Course Descriptions

Term 1

Advanced Lie Algebras: Dr Matthew Westaway, University of Bath

This course will explore a variety of topics within the theory of Lie algebras which one would not normally have seen in an introductory course on the subject. The course will focus particularly on the following areas:

• Universal enveloping algebras and the Poincaré-Birkhoff-Witt Theorem
• Representation theory of semisimple Lie algebras over C, including the classification of finite-dimensional simple modules
• Harish-Chandra’s Theorem describing the centre of the universal enveloping algebra
• A brief introduction to category O and the Kazhdan-Lusztig conjecture
• Lie algebras over fields of positive characteristic

Course notes will be available here in due course.

Introduction to Quantum Lattice Systems: Professor Daniel Ueltschi, University of Warwick

Noncommutative differential geometry: Dr Edwin Beggs (Swansea University)

This course is about applying the usual methods of differential geometry (forms, vector fields, connections) to noncommutative algebras. We will also consider Hopf algebras as as symmetries and positive maps on C* algebras. It is taken from the book `Quantum Riemannian Geometry’ (Springer Grundlehren 355) by S. Majid and myself. 

Syllabus: 

• Differential calculi on noncommutative algebras
• Introduction to Hopf algebras and their calculi
• Calculi on graphs and finite groups
• Covariant derivatives on modules and bimodules, curvature
• Monoidal categories
• Noncommutative vector fields, states and divergences
• CP maps and the KSGNS construction, Hilbert C* bimodules
• The flow of states generated by vector fields
• Parallel transport

Transitions over a saddle: Professor Robert MacKay, University of Warwick

Singularity categories: Dr Matt Booth, Imperial College, University of London

The singularity category of a ring is a triangulated category which contains lots of interesting geometric and homological information. We'll begin with some introductory material on triangulated categories and commutative algebra. Then we'll see three perspectives on singularity categories - firstly as a quotient of the bounded derived category, secondly as the stable category of maximal Cohen-Macaulay modules, and thirdly as matrix factorisations. We'll see some nontrivial examples coming from Kleinian singularities, and, if time permits, some aspects of the dg theory, with a focus on Hochschild (co)homology. This course will be a mixture of representation theory and algebraic geometry.

Fourier Analysis in Phase Space: Professor Jens Marklof, University of Bristol

There is a beautiful mathematical world hidden underneath the humble Fourier transform, connecting the representation theory of the symplectic group with important applications in the analysis of PDE, number theory, quantum mechanics and signal processing. This introductory course is aimed at postgraduate students from a broad mathematical background, requiring few prerequisites beyond the standard foundational undergraduate lectures. 

Topics to be covered include: 

1. Fourier integrals and pseudodiGerential operators 

2. Oscillatory integrals and stationary phase 

3. Symplectic geometry and Hamiltonian dynamics 

4. Heisenberg group and Schrödinger representation; Stone-von Neumann theorem 

5. The Shale-Weil representation of the metaplectic group 

6. Bargmann representation and Fock space 

7. Poisson summation, theta functions and Gauss sums 

8. Semiclassical measures and Egorov’s theorem; quantum ergodicity 

Course notes available here

Numerical Solutions to Differential Equations: Dr Waleed Ali, University of Bath

This unit will cover some of the techniques for solving problems involving differential equations numerically and implementing these techniques on MATLAB.  The differential equations will range from initial value to problems to more complicated partial differential equations with many different implemented conditions.  The unit layout will be as follows:

1. Solving Linear Systems of Equations
   1. Direct Methods
   2. Iterative Methods
2. Solving Initial Value Problems
   1. Euler Method
   2. Modified Euler Method
   3. Runge-Kutta Methods
   4. Implicit Methods
   5. MATLAB’s In-Built Mechanisms
3. Solving Boundary Value Problems
   1. Finite Difference Approximations
   2. Mixed Value Problems
   3. Symmetric Boundary Value Problems
4. Solving Partial Differential Equations
   1. Method of Lines
   2. Accuracy & Convergence

Term 2 

Survey of Supersymmetric Quantum Mechanics and Applications: Dr Niklas Garner, University of Oxford

We will discuss aspects of supersymmetric quantum mechanics, focusing on the cohomological descriptions of supersymmetric ground states and their connections to topics in geometry and topology. We will then describe more advanced topics including gauge
theories/equivariance and appearances of various ideas in supersymmetric quantum
mechanics to higher-dimensional supersymmetric quantum field theories.
No previous knowledge of supersymmetry is necessary, but we will assume a familiarity with classical/quantum mechanics and basics aspects of differential geometry and algebraic topology.

Formalising Fermat: Professor Kevin Buzzard, Imperial College University of London

I am teaching a computer a proof of Fermat's Last Theorem. This boils down to writing computer code in a programming language called Lean which corresponds to a modern mathematical proof of the theorem. This course will be about the mathematics which goes into the proof, and also about how one teaches this sort of mathematics to a computer.

