PhD and MSc Projects
For general information on postgraduate study, see postgraduate studies on the departmental website
Current PhD Students
| Shannon Ezzat | Representation growth of finitely generated nilpotent groups |
| Adam Gillard | Dark matter and Elko fields |
| Daniel Lond | A geometric approach to complete reducibility |
| Ewan Orr | Turing's A-types and machine learning |
Possible PhD and MSc/MA Projects
My main research is in group theory, especially the geometry associated to spaces of representations of finitely generated groups. I am also interested in quantum mechanics and cryptography.
PhD projects
Reductive algebraic groups
Reductive algebraic groups are certain groups of matrices with entries from a field k. In the classical case when k has characteristic zero - for example, when k is the field of complex numbers - much is known; I am interested in the case when k has positive characteristic (for example, when k is the algebraic closure of a finite field). The project is to study so-called completely reducible subgroups of a reductive algebraic group G; this is part of a broad programme to understand all the closed subgroups of G. It involves group theory and algebraic geometry as well as some recent ideas from geometric invariant theory.
Representation growth zeta functions of nilpotent groups
This project is on some counting problems involving finitely generated groups. One takes a sequence of numbers associated to some finitely generated group and uses them as coefficients in a formal power series or zeta function; the usual Riemann zeta function can be interpreted in this way. The aim is to calculate some explicit examples of representation growth zeta functions for finitely generated nilpotent groups — here one counts the number of irreducible complex characters of the group in each dimension. This involves ideas from group theory, number theory, algebraic geometry and logic.
Lattices in automorphism groups of trees
A tree is a graph with no loops. Infinite trees and their automorphism groups are useful in understanding finite graphs: a finite graph arises as the quotient of an infinite tree X by a lattice subgroup of Aut(X), the group of automorphisms of X. This project involves investigating the structure and properties of lattices in Aut(X). There are applications to lattice subgroups of Lie groups over non-Archimedean local fields (the field of p-adic numbers Qp, for example).
Quantum field theory
Symmetries of a physical system give rise to representations of the corresponding symmetry groups: in quantum field theory, the relevant groups are the Poincaré group and the Lorentz group of transformations of space-time. The project is to investigate the interplay between representation theory and the structure of quantum field theories, especially some theories recently put forward as candidates for dark matter. This would probably involve joint supervision with colleagues in the Department of Physics and Astronomy.
Cryptography
I am interested in applications of algebra and number theory to cryptography. Possible projects include an investigation of cryptosystems based on group theory. This may involve joint supervision with colleagues in the Department of Computer Science and Software Engineering.
MSc/MA projects
All of the above are available as Masters projects as well. I will also consider supervising a Masters project on other topics in group theory, algebra, geometry or number theory: please contact me if you want to discuss ideas.
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