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.
- 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.