Department of

Mathematics


Seminar Calendar
for Probability Seminar events the year of Saturday, November 9, 2019.

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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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Thursday, January 24, 2019

2:00 pm in 347 Altgeld Hall,Thursday, January 24, 2019

Introduction to Percolation Theory

Grigory Terlov (UIUC Math)

Abstract: This is the first part of two talks designed to introduce students to Percolation Theory. We will describe the model, talk about infinite clusters, prove the existence of the phase transition, introduce the universality principle and more.

Thursday, January 31, 2019

2:00 pm in 347 Altgeld Hall,Thursday, January 31, 2019

Introduction to Percolation Theory (Part 2)

Grigory Terlov (UIUC Math)

Abstract: This is the second part of two talks designed to introduce students to Percolation Theory. We will discuss an upper bound for critical probability for $\mathbb{Z}^d$ via cut-sets and duality. This talk should be accessible for people who missed the first part.

Thursday, February 7, 2019

2:00 pm in 347 Altgeld Hall,Thursday, February 7, 2019

An Introduction to Dyson Brownian Motion and Universality

Kesav Krishnan (UIUC Math)

Abstract: We define Brownian motion on the space of N×N Hermitian Matrices, and derive an SDE for the corresponding process of the eigenvalues. We then establish that the eigenvalue process is identical to Brownian motion in R^n confined to the Weyl Chamber.

Thursday, February 14, 2019

2:00 pm in 347 Altgeld Hall,Thursday, February 14, 2019

An Introduction to Dyson Brownian Motion and Universality (Part 2)

Kesav Krishnan (UIUC Math)

Abstract: We will discuss the connections of Dyson Brownian Motion and the Totally Asymmetric Simple Exclusion Process (TASEP). This will be the first glimpse of the Kardar Parisi Zhang Universality class.

Tuesday, March 26, 2019

1:00 pm in Altgeld Hall,Tuesday, March 26, 2019

To Be Announced

Thursday, April 4, 2019

2:00 pm in 347 Altgeld Hall,Thursday, April 4, 2019

On the range of lattice models in high dimensions

Ed Perkins (University of British Columbia)

Abstract: We investigate the scaling limit of the {\em range} (the set of visited vertices) for a general class of critical lattice models, starting from a single initial particle at the origin. Conditions are given on the random sets and an associated ``ancestral relation" under which, conditional on longterm survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball, that is for the {\em extrinsic one-arm probability}. We show that these conditions are satisfied by the voter model in dimensions $d\ge2$, sufficiently spread out critical oriented percolation and critical contact processes in dimensions $d>4$, and sufficiently spread out critical lattice trees in dimensions $d>8$.

Monday, April 8, 2019

1:00 pm in 145 Altgeld Hall,Monday, April 8, 2019

Uniform dimension results for the inverse images of symmetric Levy processes.

Hyunchul Park (SUNY New Paltz)

Abstract: In this talk, we prove the uniform Hausdorff dimension of the inverse images of a large class of symmetric Levy processes with weak scaling conditions on their characteristic exponents. Along the way we also prove an upper bound for the uniform modulus of continuity of the local times of these processes. This result extends a result of Kaufman (1985) for Brownian motions and of Song, Xiao, and Yang (2018) for stable processes. We also establish the packing dimension results as a byproduct.

Tuesday, April 9, 2019

2:00 pm in 345 Altgeld Hall,Tuesday, April 9, 2019

Quantitative inequalities for the expected lifetime of the Brownian motion

Daesung Kim (Purdue University)

Abstract: The isoperimetric-type inequality for the expected lifetime of the Brownian motion state that the $L^p$ norm of the expected lifetime in a region is maximized when the region is a ball with the same volume. In particular, if $p=1$, it is called the Saint-Venant inequality and has a close relation to the classical Faber—Krahn inequality for the first eigenvalue. In this talk, we prove a quantitative improvement of the inequalities, which explains how a region is close to being a ball when equality almost holds in these inequalities. We also discuss some related open problems.

Thursday, April 11, 2019

2:00 pm in 347 Altgeld Hall,Thursday, April 11, 2019

Local Limit Theorem

Qiang Wu (UIUC Math)

Abstract: This talk is an introduction to some classical CLT variants, specifically on local limit theorem (LLT). The proof of classical LLT for lattice and non-lattice distribution will be discussed using the characteristic approach. Other various generalizations of LLT will be pointed out. Finally, a concise combinatorial approach for LLT of simple random walk will be sketched. Time permits, I will talk about the generalized Berry-Esseen Inequality.

