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.

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.

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.

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.

Science Talk for Families - Stars and Supernovae

Boaz Katz
John N. Bahcall Fellow, School of Natural Sciences
April 29, 2014
In this Science for Families talk, Boaz Katz, John N. Bahcall Fellow in the School of Natural Sciences explores stars and supernovae. One of the most striking facts that we know about the cosmos is that most of the chemical elements that ­constitute matter, including our bodies, were produced inside stars and ­distributed by their explosions. It is amazing how much we know about these distant objects that look like dots in the sky. It is ­mind-­blowing that the distant stars and their ­explosions turned out to be crucial for our ­existence.

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.

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.

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.

Climate, Conflict, and Historical Method

Nicola Di Cosmo
Luce Foundation Professor, School of Historical Studies
May 2, 2014
How can historians contribute to investigating the ­relationships between climate change, ecology, and human activity? Scientific research is making available ­volumes of data on the possible correlations between ­environmental change and social transformations over long periods of time. Yet, how strong and how precise a ­correlation one might be able to establish between ­phenomena like droughts, floods, and volcanic eruptions and the emergence of conflict, the migration of peoples, or the collapse of civilizations remain open questions.

(dis)Honesty

Dan Ariely
Duke University
May 9, 2014
In this lecture, Dan Ariely, James B. Duke Professor of Psychology and Behavioral Economics at Duke University and former Member (2005-07) in the School of Social Science, will discuss how the principles of behavioral economics can help us understand some of our irrational tendencies, specifically the mechanisms at work behind dishonest behavior. According to Ariely, one of the most interesting lessons is understanding our capacity to think of ourselves as honest even when we act dishonestly.