Department of

# Mathematics

Seminar Calendar
for Probability Seminar events the year of Tuesday, February 19, 2019.

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events for the
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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2019          February 2019            March 2019
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31


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.