Last year I gave a course with the same title. Back then I was very much focussed on "bottom-up" arguments; we talked about basic things such as adeles and division algebras. This year my focus will be more on a "top-down" approach. In the lectures I will explain a high-level overview of the proof of Fermat's Last Theorem which I'm formalizing, showing how it follows from difficult and deep results in the theory of Galois representations. I will then go into the details of how some of these results are proved. When we get to things which have not been formalized, I will be formalizing the material live in Lean. Prerequisites: Galois theory, arithmetic of elliptic curves, p-adic numbers.

Sturm-Liouville and Dirac Operators: Dr Thomas Bothner, University of Bristol

Many important directions of mathematics and physics are in-debted to concepts and methods which evolved during the investigation of the Sturm- Liouville equation y”+ q(x)y = λy, and the allied Sturm-Liouville operator L= — d^{2 /} dx² + q(x).

These have provided a constant source of new ideas and problems in the spectral theory of operators and close-by areas of analysis. In fact, the sources go back to the first studies of D. Bernoulli and L. Euler on the solution of the equation describing the vibrations of a string, and have lasted at least till the 1960s when an unexpected connection between the spectral theory of Sturm-Liouville operators and certain nonlinear partial differential equations was made. In this module we will familiarise ourselves with the basic techniques used in the analysis of Sturm-Liouville and closely associated Dirac equations. For instance we will be investigating transformation operators, the spectral problem on the half and full line as well as nonlinear equations tied in with Sturm-Liouville equations. This will prepare students for reading literature of a wide variety of subjects in which
Sturm-Liouville equations and techniques occur.
Text: Lecture notes will be made available online on my homepage.
Prerequisites: A solid working knowledge in Real, Functional and Complex Analysis. 

I look forward to working with and getting to know you all, albeit via Microsoft TEAMS. It will be an interesting, fun and rewarding term. Please keep in mind, in this course expect to work hard, read the class notes, and ask questions when things don't make sense.  In exchange you may expect me to do all I can to help you to learn.

Random Matrix Theory and its applications: Dr Joseph Baron, University of Bath

Suppose we populate a large matrix with random numbers. What can we say about its eigenvalues and eigenvectors?  In answering this deceptively simple question, Random Matrix Theory (RMT) provides an invaluable tool with which to characterise systems of many interacting components.  Examples of the resounding success of RMT can be found in such disparate research areas as heavy nuclei, theoretical ecology, neural networks, inference problems, finance and spin-glass physics.
In this course, using tools from theoretical physics, we will explore some classic results in RMT, with a particular focus on their most salient applications.  Specifically, we will discuss the Wigner semi-circle law, the Wigner surmise, the Marchenko-Pastur Law, the Girko circular law and the generalisation to the elliptical law.  For each case, we will give concrete examples where the need for the results arises.  As an extension, we will consider additional structure in the random matrix, which is necessitated by many modern applications, modifying the classic results.

An Introduction to Infinite Dimensional Analysis: Professor Eugene Lytvynov, Swansea University

The syllabus for this course is:

• Locally convex topological vector space (l.c.s.) and its dual. Examples: rigged Hilbert spaces, Gelfand triple, space of continuous functions with compact support and signed Radon measures. Configuration space.
• Tensor product of vector spaces, topological tensor product of l.c.s.'s, polynomials on the dual of a l.c.s.  Polynomial sequence on the dual of a l.c.s.
  Cylinder algebra and cylinder sigma-algebra on the dual of a l.c.s. Borel sigma-algebra. Probability measure on a Hilbert space and on the dual of a nuclear space (generalised stochastic processes). Probability measure on the space of Radon measures (random measure), probability measure on the configuration space (point process).
• Gaussian measure on an infinite dimensional space, white noise measure.
• Gaussian analysis: multiple stochastic integral, Hermite polynomial sequence over infinite dimensional space, Segal-Wiener-Ito isomorphism between  the Gaussian L^2-space and the symmetric Fock space. Space of test functionals (Hida space), space of generalised functionals (white noise). Segal-Bargmann transform in infinite dimensions.
• Elements of analysis on configuration spaces: Poisson point process, its Wiener-Ito decomposition, Charlier polynomial sequence over configuration space. Correlation measure of a point process. Determinantal point processes.
• Random measures, gamma random measure, Laguerre polynomials over the space of Radon measures.

Advanced Plasma Simulation: Wayne Arter

Twelve lectures, designed for people familiar with the concepts of physics modelling. The content will cover numerical methods
employed in nuclear fusion plasma codes by means of examples, with particular reference to ‘gotcha’s’ in the legacy software that remains in use. Since the material will be based largely on the lecturer’s own publications, the content will be novel and it is hoped that ultimately an advanced textbook for general sale will result. Notes and course material will in any event eventually be posted on the web.

Suggested general reading
“Think before you compute”, E.J.Hinch 2020, CUP
“Numerical simulation of magnetic fusion plasmas”, W.Arter 1995, Reports on Progress in Physics 58, 1.
“Challenges for modelling fusion plasmas”, W.Arter 2019 [W. Arter. Presentation at Isaac Newton Institute Plasma
Physics Day, UKAEA, 2019. ht t ps: / / www. newt on. ac. uk/ semi nar /
20191028140014452.].