Research

My research interests span a range of topics at the interface between algebraic geometry (the study of spaces defined by solutions of polynomial equations) and representation theory (the study of symmetry through linear algebra).

A list of current and past research projects and associated works is given below.

Mixed Hodge modules and real groups

Real reductive groups are the most fundamental examples of Lie groups, the mathematical objects controlling continuous symmetry. Their representation theory, which studies concrete realisations as symmetries of a vector space, is subtle, but many features become more transparent after localising over a space called the flag variety and transforming to geometry. I am currently working on a program, joint with Kari Vilonen, that aims to take advantage of deep structures in geometry, called mixed Hodge modules, to gain subtle information about various aspects of this representation theory.

Works

Unitary representations of real groups and localization theory for Hodge modules, with Kari Vilonen, arXiv:2309.13215

Hodge filtrations on tempered Hodge modules, with Kari Vilonen, arXiv:2206.09091

Mixed Hodge modules and real groups, with Kari Vilonen, arXiv:2202.08797

Elliptic Springer theory

Classical Springer theory is concerned with various aspects of the Grothendieck-Springer resolution, a geometric space providing a bridge between Lie theory and the theory of singular spaces. Elliptic Springer theory is a more intricate version, concerning a space called the elliptic Grothendieck-Springer resolution built from moduli of principal bundles on an elliptic curve. The elliptic theory exhibits two new phenomena: global structure and an unstable locus.  In my thesis work, I established basic facts about this space, and showed how these new phenomena lead to new examples of singular spaces linked to Lie theory.

Works

On subregular slices of the elliptic Grothendieck-Springer resolution, Pure and Applied Mathematics Quarterly, 17 (5), 2021, pp 1913-2004. DOI

The elliptic Grothendieck-Springer resolution as a simultaneous log resolution of algebraic stacks, arXiv:1908.04140

Elliptic Springer theory and singularities, PhD thesis, King's College London, 2019.  PDF

Elliptic quantum groups

Elliptic quantum groups arise as quantisations of the elliptic symmetries connected with principal bundles on elliptic curves. They have shown themselves to be somewhat mysterious objects, with a number of different definitions and ad hoc constructions. In work in progress with Yaping Yang and Gufang Zhao, I aim to give a systematic and accessible construction of these objects by viewing them locally as sheaves on a space, building on ideas from Yang and Zhao's preprint arXiv:1708.01418.