# Columbia Probability Seminar

The seminar covers a wide range of topics in pure and applied probability. The seminar is organized jointly by the Mathematics and Statistics departments and is run by Guillaume Barraquand, Ivan Corwin, Julien Dubedat, Ioannis Karatzas, Jeffrey Kuan, Marcel Nutz, Philip Protter, and Hao Shen .

E-mail the organizers at probability_seminar -- at -- math.columbia.edu.

The seminar usually takes place in the Mathematics Department (Math 520), on Fridays at 12 noon - 1 p.m.Directions to the Mathematics Department.
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Here is a list of upcoming probability conferences and meetings.

## Friday, February 5, 2016

• Columbia—Courant Probability Day
• Time: 9:00am, Feb. 5th, Math 417 Columbia University

• 9:00-9:30, Coffee, tea and light breakfast

• 9:30-10:30, Ajay Chandra, An analytic BPHZ theorem for Regularity Structures

Abstract: When trying to tame divergences using counterterms within regularity structures there are two key things one has to verify: (i) the insertion of the counter-term corresponds to a renormalization of the equation and is allowed by the algebraic structure of regularity structures, (ii) there is a way to choose the value of counterterms which yield the right stochastic estimates. This verification is difficult when the divergences become numerous and are nested/overlapping.

Recent work by Bruned, Hairer, and Zambotti provides a robust framework to systematically handle the first issue, I will describe how this can be combined with multiscale techniques from constructive field theory in order to handle the second.This is joint work with Martin Hairer.

• 10:45-11:45, Hendrik Weber, Global well-posedness for the dynamic Phi^4_3 model the torus

Abstract: The theory of non-linear stochastic PDEs has recently witnessed an enormous breakthrough when Hairer and Gubinelli devised methods to give an interpretation and show local well-posedness for a class of very singular SPDEs from Mathematical Physics.

In this talk I will discuss how to extend their method to get global bounds in a prominent example, the dynamic Phi^4 model. I will first show how to use a simple PDE argument to show global in time well-posedness for the dynamic Phi^4 equation on the two-dimensional plane. The emphasis of the talk will be on an extension of this method which yields global in time solutions for the three dimensional Phi^4 model on the torus. This is joint work with Jean-Christophe Mourrat.

• 12:00-1:00, Weijun Xu, Large scale behaviour of phase coexistence models

Abstract: The solutions to many interesting stochastic PDEs are often obtained after suitable renormalisations. These renormalisations often change the original equation by a quantity which is infinity, but they do have concrete physical meanings. We will explain the meaning of the infinities in the context of the Phi^4_3 equation. As a consequence, we will see how this equation, interpreted after suitable renormalisations, arises naturally as the universal limit for symmetric phase coexistence models. We will also see how this universality can be lost when asymmetry is present. Based on joint works with Martin Hairer and Hao Shen.

## March 21—23, 2016

• Minerva Lectures by Scott Sheffield

• ## Columbia / Courant Joint Probability Seminar Series:

• Time: 9:30 am, Friday October 9th, Warren Weaver Hall, Room 512 (at Courant)

• 9:30-10:30 Hoi Nguyen, Anti concentration of random walks and eigenvalue repulsion of random matrices.

Abstract: I will survey recent characterization results on random walks (in both abelian and non-abelian groups) which sticks to a small region unusually long. As an application, we demonstrate a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for Wigner matrices of discrete type.

• 10:30-11 Coffee break

• 11-12 Nina Snaith, Combining random matrix theory and number theory.

Abstract: Many years have passed since the initial suggestion by Montgomery (1973) that in an appropriate asymptotic limit the zeros of the Riemann zeta function behave statistically like eigenvalues of random matrices, and the subsequent proposal of Katz and Sarnak (1999) that the same is true of families of more general L-functions. While this limiting behaviour is very informative, even more interesting are the intricacies involved in the approach to this limit, the understanding of which allows us to use random matrix theory in novel ways to shed light on major open questions in number theory.

• 12-1 Ramon van Handel, The norm of structured random matrices.

Abstract: Understanding the spectral norm of random matrices is a problem of basic interest in several areas of pure and applied mathematics. While the spectral norm of classical random matrix models is well understood, existing methods almost always fail to be sharp in the presence of nontrivial structure. In this talk, I will discuss new bounds on the norm of random matrices with independent entries that are sharp under mild conditions. These bounds shed significant light on the nature of the problem, and make it possible to easily address otherwise nontrivial phenomena such as the phase transition of the spectral edge of random band matrices. I will also discuss some conjectures whose resolution would complete our understanding of the underlying probabilistic mechanisms.