Summer Research Scholars — David Ridout

I'm always keen to supervise vacation scholars, whether they be interested in mathematical physics, pure mathematics or applied mathematics. Here, you can find some ideas for projects, though in practice it's always better to discuss in person so as to make sure the project is suitable. And I've probably got other ideas that I haven't remembered to add to the list. You may notice that most of the list consists of further study, extending (or filling gaps in) what you've seen in your undergraduate subjects. In my opinion, this is as it should be — you cannot have too broad a mathematical education.

• Conformal field theory. One of the centrepoints of modern mathematical physics, conformal field theory underlies our understanding of both critical statistical models and string theory. It also connects to an incredible amount of beautiful mathematics including representation theory, number theory, combinatorics, category theory, differential and algebraic geometry, etc. In this project, we study the simplest example: that describing a spinless massless noninteracting bosonic string. Some complex analysis (residue calculus) and exposure to basic lagrangian mechanics would be useful, but isn't essential.
• Lie algebras. Continuous symmetries are generally modelled by Lie groups. However, Lie groups are profitably analysed by studying their Lie algebras. Here, we look at some simple examples of Lie algebras and see what their representation theories have to do with quantum physics. If time permits, this could be extended to studying examples of Lie superalgebras, affine Kac–Moody algebras or quantum groups, potentially with some physical applications. A strong appreciation of linear algebra is essential here, though some abstract algebra wouldn't hurt too.
• Perturbation theory. In most applications, you can't compute things exactly and so you have to come up with ways to approximate them. Perturbation theory covers cases in which your application is sufficiently close to one where the computations can be done exactly. It studies what happens to the answers when you change the exactly solvable case by a small amount. In this project, we look at perturbation theory for eigenvalues and eigenvectors of matrices. Not only does this have many applications (eg quantum physics), it's also a great excuse to study the extremely beautiful confluence of eigentheory and complex analysis. You'll need to know some complex analysis and be very comfortable with eigenthings.
• Chaos. I'm sure you've seen pretty pictures advertising the wonders of chaotic dynamical systems, but these pictures hide a lot of cool and essential mathematics. I'm not an expert in this area, but everyone should know a little dynamical systems theory, if only to get a better understanding of what all these pretty pictures actually mean. Here, we'll learn some basic dynamical systems theory and see how chaos naturally arises in very simple systems. If time permits, we'll also look at how periodic points in chaotic systems can inform us of the dynamics in general.
• Eigenfunction expansions. If you took Differential Equations, then you'll already appreciate the interpretation of Fourier series as the expansion of a (suitably nice) function with respect to an orthogonal basis of eigenfunctions of the 1D laplacian operator. If you've taken Methods of Mathematical Physics, you'll know that this isn't an isolated nugget of joy — there are other orthogonal bases that are intimately tied up with natural physical problems including Legendre polynomials and Bessel functions. This project looks at a general formalism underlying all these examples: Sturm–Liouville theory. We'll also look at some other examples of orthogonal bases that arise in physics.
• Representations and characters. When you first meet finite groups, you usually get hit with lots of beautiful results about their structures and relations (homomorphisms) between them. However, you usually don't see what they're good for or where they come from. The idea that groups model symmetries is well known, but the key fact is that when you meet groups in the wild (ie in science), they generally come disguised as matrices (or linear transformations). This is called a representation of the group and representation theory is one of the most important (and powerful!) tools in the math physics arsenal. Here, we look at representations of finite groups and some beautiful (and imaginatively named) functions called characters that, funnily enough, help to characterise the representation.

Past projects

• David Chen, finite group representations, pure math, 2021.
• Bridget Gatt, chaotic dynamical systems, applied math, 2020.
• Eric Ma, Lie groups in physics, mathphys, 2020.
• Steven Xu, Temperley–Lieb algebras and their representations, pure math, 2020.
• Steven Xu, coadjoint orbits of Lie groups, pure math, 2019.
• Lukas Anagnostou, conformal field theory, mathphys, 2018.
• Daniel Tan, semisimple Lie algebras, pure math, 2018.
• Madeleine Johnson, Lie algebra cohomology, pure math, 2017.