The seminar is organized by
Date and location  Speaker  Title  

November 1, 5.30 pm  6.30 pm, Room 528  Shalin Parekh  Introduction to the stochastic heat equation  
The stochastic heat equation with additive noise is one of the simplest yet most fundamental examples of a stochastic PDE. I will first motivate why this object is important, and talk about its invariance under scaling. Then I will give the basic solution theory and regularity properties for this equation, as well as its invariant measure and ergodicity properties. I will also briefly mention more interesting PDEs which arise as nonlinear perturbations of this equation, such as stochastic quantization equations and KPZ.


November 8, 5.30 pm  6.30 pm, Room 528  Mark Rychnovosky  GUE and TracyWidom distribution  
We will introduce the Gaussian Unitary Ensemble and show that the distribution of its eigenvalues can be described by a Fredholm determinant. Using this expression, we will outline a steep descent argument to show that the largest eigenvalue follows the Tracy Widom distribution.


November 15, 5.30 pm  6.30 pm, Room 528  Mark Rychnovosky  GUE and TracyWidom distribution (bis)  
We will use the steepest descent to show that the largest eigenvalue in GUE follows the TracyWidom distribution.


November 22, 5.30 pm  6.30 pm, Room 528  Promit Ghosal  Convergence of holonomy field to master field on the sphere  
I will introduce YangMills holonomy field on a compact Riemannian surface and state the result on the convergence of holonomy field to master field on the sphere. This will be followed by some characterization of the master field. Next, I will talk about how discrete betaensemble is connected to YangMills on sphere and show the proof of one proposition (proposition 2.5 of [DN17]) where this connection is helpful.


November 29, 5.30 pm  6.30 pm, Room 528  Hugo Falconet  Ito's excursion theory and applications  
I will give an introduction to Ito’s excursion theory. After a short detour with computations in the discrete setting, I will introduce local times for continuous diffusions and explain Ito's theorem : labelled by the local time at zero, the excursions away from zero form a Poisson point process. Then I will explain how to use it to perform computations and we will see for instance how to obtain the law of the maximum of the absolute value of the standard brownianbridge between time zero and one using this theory, which is based on [PW]).


December 6, 5 pm  6 pm, Room 622  Hugo Falconet  Bernoulli Percolation  
I will introduce the model of Bernoulli percolation and explain some of its classical properties together with the main ingredients of their proofs (following Percolation et modèle d'Ising by W.Werner). In particular we will see the existence of a phase transition, the uniqueness of the infinite component at supercriticality and exponential decay of the probability that zero is connected to distance n at subcriticality.


February 8, 3 pm  4 pm, Room 528  Yier Lin  KestenStigum theorem  
For Galton Watson process Z_n with mean m>1, it is well known Z_n / m^n \to Y for some nonnegative random variable Y. The Kesten Stigum theorem gives equivalent condition on whether P(Y=0) = 1 or P(Y=0)=q, where q is the probability Z_n finally extincts. We will introduce the size biased GaltonWatson tree and prove this theorem.


February 15, 5 pm  6 pm, Room 528  Maithreya Sitaraman  The Upper Bound Lemma  
The dual of an abelian group is just the thing we expect: the collection of characters of the group, and is therefore a group itself. When we move to the nonabelian case, we would like to extend this notion of duality in a way that allows us to take Fourier transforms.. and the way we do this is to call the set of irreducible representations of our group as it's dual. This, however, fails to be a group. In this talk, I will set up and prove Fourier inversion on the dual of a (possibly nonabelian) finite group, and then prove a key corollary of this setup: The Upper Bound Lemma (introduced by Diaconis and Shahshahani), which allows one to bound how quickly a random walk on a group converges to the uniform distribution. I will then discuss some applications of the Upper Bound Lemma to some concrete problems that are of interest to probabilists. Apart from concrete problems one can solve, the Upper Bound Lemma is interesting on a philosophical level, since it forms a connection between representation theory and probability theory.


February 22, 5 pm  6 pm, Room 528  Yier Lin  Kesten Stigum theorem (bis)  
We introduce Galton Watson process with immigration (GWI), it turns out that for Galton Watson process, consider its behavior under a change of measure, it's almost a GWI. We will use this observation to prove the KestenStigum theorem.


March 8, 5 pm  6 pm, Room 528  Shalin Parekh  CameronMartin Theory  
We will give a short introduction to the abstract theory of Gaussian measures on infinitedimensional vector spaces, with applications to the study of Brownian motion and the Gaussian free field.


March 22, 5 pm  6 pm, Room 528  Shalin Parekh  CameronMartin Theory (bis)  
We will continue to discuss the theory Gaussian measures on Banach spaces, specifically the CameronMartin space and absolute continuity of translation maps.


April 5, 5 pm  6 pm, Room 528  Xuan Wu  On the qTASEP model  
Let's look at the qTASEP model, which is continuous Markov chain with state space to be the location of particles and let's derive the exact moment integral formula with nested contours using Duality and Bethe ansatz. If time admits, we could talk about how the moment formula turn into a Fredholm determinant integral formula by deforming to contours.


April 12, 5 pm  6 pm, Room 528  Xuan Wu  On the qTASEP model (bis)  
Xuan will continue her previous lecture.
