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).
On the algebraic geometry side, my interests include Hodge theory, singularity theory and moduli theory. On the representation theory side, I think mostly about Lie theory, especially the representation theory of real reductive Lie groups.
A list of current and past research projects and associated works is given below.
Real reductive groups are the most fundamental examples of Lie groups, the mathematical objects controlling continuous symmetry. One of the biggest unsolved problems in their study is the classification of irreducible unitary representations (concrete realisations as unitary operators on a Hilbert space), which arise naturally in fields as diverse as harmonic analysis, number theory and mathematical physics. In recent work with Kari Vilonen, I proved a conjecture that unitary representations can be detected using a refined structure, the Hodge filtration, defined geometrically using Saito's theory of mixed Hodge modules. Part of my ongoing work is to determine what progress can be made on the classification using these new tools.
Hodge theory, intertwining functors, and the Orbit Method for real reductive groups, with Lucas Mason-Brown, arXiv:2503.14794
The FPP conjecture for real reductive groups, with Lucas Mason-Brown, arXiv:2411.01372
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, Advances in Mathematics, 470, 2025, article no 110255. DOI
Singularities are points where a geometric space becomes pinched or folded. An important part of singularity theory is to measure how bad a given singularity is, and to what extent it may violate theorems about smooth spaces. Hodge theory provides a powerful set of tools for doing just this, using invariants called Hodge ideals and higher (aka microlocal) multiplier ideals. I have recently become interested in what abstract theorems about mixed Hodge modules can say about this theory.
On the Hodge and V-filtrations of mixed Hodge modules, with Ruijie Yang, arXiv:2503.16619
Archimedean zeta functions, singularities and Hodge theory, with András Lőrincz and Ruijie Yang, arxiv:2412.07849
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.
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