#
Integral Operators in Random Matrix Theory

## Harold Widom

**Abstract: **

Lecture 1. (Monday, July 4) Hilbert-Schmidt and trace class operators.
Definitions and fundamental properties of the trace and determinant. Fredholm
determinants of integral operators.

Lecture 2. (Wednesday, July 6) How integral operators arise in random matrix
theory. Correlation functions and kernels of integral operators. Spacing
distributions as operator determinants. Scaling of the Gaussian ensembles. The
sine and Airy kernels.

Lecture 3. (Friday, July 8) Use of operator determinants to derive differential
equations for distribution functions arising in random matrix theory. The
representations of basic distributions in terms of Painlevé functions.