## Distinguished Lecture Series

Every year, the Distinguished Lecture Series (DLS) brings two to four eminent mathematicians to UCLA for a week or more to give a lecture series on their field, and to meet with faculty and graduate students.

The first lecture of each series is aimed at a general mathematical audience, and offers a rare opportunity to see the state of an area of mathematics from the perspective of one of its leaders. The remaining lectures in the series are usually more advanced, concerning recent developments in the area.

*Previous speakers of the DLS include:* Peter Sarnak, Peter Schneider, Zhenghan Wang, Pierre Colmez, Etienne Ghys, Goro Shimura, Jean Bellissard, Andrei Suslin, Pierre Deligne, Michael Harris, Alexander Lubotzky, Shing-Tau Yau, Hillel Furstenberg, Robert R. Langlands, Clifford Taubes, Louis Nirenberg, Oded Schramm, Louis Nirenberg, I.M. Singer, Jesper Lutzen, L.H. Eliasson, Raoul Bott, Dennis Gaitsgory, Gilles Pisier, Gregg Zuckerman, Freydoon Shahidi, Alain Connes, Jöran Friberg, David Mumford, Sir Michael Atiyah, Jean-Michel Bismut, Jean-Pierre Serre, G. Tian, N. Sibony, C. Deninger, Peter Lax, and Nikolai Reshetikhin, Horng-Tzer Yau, Ken Ono, Leonid Polterovich, Barry Mazur, Grigori Margulis, Mario Bonk, Avi Wigderson, John Coates, Charles Fefferman, C. David Levermore, Shouwu Zhang.

**The DLS is currently supported by the Larry M. Wiener fund.**

## Past Lectures

University of Sydney Visit: 05/30/2017 to 06/01/2017 |
College de France Visit: 05/09/2017 to 05/11/2017 |
Princeton University Visit: 04/04/2017 to 04/06/2017 |

Massachusetts Institute of Technology Visit: 01/24/2017 to 01/26/2017 |
Stanford University Visit: 11/14/2016 to 11/18/2016 |
Columbia University Visit: 05/17/2016 to 05/19/2016 |

Harvard University Visit: 04/25/2016 to 04/28/2016 |
Princeton University Visit: 05/19/2015 to 05/21/2015 |
Columbia University Visit: 02/17/2015 to 02/19/2015 |

Microsoft Research Visit: 11/03/2014 to 11/06/2014 |
Cambridge University Visit: 10/04/2014 to 10/10/2014 |
Duke University / UC Berkeley Visit: 05/19/2014 to 05/23/2014 |

IAS, Princeton Visit: 10/30/2013 to 11/06/2013 |
Texas A&M Visit: 10/22/2013 to 10/26/2013 |
Eötvös Loránd University Visit: 05/28/2013 to 05/30/2013 |

Rheinische Friedrich-Wilhelms-Universität Bonn Visit: 05/07/2013 to 05/09/2013 |
University of Strasbourg Visit: 04/03/2013 to 04/20/2013 |
Massachusetts Institute of Technology Visit: 01/24/2012 to 01/26/2012 |

Hebrew University Visit: 04/26/2011 to 04/28/2011 |

## Upcoming Lectures

### Alice Guionnet

#### MIT

**Visit:**

**Lectures:**

**Random matrix theory and Dyson-Schwinger equations**

**Lecture 1 (5/8) - The uses of Dyson-Schwinger equations**

The most well known results from probability theory, such as central limit theorem and the law of large numbers, deal with independnet random variables. Such theorems do not apply to strongly correlated variables such as the eigenvalues of random matrices. To study the fluctuations of such systems the analysis of Dyson-Schwinger equations have been central in the recent years. These equations describe relations between the moments of the observables. they are often derived thanks to the underlying symmetries of the model. In this talk we will introduce Dyson-Schwinger equations in random matrix theory, and their analogue in random tilings given by Nekrasov's equations. We will describe how to analyze them in order to derive central limit theorems, the universality of local fluctuations or large deviations estimates.

**Lecture 2 (5/9) - Universality in random matrix theory**

In this talk I will focus on universality in Random matrices and joint works with Bekerman and Figalli. To this end, I will introduce the concept of approximate transport maps and show how to construct it thanks to Dyson-Schwinger equations

**Lecture 3 (5/10) - Several matrix questions**

In this last talk I will discuss Dyson-Schwinger equations for several matrix problems and their applications to obtain central limit theorems and large deviations estimates.

#### About Alice Guionnet :

**Poster:** Guionnet.2.pdf

### Amie Wilkinson

#### University of Chicago

**Visit:**

**Lectures:**

**The Ergodic Hypothesis and Beyond**

**Lecture 1 (5/22) - "The general case" **

Abstract: The celebrated Ergodic Theorems of George Birkhoff and von Neumann in the 1930's gave rise to a mathematical formulation of Boltzmann's Ergodic Hypothesis in thermodynamics. This reformulated hypothesis has been described by a variety of authors as the conjecture that ergodicity -- a form of randomness of orbit distributions -- should be``the general case" in conservative dynamics. I will discuss remarkable discoveries in the intervening century that show why such a hypothesis must be false in its most restrictive formulation but still survives in some contexts. In the end, I will begin to tackle the question, "When is ergodicity and other chaotic behavior the general case?"

**Lecture 2 (5/23) - Robust mechanisms for chaos, I: Geometry and the birth of stable ergodicity**

Abstract: The first general, robust mechanism for ergodicity was developed by E. Hopf in the 1930's in the context of Riemannian geometry. Loosely put, Hopf showed that for a negatively curved, compact surface, the ``typical" infinite geodesic fills the manifold in a very uniform way, a property called equidistribution. I will discuss Hopf's basic idea in both topological and measure-theoretic settings and how it has developed into a widely applicable mechanism for chaotic behavior in smooth dynamics

**Lecture 3 (5/24) - Robust mechanisms for chaos, II: Stable ergodicity and partial hyperbolicity**

Abstract: Kolmogorov introduced in the 1950's a robust mechanism for {\em non-ergodicity}, which is now known as the KAM phenomenon (named for Kologorov, Arnol'd and Moser). A current, pressing problem in smooth dynamics is to understand the interplay between KAM and Hopf phenomena in specific classes of dynamical systems. I will describe a class of dynamical systems, called the {\em partially hyperbolic systems}, in which the two phenomena can in some sense be combined. I'll also explain recent results that give strong evidence for the truth of a modified ergodic hypothesis in this setting, known as the Pugh-Shub stable ergodicity conjecture.

#### About Amie Wilkinson :

### Peter Oszvath

#### Princeton University

**Visit:**