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 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: Introduction to matrix ensembles, including Wigner and Wishart random matrices, and the Gaussian and Circular Ensembles. Overview of connections with Data Science, Complex Quantum Systems, Mathematical Finance, Network Models, Numerical Linear Algebra, and Population Dynamics (1 Lecture)

Statement and proof of Wigner’s Semicircle Law; statement of Girko’s Circular Law; applications to Population Dynamics (May’s model). (3 lectures)

Statement and proof of the Marchenko-Pastur Law using the Stieltjes and R-transforms; applications to Data Science and Mathematical Finance. (3 lectures)

Derivation of the Joint Eigenvalue Probability Density for the Gaussian and Circular Ensembles;

method of orthogonal polynomials; applications to eigenvalue statistics in the large-matric limit; behaviour in the bulk and at the edge of the spectrum; universality;

applications to Numerical Linear Algebra and Complex Quantum Systems (5 lectures)

Dyson Brownian Motion (2 lectures)

Connections to other problems in mathematics, including the longest increasing subsequence problem; distribution of zeros of the Riemann zeta-function; topological genus expansions. (2 lectures)