Department of

Mathematics


Seminar Calendar
for Probability Seminar events the next 12 months of Wednesday, August 1, 2012.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, August 28, 2012

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

Computational methods for stochastically modeled biochemical reaction networks

David Anderson (UW-Madison)

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

Sebastien Roch (U Wisconsin-Madison)

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.

Tuesday, January 29, 2013

2:00 pm in Altgeld Hall 347,Tuesday, January 29, 2013

Edge universality of Wigner matrices

Ji Oon Lee (Korea Advanced Institute of Science and Technology)

Abstract: For Gaussian ensembles, the probability distribution of the largest eigenvalue converges to the Tracy-Widom distribution. It is also well known that the same result holds for many random matrices other than Gaussian ensembles, which is known as the edge universality. In this talk, a necessary and sufficient condition for the edge universality will be discussed. Tools and key ideas of random matrix theory will also be covered. This is a joint work with Jun Yin.

Tuesday, February 5, 2013

2:00 pm in Altgeld Hall 347,Tuesday, February 5, 2013

Belief propagation for optimal edge-cover in the random complete graph

Rajesh Sundaresan (Indian Institute of Science, ECE)

Abstract: In this talk, we will highlight the main steps in applying Aldous's objective method to the problem of finding the minimum cost edge-cover of the complete graph with random independent and identically distributed edge-costs. We will also discuss a belief propagation algorithm that converges to the optimal solution as the network size grows. The belief propagation algorithm is useful because it yields a near optimal solution with lesser complexity in the above random context than the best known algorithms designed for optimality in worst-case settings. The talk will be based on joint work with Mustafa Khandwawala.

Wednesday, February 13, 2013

2:00 pm in Altgeld Hall 241,Wednesday, February 13, 2013

Operator limits of random matrices

Benedek Valko (UW-Madison)

Abstract: By the Hilbert-Polya conjecture the critical zeros of the Riemann zeta function correspond to the eigenvalues of a self adjoint operator. By a conjecture of Dyson and Montgomery the critical zeros (after a certain rescaling) look like the bulk eigenvalue limit point process of the Gaussian Unitary Ensemble. It is natural to ask if this point process can be described as the spectrum of a random self adjoint operator. We show that this is indeed the case: in fact for any $\beta>0$ the bulk limit of the Gaussian beta ensemble can be obtained as the spectrum of a self adjoint random differential operator. (NOTE: unusual time and place of the seminar.)

Tuesday, February 19, 2013

2:00 pm in Altgeld Hall 347,Tuesday, February 19, 2013

Trace estimates for relativistic stable processes

Hyunchul Park (UIUC Math)

Abstract: A classical question in analysis asks the relationship between the spectrum of the Laplacian on a domain and the geometry of the domain. One way to study the relationship is to study the heat trace(partition function) of the domain. For Brownian motions(whose infinitesimal generators are the Laplacian), the coefficients of the trace of a domain contain important features of the domain such as an area, a perimeter, and the Euler characteristic. A natural question is what happens when Brownian motions are replaced by other Levy processes. In this talk, we study the asymptotic behavior, as the time $t$ goes to zero, of the trace of the semigroup of a killed relativistic $\alpha$-stable process in bounded $C^{1,1}$ open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of $t$ of the trace with an error bound of order $t^{2/\alpha}t^{-d/\alpha}$ for $C^{1,1}$ open sets and of order $t^{1/\alpha}t^{-d/\alpha}$ for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders $t^{-d/\alpha}$ and $t^{(2-d)/\alpha}$ for $C^{1,1}$ open sets, and, when $\alpha\in (0,1]$, between the orders $t^{-d/\alpha}$ and $t^{(1-d)/\alpha}$ for Lipschitz open sets. This is a joint work with R. Song.

