EVENT DETAILS AND ABSTRACT

Title: Random matrices, free probability and the enumeration of maps
Speaker: Alice Guionnet
Speaker Info: MIT
Brief Description:
Special Note:
Abstract:

In these three lectures, I will describe relations between three domains of mathematics, namely probability theory, operator algebra and combinatorics based on -Random matrices with size going to infinity, -Free probability (which is a non-commutative probability theory equipped with a notion of freenees analoguous to independence), -the enumeration of maps which are connected graphs sorted by their genus. In the first lecture I will introduce all concepts and detail the classical relations due to the work of Brezin-Parisi-Itzykson and Zuber on one hand, and Voiculescu on the other. The next two lectures will emphasize the uses of such relations, and in particular how simple ideas from probability theory such as integration by parts can be used in combinatorics and operator algebra. In the second lecture I will describe how this leads to the so-called loop-equations, which can be seen as induction relations for the enumeration of maps, and how this can be used to solve the combinatorial problem. In the last lecture I will focus on free probability and show how integration by parts can be used to define transport maps and prove isomorphisms between algebras. These lectures are based on join works with G. Borot, B.Collins, E. Maurel-Segala, and D. Shlyakhtenko.