Graduate student reading seminar
This semester we will focus on tools and methods.
9/15, 9/22: Elnur
9/29, 10/6 Dae Han
10/13, 10/20: Jessica
10/27, 11/3: Hao Kai
11/10, 11/17: Chris
11/24, 12/1: Louis
12/8, 12/15: Jinsu
1/26, 2/2: Hans
2/9, 2/16: Fan
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.
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
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
2/04, 2/11: Scott
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
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
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times.
A brief introduction to enlargement of filtration and the Dufresne identity Notes
3/13: Dae Han
An introduction to random polymers
3/20: Dae Han
Directed polymers in a random environment: path localization and strong disorder
Scale and Speed for honest 1 dimensional diffusions
Rogers & Williams - Diffusions, Markov Processes and Martingales
Ito & McKean - Diffusion Processes and their Sample Paths
Breiman - Probability
Introduction to stochastic interface models
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  for a nice overview.
5/15: Greg, Maso
Rigorous use of the replica trick.