School of Mathematics

Measures on spaces of Riemannian metrics

Dmitry Jakobson
McGill University
July 21, 2014
This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We construct Gaussian measure on the manifold of Riemannian metrics with the fixed volume form. We show that diameter and Laplace eigenvalue and volume entropy functionals are all integrable with respect to our measures. We also compute the characteristic function for the \(L^2\) (Ebin) distance from a random metric to the reference metric.

A central limit theorem for Gaussian polynomials and deterministic approximate counting for polynomial threshold functions

Anindya De
Institute for Advanced Study; Member, School of Mathematics
May 13, 2014
In this talk, we will continue, the proof of the Central Limit theorem from my last talk. We will show that that the law of "eigenregular" Gaussian polynomials is close to a Gaussian. The proof will be based on Stein's method and will be dependent on using techniques from Malliavin calculus. We will also describe a new decomposition lemma for polynomials which says that any polynomial can be written as a function of small number of eigenregular polynomials. The techniques in the lemma are likely to be of independent interest. Based on joint work with Rocco Servedio.

Recovering elliptic curves from their \(p\)-torsion

Benjamin Bakker
New York University
May 2, 2014
Given an elliptic curve \(E\) over a field \(k\), its \(p\)-torsion \(E[p]\) gives a 2-dimensional representation of the Galois group \(G_k\) over \(\mathbb F_p\). The Frey-Mazur conjecture asserts that for \(k= \mathbb Q\) and \(p > 13\), \(E\) is in fact determined up to isogeny by the representation \(E[p]\). In joint work with J.

Geometry of metrics and measure concentration in abstract ergodic theory

Tim Austin
New York University
April 30, 2014
Many of the major results of modern ergodic theory can be understood in terms of a sequence of finite metric measure spaces constructed from the marginal distributions of a shift-invariant process. Most simply, the growth rate of their covering numbers gives the entropy of the process, and then one finds that more refined geometric invariants determine other properties of the process.

Landau damping: Gevrey regularity and paraproducts

Clément Mouhot
University of Cambridge
April 30, 2014
We present the key ideas of a new proof of Landau damping for the Vlasov-Poisson equation obtained in a joint work with Bedrossian and Masmoudi. This nonlinear transport equation is a fundamental model for describing self-interacting plasmas or galaxies, and Landau damping is a nonlinear stability mechanism based on phase mixing. The new method does not use a Newton scheme and seems to capture a critical Gevrey regularity for the damping. Moreover its greater flexibility should open the way to further work, and we will finish by sketching some open questions.

A central limit theorem for Gaussian polynomials and deterministic approximate counting for polynomial threshold functions

Anindya De
Institute for Advanced Study; Member, School of Mathematics
April 29, 2014
In the last few years, there has been a lot of activity in the area of structural analysis and derandomization of polynomial threshold functions. Tools from analysis and probability have played a significant role in many of these works.

The focus of the talk will be towards achieving the following goal:
Given a degree-d polynomial threshold function, deterministically approximating the fraction of satisfying assignments up to o(1) error in polynomial time. Along the way, we'll first survey some important existing results in this area.

Search games and Optimal Kakeya Sets

Yuval Peres
Microsoft Research
April 28, 2014
A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1919); we find a new connection to game theory and probability. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time.

Nonlinear Brownian motion and nonlinear Feynman-Kac formula of path-functions

Shige Peng
Shandon University
April 23, 2014
We consider a typical situation in which probability model itself has non-negligible cumulated uncertainty. A new concept of nonlinear expectation and the corresponding non-linear distributions has been systematically investigated: cumulated nonlinear i.i.d random variables of order \(1/n\) tend to a maximal distribution according a new law of large number, whereas, with a new central limit theorem, the accumulation of order \(1/\sqrt{n}\) tends to a nonlinear normal distribution.

Free entropy

Philippe Biane
Université Paris-Est Marne-la-Vallée
April 22, 2014
Free entropy is a quantity introduced 20 years ago by D. Voiculescu in order to investigate noncommutative probability spaces (e.g. von Neumann algebras). It is based on approximation by finite size matrices. I will describe the definition and main properties of this quantity as well as applications to von Neumann algebras. I will also explain a new approach based on work with Y. Dabrowski, using random matrices, which leads to the solution of some problems concerning this quantity.