Tuesday, April 16, 2019

2:00 pm in 345 Altgeld Hall,Tuesday, April 16, 2019

Large deviations for quasilinear parabolic stochastic partial differential equations

Rangrang Zhang (Beijing Institute of Technology and University of Tennessee)

Abstract: In this talk I will present some recent results on large deviations for quasilinear parabolic stochastic partial differential equations. More precisely, I will talk about Freidlin-Wentzell type large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are not necessarily locally monotone. Our proof is based on the weak convergence approach.

Thursday, April 18, 2019

2:00 pm in 347 Altgeld Hall,Thursday, April 18, 2019

Local Limit Theorem (Part 2)

Qiang Wu (UIUC Math)

Abstract: This talk the second part of an introduction to some classical CLT variants, specifically on local limit theorem (LLT). The proof of classical LLT for lattice and non-lattice distribution will be discussed using the characteristic approach. Other various generalizations of LLT will be pointed out. Finally, a concise combinatorial approach for LLT of simple random walk will be sketched. Time permits, I will talk about the generalized Berry-Esseen Inequality.

Thursday, April 25, 2019

2:00 pm in 347 Altgeld Hall,Thursday, April 25, 2019

Coupling and its applications

Peixue Wu (UIUC Math)

Abstract: I will define what is coupling. The beginning example is the transport problem, which leads to the concepts of optimal coupling and probability distance. We will also talk about applications of coupling to study ergodicity, gradient estimate and Harnack's inequality for Markov processes.

Thursday, August 29, 2019

2:00 pm in 347 Altgeld Hall,Thursday, August 29, 2019

Organizational Meeting

Thursday, September 5, 2019

2:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

Introduction to Random Planar Maps

Grigory Terlov (UIUC Math)

Abstract: A "typical" continuous curve on a plane looks like a path of Brownian motion. A natural next question we might ask is "what does a "typical" continuous 2d-surface looks like?" One of the ways to construct such a model is to find a discrete object and consider a scaling limit of it (analogous to considering a scaling limit of a random walk to construct Brownian motion). Such objects are called random planar maps - planar multi-graphs embedded in a sphere or a plane. Of course, similarly to random walks, there are many other reasons why these discrete objects are interesting. In these two talks we will consider several ways of defining random planar maps and a measure on them, connections with random walks and random trees. Finally, in the remaining time I will try to mention several highlights of the field in connection with combinatorics, percolation theory, scaling limits, and Ergodic theory.

Thursday, September 12, 2019

2:00 pm in 347 Altgeld Hall,Thursday, September 12, 2019

Introduction to Random Planar Maps part 2

Grigory Terlov (UIUC Math)

Abstract: The main focus of the second part of this talk is to discuss bisections between random bipartite planar maps and decorated Galton Watson trees. Then if time permits we will continue connecting this model with other areas of probability that audience might be familiar with and/or interested in exploring.

Thursday, September 19, 2019

2:00 pm in 347 Altgeld Hall,Thursday, September 19, 2019

The Yang Mills Problem for Probabilists

Kesav Krishnan (UIUC math)

Abstract: We aim to introduce the problem of rigorously defining the Yang-Mills field from the probability perspective. In this first talk, we will introduce lattice guage theory, and some geometric preliminaries

Tuesday, September 24, 2019

2:00 pm in 347 Altgeld Hall,Tuesday, September 24, 2019

On the potential theory of Markov processes with jump kernels decaying at the boundary

Zoran Vondracek (University of Zagreb)

Abstract: Consider a $\beta$-stable process in the Euclidean space $\mathbb{R}^d$, $0<\beta\le 2$, which is killed upon exiting an open subset $D$. The killed process is then subordinated via an independent $\gamma$-stable subordinator. The resulting process $Y^D$ is called a subordinate killed stable process. In two recent papers, it has been shown that the potential theory of this process exhibits some interesting features. The first one is the form of the jumping kernel which depends on the distance of points to the boundary in a novel way. The second and unexpected feature is the fact that for some values of the stability index $\gamma$, the boundary Harnack principle fails. In the first part of the talk, I will review these results. The second part of the talk will be devoted to ongoing work on potential theory of jump processes in open subset $D$ of $\mathbb{R}^d$ defined through their jumping kernels that depend not only on the distance between two points, but also on the distance of each point to the boundary $\partial D$ of the state space $D$. Joint work with Panki Kim and Renming Song

Tuesday, October 1, 2019

2:00 pm in 347 Altgeld Hall,Tuesday, October 1, 2019

The genus of generalized sparse random graphs

Yifan Jing (UIUC)

Abstract: Determining the genus of graphs is one of the central problems in topological graph theory. In particular, it was proved by Thomassen that determining the genus of any given graph is NP-Complete. Recently, we provided a polynomial-time approximation scheme for the genus of dense graphs. One of the key ingredients there is that we approximate the genus of dense multipartite quasirandom graphs. Motivated by that project, we study the genus of generalized sparse random graphs, that is graphs can be decomposed by finite sum of sparse random graphs. This is joint work with Bojan Mohar.