Tuesday, February 26, 2013

2:00 pm in Altgeld Hall 347,Tuesday, February 26, 2013

Accumulation and spread of advantageous mutations in a spatially structured tissue

Jasmine Foo (U Minnesota)

Abstract: I will discuss a stochastic model of mutation accumulation and spread in a spatially-structured population. This situation arises in a variety of ecological and biological problems, including the process of cancer initiation from healthy tissue. Cancer arises via the accumulation of mutations to the genetic code. Although many mathematical models of cancer initiation have assumed `perfect mixing' or spatial homogeneity, solid tumors often initiate from tissues with well-regulated spatial architecture and dynamics. Here, we study a stochastic model to investigate the temporal dynamics and patterns of mutation accumulation (i.e. how they depend on system parameters such as mutation rate, population size, and selective fitness advantage of mutations).

Tuesday, March 26, 2013

2:00 pm in Altgeld Hall 347,Tuesday, March 26, 2013

Widder's representation theorem for Dirichlet forms

Nate Eldredge (Cornell)

Abstract: In classical PDE theory, a well-known annoyance is that solutions of the initial-value problem for the heat equation on R^n need not be unique. There is an "obvious" solution given by convolution with the Gaussian heat kernel, but there are other pathological solutions as well. Widder's Theorem asserts that by restricting our attention to nonnegative solutions, we can exclude these pathological solutions and recover uniqueness. This is a reasonable restriction to make, since, for instance, temperatures below absolute zero don't make physical sense. I will discuss an extension of this theorem to the context of a metric measure space equipped with a local Dirichlet form. This provides a weak sort of geometry, and gives us a "Laplacian" that we can use to define a notion of a "solution of the heat equation", as well as a continuous Markov process that plays the role of Brownian motion. I'll begin with a primer on Dirichlet forms with some examples, describe how Widder's theorem looks in this context, and give a sketch of the proof and some of the ingredients it requires. This is joint work with Laurent Saloff-Coste.

Tuesday, April 2, 2013

2:00 pm in Altgeld Hall 347,Tuesday, April 2, 2013

Invariance Principle for theta sums

Francesco Cellarosi (UIUC Math)

Abstract: Theta sums are very classical objects in Number Theory and Physics and can be seen as sums of strongly dependent random variables. I will present several results concerning these sums, such as a non-standard CLT (exhibiting a form of anomalous diffusion), and a weak invariance principle. The standard probabilistic methods for sums of weakly dependent random variables fail in this situation. I will present an approach that uses ergodic theory and homogeneous dynamics. Joint work with Jens Marklof (Bristol, U.K.)

Tuesday, April 16, 2013

2:00 pm in Altgeld Hall 347,Tuesday, April 16, 2013

An Identity of Hitting Times and Its Application to the Valuation of Guaranteed Minimum Withdrawal Benefit

Runhuan Feng (UIUC Math)

Abstract: In this work, we explore an identity in distribution of hitting times of a finite variation process (Yor's process) and a diffusion process (geometric Brownian motion with affine drift), which arise from applications in financial and actuarial mathematics. It turns out that this rather simple probabilistic technique is very useful in certain calculations in conjunction with a spectral expansion method, which would otherwise be very difficult. As a result, we provide an analytical solution to the fair price of variable annuity guaranteed minimum withdrawal benefit, which was only known by numerical PDE solutions in the existing literature. A short introduction to variable annuity with guaranteed minimum withdrawal benefits will be provided. No previous background in finance or actuarial science is required.

Tuesday, April 23, 2013

2:00 pm in Altgeld Hall 347,Tuesday, April 23, 2013

Emergent metastability for dynamical systems on networks

Lee DeVille (Illinois Math)

Abstract: We will consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior. We will concentrate on the case of SDE with small white noise for concreteness. We also present some specific results relating to stochastic perturbations of the Kuramoto system of coupled nonlinear oscillators. Along the way, we show that there is a non-standard spectral problem that appears naturally, and that the important features of this spectral problem are determined by a certain homology group.