Department of

# Mathematics

Seminar Calendar
for Graduate Probability Seminar events the year of Tuesday, October 26, 2021.

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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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Thursday, September 23, 2021

2:00 pm in Altgeld Hall 347,Thursday, September 23, 2021

#### Coupling from the Past 1

###### Aditya Suresh Gopalan (UIUC ISE )

Abstract: Abstract: The Coupling from the Past algorithm of Propp and Wilson is a good technique for sampling from and proving the existence of stationary distributions for Markov chains. In this talk, we discuss the proof of the technique. Briefly, the system works by constructing sequences sample paths whose initial states are determined by the previous element in the sequence, so that the original states are pushed to minus infinity'', thus establishing the coupling and convergence. This technique has applications to a wide variety of fields; one such application to interference queueing will be discussed in the next talk.

Thursday, September 30, 2021

2:00 pm in Altgeld Hall 347,Thursday, September 30, 2021

#### Coupling from the Past 2

###### Aditya S. Gopalan (UIUC ISE)

Abstract: We give a coupling from the past proof for the existence of a stationary distribution for interference queueing networks on grids. This talk highlights the use of this technique for purposes other than sampling, for which it was developed.

Thursday, October 7, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 7, 2021

#### Use of couplings in distributional convergence

###### Grigory Terlov (UIUC MATH)

Abstract: It was mentioned in previous talks that bounding distance between two distributions in appropriate metric is often equivalent to finding a good coupling of random variables. In this talk we will build up on that idea and study the use of couplings in bounding total variation distance between two counting random variables. We will apply these techniques to head runs in a sequence of independent coin tosses and occupancy problem as well as other models if time permits. This talk serves as introduction to the Chen-Stein method for Poisson approximation, on which we will focus in the second talk.

Thursday, October 14, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 14, 2021

#### Use of couplings in distributional convergence (Part2: Stein-Chen method)

###### Grigory Terlov (UIUC Math )

Abstract: At the end of the last lecture I presented two results due to C. Stein and L.H.Y. Chen that utilize couplings to prove convergence to Poisson distribution. These techniques allow for dependence among random variables and give explicit rate of convergence. In this talk I will explain the main idea behind Stein-Chen method and prove both of the theorems stated last time.

Thursday, October 21, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 21, 2021

#### Coupling method for diffusion processes

###### Peixue Wu (UIUC Math)

Abstract: In my first part of my talk, first I will introduce the notion of ergodicity for continuous-time stochastic processes. Then I will briefly discuss the relation between the ergodicity of a one-dimensional diffusion process and its coupling under some assumptions on the diffusion process.

2:00 pm in Altgeld Hall 347,Thursday, October 21, 2021

#### Coupling method for diffusion processes

###### Peixue Wu (UIUC Math)

Abstract: In my first part of my talk, first I will introduce the notion of ergodicity for continuous-time stochastic processes. Then I will briefly discuss the relation between the ergodicity of a one-dimensional diffusion process and its coupling under some assumptions on the diffusion process.

Thursday, October 28, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 28, 2021

#### Coupling for diffusion processes II

###### Peixue Wu (UIUC math )

Abstract: First I will continue talking about the examples of ergodic&non-ergodic one-dimensional diffusion processes, using the fundamental theorem of ergodicity and coupling. Second, I will introduce some functional inequalities using coupling method, which is undergoing active research nowadays.

Thursday, November 4, 2021

2:00 pm in Altgeld Hall 347,Thursday, November 4, 2021

#### introduction to optimal transport

###### Kesav Krishnan (UIUC math )

Abstract: I will introduce the notion of optimal transport and the interpretation of coupling of measure in this regard.

Thursday, November 11, 2021

2:00 pm in Altgeld Hall 347,Thursday, November 11, 2021

#### Intro to optimal transport II

###### Kesav Krishnan (UIUC math )

Abstract: Having introduced the notion of optimal transport, I will discuss some of its uses, notably in proving the isoperimetric inequality.

Thursday, November 18, 2021

2:00 pm in Altgeld Hall 347,Thursday, November 18, 2021

#### Non-concentration of the chromatic number of a random graph

###### Bob Krueger (UIUC math )

Abstract: In a 2019 breakthrough, Heckel proved that the chromatic number of a uniformly random graph on n vertices is not concentrated on $n^{1/4-o(1)}$ values for all sufficiently large n. I will outline an improvement by Heckel and Riordan which improves the 1/4 to an optimal 1/2, focusing on their use of coupling.