Thursday, October 3, 2019

2:00 pm in 347 Altgeld Hall,Thursday, October 3, 2019

Introduction to Spin Glasses Part I

Qiang Wu (UIUC Math)

Abstract: I will introduce some spin glass models in the first talk, such as Curie-Weiss(CW) model, Sherrington-Kirkpatrick(SK) model etc, then particularly discuss the high temperature analysis of SK model by Guerra’s interpolation.

Tuesday, October 8, 2019

2:00 pm in 347 Altgeld Hall,Tuesday, October 8, 2019

Introduction to Spin Glasses Part II

Qiang Wu (UIUC Math)

Abstract: This time we will discuss the parisi formula of free energy, I will describe how to derive the formula along with the parisi PDE. If time permits, ultrametricity of asymptotic Gibbs measure will be briefly introduced from probabilistic and geometric view.

Tuesday, October 15, 2019

2:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2019

On the spectral heat content for subordinate killed Brownian motions with respect to a wide class of subordinators

Hyunchul Park (SUNY New Paltz)

Abstract: In this talk, we study the asymptotic behavior of the spectral heat content for subordinate killed Brownian motions (SKBM) with respect to a wide class of subordinators. Previously, the spectral heat content for SKBM via stable subordinators was studied by the author and R. Song. This result gives an upper bound for the heat loss for the spectral heat content for killed Levy processes, whose asymptotic limit is not available for $\mathbb{R}^d$, $d\ge 2$, even for killed $\alpha$-stable processes when $\alpha\in [1; 2)$. This is a joint work with R. Song and is in progress.

Thursday, October 17, 2019

2:00 pm in 347 Altgeld Hall,Thursday, October 17, 2019

Branching Processes Part 1

Peixue Wu (UIUC Math)

Abstract: The first part of this talk is very introductory, I will talk about the basic ideas of branching mechanism (originated from random walk) and some generalizations of the simple branching process, e.g., age-dependent processes, multi-type branching process. ​I will focus on the limit theorem of branching processes.​ ​In the second part, I will talk about the superprocess (which is measure-valued branching processes) and some recent works about it.

Thursday, October 24, 2019

2:00 pm in 347 Altgeld Hall,Thursday, October 24, 2019

Branching Processes Part 2

Peixue Wu (UIUC Math)

Abstract: The first part of this talk is very introductory, I will talk about the basic ideas of branching mechanism (originated from random walk) and some generalizations of the simple branching process, e.g., age-dependent processes, multi-type branching process. ​I will focus on the limit theorem of branching processes.​ ​In the second part, I will talk about the superprocess (which is measure-valued branching processes) and some recent works about it.

Tuesday, November 5, 2019

2:00 pm in 347 Altgeld Hall,Tuesday, November 5, 2019

A Probabilistic Intuitive Boundary Construction

Peter Loeb (Illinois Math)

Abstract: Robert Martin’s 1941 generalization of the boundary of the unit disk is now a fundamental tool in potential theory and probability theory. After an introduction for the non-specialist, I will present an alternative approach to Martin’s construction and integral representation. The approach looks inside a domain using Brownian paths starting at nonstandard points that merge to form boundary points.

Thursday, November 14, 2019

2:00 pm in 347 Altgeld Hall,Thursday, November 14, 2019

Probabilistic solutions for fluid systems and stochastic Noether-Kelvin theorem

Jianyu Hu (UIUC Math)

Abstract: In this talk, we will study a probabilistic representations for the fluid systems based on stochastic Lagrangian paths. Unlike the Feynman-Kac formula, this theorem is a representation for nonlinear equations. When the systems have some noise, we will study the stochastic Noether-Kelvin theorem for a class of stochastic fluid equations. In general, the solutions of these systems do not preserve energy conservation, but rather than circulation conservation.

Tuesday, November 19, 2019

2:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2019

Absolute continuity and singularity for probability measures induced by a drift transform of the independent sum of Brownian motion and symmetric stable process

Ruili Song (Nanjing University of Finance and Economics, and UIUC Math)

Abstract: We consider a Levy process $X$ which is the independent sum of a Brownian motion and a symmetric $\alpha$-stable process in $R^d$. The probability measure $P^b$ is induced by the drift transform of $X$ via the vector valued function $b$. We study mutual absolute continuity and singularity of $P$ and $P^b$ on the path space. We also investigate the problem of finiteness of the relative entropy of these measures on $R^d$ ($d\ge 3$).