Students

I plan to investigate foundational questions regarding the role of quantum fields in cosmology. As is well known, the gauge and fermion fields in the standard model of particle physics contribute divergent “zero-point” vacuum energy. Furthermore, in a curved background spacetime the local scale symmetry of the classical theory is broken by quantum stress-energy “trace anomalies.” A possible resolution of these puzzles involves including four-derivative dimension zero scalar fields. These fields may also explain the large scale density density perturbations observed in cosmology. Their quantization is subtle due to potential ghosts and classical instabilities. In the free theory and conformally flat spacetime, these are removed with a physical state constraint, leaving the vacuum as the only physical state. My initial project will be to investigate interacting dimension zero fields and their effect when coupled to gravity, quantum mechanically.

My general interest lies in geometric representation theory and algebraic geometry. In particular, I am studying moduli spaces of sheaves and derived equivalences.

Fun Fact: I enjoy baking in my spare time and am trying to become a decent bartender.

I am a first year PhD student working with Antony Maciocia on computing the walls and moduli spaces for Bridgeland stability in higher dimensions.

Fun Fact: A fun fact about myself is that I have driven an Edinburgh tram!

I am interested in the mathematics describing modern Quantum Field Theory, especially in tackling problems present in String Theory using tools from Algebraic- and specifically Enumerative Geometry.

Fun Fact: Outside of academia, I enjoy playing chess, tennis and making music.

I’m interested in geometric and low-dimensional topology, particularly in dimension 4. Right now, I’m looking at embedded surfaces in 4-manifolds, which is like a higher dimensional version of knot theory.

Fun fact: I’m also very involved in Linguistics Olympiads!

My research interests lie in some interaction between category theory, logic and geometry.

Fun Fact: Outside of maths, I like linguistics, going scuba diving and singing in choirs.

My project will use combinatorial techniques from algebraic topology and geometric group theory to study discrete groups. It will start with using discrete more theory to study the topology of interval posets. These methods will then be adapted to investigate computational and word processing problems for discrete groups such as Artin, Coxeter, and mapping class groups.

My project will initially look at the interplay of lattice models from statistical mechanics (e.g. the six-vertex or ice model) and integrable PDEs, such as the KP hierarchy, using the boson-fermion correspondence. The aim is to find new algebraic and combinatorial descriptions of the solutions of these hierarchies, called tau-functions, using the combinatorics of lattice models in the fermionic Fock space. This will involve techniques from algebra and combinatorics, such as the ring of symmetric functions, vertex operators and infinite rank Lie algebras, but also determinant formulae and their combinatorics needed in the description of soliton solutions. Possible advanced topics include the role of cluster algebras describing the n-soliton solutions and/or the extension to non-commutative integrable systems.

In my project, we study category theoretic and topological approaches to developing more general models in causal inference. We aim to extend the causal inference framework within categorical probability theory to time-varying data, and investigate how tools from applied topology such as sheaves on graphs or quiver representations can be used to model causal phenomena.

TBC

My interests are in geometric group theory. In particular, I am looking forward to learning more about negatively curved spaces and their boundaries, as well as generalisations of hyperbolic spaces.

Fun fact: I used to do rifle shooting competitively (and hoping to get back into it in Edinburgh!)

This project will study homotopical and higher algebraic structures that appear in quantum field theories.

 My research interests are in low dimension topology, quantum invariants, group theory, and representation theory. I am interested in areas where these topics come together in various ways.
At the moment I’m looking into Thompson knot theory, and I am always happy to talk about different topics in math!

The aim of my project is to classify the Boolean inverse monoids, these are inverse semigroups which generalise Boolean algebras. The main inspiration in working on this classification comes from the non-commutative Stone duality between the category of Boolean inverse monoids and the category of étale groupoids. In recent years there has been plenty of results in étale groupoid homology and I wish to use these to develop a homology theory for the Boolean inverse monoids. To this end, I am interested in the representation theory of Boolean inverse monoids and étale groupoids, and their relation to sheaves. This work will also have applications in C*-algebra theory.

Klein’s Erlanger Program was an influential approach to geometry via symmetry. Klein geometries were “static”. Adding time into the fold, results in a kinematical Erlanger program which, roughly speaking, exploits the beautiful interplay between different homogeneous spaces of kinematical Lie groups to describe the dynamics of particles and fields. The precise project, to be determined later in conversation with Ioannis, will be set in this context.

Research Interests: My primary research interests are universal algebra and category theory.

The fundamental objects of algebraic geometry are spaces defined by polynomial equations, called algebraic varieties. The main goal of algebraic geometry is ultimately the classification of algebraic varieties. Recent developments have explained that in order for this question to be tractable, one must restrict to a nicer class of “stable” varieties. For this more well-behaved class of varieties, it is expected that one can classify them into parameter spaces called moduli spaces. An important aspect of the theory is then to understand of the geometry of these moduli spaces.