# Graduate student reading seminar

## Contents

## 2016 Spring

Tuesday, 2:25pm, B321 Van Vleck

## 2015 Fall

This semester we will focus on tools and methods.

9/15, 9/22: Elnur

I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.

9/29, 10/6, 10/13 :Dae Han

10/20, 10/27, 11/3: Jessica

I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure.

11/10, 11/17: Hao Kai

11/24, 12/1: Chris

12/8, 12/15: Louis

2016 Spring:

1/26, 2/2: Jinsu

2/9, 2/16: Hans

2/25, 3/3: Fan

## 2015 Spring

2/3, 2/10: Scott

An Introduction to Entropy for Random Variables

In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.

2/17, 2/24: Dae Han

3/3, 3/10: Hans

3/17, 3/24: In Gun

4/7, 4/14: Jinsu

4/21, 4/28: Chris N.

## 2014 Fall

9/23: Dave

I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology

9/30: Benedek

A very quick introduction to Stein's method.

I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:

Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293.

The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method

Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year.

10/7, 10/14: Chris J. An introduction to the (local) martingale problem.

10/21, 10/28: Dae Han

11/4, 11/11: Elnur

11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras

12/2, 12/9: Yun Zhai

## 2014 Spring

1/28: Greg

2/04, 2/11: Scott

Reflected Brownian motion, Occupation time, and applications.

2/18: Phil-- Examples of structure results in probability theory.

2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains

3/11, 3/25: Chris J Some classical results on stationary distributions of Markov processes

4/1, 4/8: Chris N

4/15, 4/22: Yu Sun

4/29. 5/6: Diane

## 2013 Fall

9/24, 10/1: Chris A light introduction to metastability

10/8, Dae Han Majoring multiplicative cascades for directed polymers in random media

10/15, 10/22: no reading seminar

10/29, 11/5: Elnur Limit fluctuations of last passage times

11/12: Yun Helffer-Sj?ostrand representation and Brascamp-Lieb inequality for stochastic interface models

11/19, 11/26: Yu Sun

12/3, 12/10: Jason

## 2013 Spring

2/13: Elnur

Young diagrams, RSK correspondence, corner growth models, distribution of last passage times.

2/20: Elnur

2/27: Chris

A brief introduction to enlargement of filtration and the Dufresne identity Notes

3/6: Chris

3/13: Dae Han

An introduction to random polymers

3/20: Dae Han

Directed polymers in a random environment: path localization and strong disorder

4/3: Diane

Scale and Speed for honest 1 dimensional diffusions

References:

Rogers & Williams - Diffusions, Markov Processes and Martingales

Ito & McKean - Diffusion Processes and their Sample Paths

Breiman - Probability

http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf

4/10: Diane

4/17: Yun

Introduction to stochastic interface models

4/24: Yun

Dynamics and Gaussian equilibrium sytems

5/1: This reading seminar will be shifted because of a probability seminar.

5/8: Greg, Maso

The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two approaches. See [1] for a nice overview.

5/15: Greg, Maso

Rigorous use of the replica trick.