating. The existence of a finite moment generating function turns out to be crucial for the development of the theory in that it allows to define suitable positive martingales which, in turn, generate changes of measure. In important areas of application such as risk theory and operations research, however, stochastic processes with heavy-tails arise naturally; so no moment generating function exists. Moreover, it is well known (as we shall discuss) that the qualitative features of most likely paths given rare events are completely different in light and heavy tailed settings. One of the most advantageous features of the Donsker-Varadhan theory is that it suggests, via changes of measure, efficient Monte Carlo simulation methods for sampling rare events. In this talk, we present an approach for the large deviations analysis of heavy-tailed processes that is based on changes of measure and thus it is parallel to the Donsker-Varadhan approach in that the analysis suggests efficient Monte Carlo simulation methods for rare events as well. (This talk builds on joint work with P. Glynn and J. C. Liu).

Tuesday, August 28, 2012

2:00 pm in Altgeld Hall 347,Tuesday, August 28, 2012

#### Computational methods for stochastically modeled biochemical reaction networks

Abstract: I will focus this talk on computational methods for stochastically modeled biochemical reaction networks. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. I will show how different computational methods can be understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.

Tuesday, September 4, 2012

2:00 pm in Altgeld Hall 347,Tuesday, September 4, 2012

#### Invasion percolation, the incipient infinite cluster and random walks in 2D

###### Michael Damron (Princeton U)

Abstract: In this talk I will define two percolation models: the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC). Both of these are closely related to critical independent percolation, and have similar critical exponents and fractal structure. I will explain recent results with Phil Sosoe and Jack Hanson (Ph. D. students at Princeton) about quenched subdiffusivity of random walks on both graphs. This extends work of H. Kesten, who showed in the 80's that a random walk on the IIC is subdiffusive in an averaged sense.

Tuesday, September 18, 2012

2:00 pm in Altgeld Hall 347,Tuesday, September 18, 2012

#### Liberation of Random Projections

###### Todd Kemp (UCSD Math)

Abstract: Given two subspaces of a finite dimensional space, there is a minimal dimension their intersection can have; when this dimension is minimized the subspaces are said to be in general position. Easy 19th Century mathematics shows that any two subspaces are almost surely'' in general position in many senses. One modern precise meaning we can give to this statement is as follows: perform a Brownian motion on the unitary group (of rotations), applied to one of the subspaces. Then for any fixed positive time, the Brownian rotated subspaces are almost surely in general position, regardless of starting configuration. What happens if the ambient space is an infinite dimensional Hilbert? While there is no unitarily invariant measure, it is still possible to make sense of the unitary Brownian motion and its action on some (but not all) subspaces. However, the easy techniques for proving the general position theorem are unavailable. Instead, one can apply stochastic analysis and free probability techniques to to analyze a spectral measure associated to the problem. In this talk, I will discuss probabilistic and PDE techniques that come into play in proving the general position theorem in infinite dimensions. This is joint work with Benoit Collins.

Tuesday, September 25, 2012

2:00 pm in 347 Altgeld Hall,Tuesday, September 25, 2012

#### Some aspects of stochastic differential equations driven by fractional Brownian motions

###### Fabrice Baudoin   [email] (Purdue Math)

Abstract: In this talk we will review several results on stochastic differential equations driven by fractional Brownian motions that the speakers obtained in a series of more or less recent works. We shall in particular focus on the study of gradients bounds, Gaussian heat kernels bounds and small time asymptotics for the operators naturally associated with such equations. The presentation will be based on joint works with L. Coutin, M. Hairer, C. Ouyang and S. Tindel.

Tuesday, October 9, 2012

2:00 pm in Altgeld Hall 347,Tuesday, October 9, 2012

#### Probabilistic Techniques in Mathematical Phylogenetics: Relating Combinatorial and Variational Distances on Trees

Abstract: I will describe recent results on a connection between the so-called reconstruction problem on Markov random fields on trees and two important problems in computational evolutionary biology: the inference of ancestral states and the estimation of phylogenies using maximum likelihood. This is joint work with Allan Sly. No biology background will be assumed.

Tuesday, October 23, 2012

2:00 pm in 347 Altgeld Hall,Tuesday, October 23, 2012

#### A Strong Law of Large Numbers for Super-stable Processes

###### Yan-Xia Ren (Peking University)

Abstract: Let $X=(X_t, t\ge 0; P_\mu)$ be a supercritical, super-stable process corresponding to the operator $-\left(-\Delta\right)^{\alpha/2 } u+\beta u-\eta u^2$ on $\mathbb{R}^d$ with constants $\beta,\eta>0$ and $\alpha\in(0,2]$, and let $\ell$ be Lebesgue measure on $\mathbb{R}^d$. Put $\hat W_t(\theta)=e^{(\beta-|\theta|^\alpha)t}X_t(e^{i\theta\cdot})$, which is a complex-valued martingale for each $\theta\in\mathbb{R}^d$ with limit $\hat W(\theta)$ say. Our main result establishes that for any starting measure $\mu$, which is a finite measure on $\mathbb{R}^d$ such that $\int_{\mathbb{R}^d}x\mu(\mathrm{d}x)<\infty$, $\frac{t^{d/\alpha}X_{t}}{e^{\beta t}}\rightarrow c_{\alpha }\hat W\left(0\right) \ell$ $P_\mu$-a.s. in a topology, termed the shallow topology, strictly stronger than the vague topology yet weaker than the weak topology. This result can be thought of as an extension to a class of superprocesses of Watanabe's strong law of large numbers for branching Markov processes. This talk is based on a joint work with Michael A. Kouritzin.

Tuesday, November 6, 2012

2:00 pm in Altgeld Hall 347,Tuesday, November 6, 2012

#### Limit theorems for one-dimensional, conservative, weakly assymetric stochastic systems

###### Milton Jara   [email] (IMPA)

Abstract: We explain how to obtain the scaling limit of various quantities of interest for one-dimensional, conservative, weakly asymmetric stochastic systems. In particular, we explain the connection between these limits and the celebrated KPZ equation.

Tuesday, November 13, 2012

2:00 pm in Altgeld Hall 347,Tuesday, November 13, 2012

#### Chaos problem in the mixed even-spin models

###### Wei-Kuo Chen (U Chicago Math)

Abstract: The Sherrington-Kirkpatrick (SK) model is one of the most important spin glasses invented by Sherrington and Kirkpatrick with the aim of understanding strange magnetic behaviors of certain alloys. In this talk we will first introduce some well-known results about this model such as the Parisi formula and the limiting behavior of the overlap. Next, we will discuss the problems of chaos in the mixed even-spin models and present mathematically rigorous results including disorder, external field, and temperature chaos.