Introductory lectures on random matrices and free probability

UCD, November 10 - December 8, 2017

Fridays 2pm - 4.30pm in the seminar room AG 1.01

Lecture courses

Fabio Deelan Cunden (UCD) - Introduction to random matrices

The study of random matrices has emerged from applications, first in data analysis and later as statistical models of complex quantum systems. Still motivated by physical problems, a mathematical theory of the spectrum of random matrices began to emerge in the 1960s, and links with other branches of mathematics, including semiclassical analysis, enumerative combinatorics and number theory were established. Over the last years, a unified body of mathematical methods has emerged, and today random matrix theory is a mature field and an active area of research. These lectures intend to provide an accessible introduction to the notions and tools used to analyse the spectral properties of random matrices. A (tentative) list of topics we shall discuss: Basic phenomena in random matrix theory: universality, concentration of measure, level repulsion; Joint distribution of the eigenvalues and Coulomb gas analogy; Determinantal processes and free fermions: correlations functions and scaling limits; Random matrices, enumeration problems and genus expansion.

Antoine Dahlqvist (UCD) - Introduction to free probability

Originally introduced by Voiculescu in the 90' as a tool in operator algebras, free probability became a field on its own at the crossroad of many others, like random matrices, combinatorics or representation theory. This series of lectures intends to give an elementary introduction to this area of research, where we shall insist on its relation with combinatorics and random matrices. A list of topics we shall discuss (that will adapt to the wishes of the audience): Unitary invariance and free probability, the combinatorics of non-crossing partitions, interplay between classical and free probability, representation theory of large symmetric groups.

Timetable

Nov 10 2.00 - 3.00 Fabio Deelan Cunden, Lecture 1
3.30 - 4.30 Antoine Dahlqvist, Lecture 1
Nov 17 2.00 - 3.00 FDC, Lecture 2
3.30 - 4.30 FDC, Lecture 3
Nov 24 2.00 - 3.00 AD, Lecture 2
3.30 - 4.30 FDC, Lecture 4
Dec 1 2.00 - 3.00 AD, Lecture 3
3.30 - 4.30 AD, Lecture 4
Dec 8 2.00 - 3.00 FDC, Lecture 5
3.30 - 4.30 AD, Lecture 5