General Prerequisites: There are no formal prerequisites, but familiarity with basic concepts and results from linear algebra and probability will be assumed, at the level of A0 (Linear Algebra) and A8 (Probability).
Course Overview: Random Matrix Theory provides generic tools to analyse random linear systems. It plays a central role in a broad range of disciplines and application areas, including complex networks, data science, finance, machine learning, number theory, population dynamics, and quantum physics. Within Mathematics, it connects with asymptotic analysis, combinatorics, integrable systems, numerical analysis, probability, and stochastic analysis. This course aims to provide an introduction to this highly active, interdisciplinary field of research, covering the foundational concepts, methods, questions, and results.
Learning Outcomes: Students will learn how some of the various different ensembles of random matrices are defined. They will encounter some examples of the applications these have in Data Science, modelling Complex Quantum Systems, Mathematical Finance, Network Models, Numerical Linear Algebra, and Population Dynamics. They will learn how to analyse eigenvalue statistics, and see connections with other areas of mathematics and physics, including combinatorics, number theory, and statistical mechanics.
Course Synopsis: The course is divided into four parts:
- Introduction (1 Lecture)
Matrix ensembles: Wigner and Wishart random matrices, the Gaussian and Circular Ensembles. Overview of applications.
- Empirical Spectral Distribution (6 Lectures)
The method of moments: Derivation of the Wigner Semicircle law and the Marchenko-Pastur law for sample covariance matrix. Non-Hermitian case: the Circular law. Introduction to the Stieltjes transform.
- Eigenvalue Statistics (6 Lectures)
Derivation of the joint distribution of eigenvalues for the Gaussian and Circular Ensembles. Eigenvalues as a point process: method of orthogonal polynomials. The spectrum in the bulk and at the edge.
- Dynamical Approach and Universality (3 Lectures)
Dyson Brownian Motion. Connections to other problems in mathematics: the longest increasing subsequence
problem, distribution of zeros of the Riemann zeta-function, topological genus expansions.
