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Random matrix theory is a diverse mathematical tool. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory, group and representation theory, differential geometry, combinatorics and mathematical physics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to time series analysis in statistics, quantum chaos and quantum field theory in physics, to the Riemann zeta function zeros and prime numbers in number theory. A self contained development of random matrix theory will be undertaken in this course from various viewpoints.
Topics to be covered include:
- Gaussian random matrix models and their application in likelihood analysis and modelling covariance matrices in time series analysis;
- eigenvalue densities and the concept of eigenvalue repulsion;
- classification of random matrices ensembles;
- derivation of Jacobians for matrix transformations such as diagonalisations;
- joint eigenvalue densities and correlation functions;
- orthogonal polynomials and the concept of determinantal point processes;
- supersymmetry and non-linear sigma-models;
- the log-gas picture;
- free probability theory and its application to matrix sums and products.
Intended learning outcomes
Upon completion of this subject, students should be able to:
- Identify the objectives of random matrix theory from the viewpoint of mathematical physics, and other areas of mathematics such as probability theory and mathematical statistics;
- Compute matrix Jacobians, apply the concepts of joint eigenvalue probability density functions, correlation functions, and spacing distributions, and understand their relevance to random matrix theory;
- Demonstrate comprehension of how the symmetry classification is related to matrix (Lie-)groups;
- Explain the basic ideas of the techniques of orthogonal polynomials, supersymmetry, loop equations, moment method and free convolutions in the analysis of random matrices; and
- Use integral transforms to study global and local statistical quantities.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 10 November 2023