Rational singularities for moment maps of totally negative quivers
10.1007/s00031-024-09873-0, Transformation Groups, 2024, pp. 37
In this article, I study counts of jets on fibres of quiver moment maps (over finite fields). The asymptotic behaviour of these counts is related to the singularities of the moment map fibre. Generalising results of Budur, I prove that moment map fibres of totally negative quivers have rational singularities. As a consequence, counts of jets on these fibres converge and I show that the limit is the p-adic volume of the fibre.
Nilpotent quiver varieties with multiplicities and symmetrisable crystals
joint with Victoria Hoskins and Joshua Jackson
arXiv:2607.17301, 2026, pp. 35
In this article, we generalise Nakajima's construction of nilpotent (framed) quiver varieties to quivers with multiplicities, i.e. over rings of truncated power series. We construct these as quotients by certain non-reductive groups, by introducing new stability conditions for framed quiver representations with multiplicities and by recursively constructing explicit slices for the group action. As an application, we show that irreducible components of nilpotent quiver varieties with multiplicities with fixed framing form the crystal of an irreducible highest-weight representation of the corresponding symmetrisable Kac-Moody algebra under the action of Hecke operators. This generalises work of Kashiwara-Saito (from symmetric to symmetrisable Kac-Moody algebras) and Geiss-Leclerc-Schröer (from the quantised enveloping algebra to its irreducible highest-weight modules).
Moduli spaces of representations of quivers with multiplicities via non-reductive GIT
joint with Victoria Hoskins and Joshua Jackson
arxiv:2510.24559, 2025, pp. 57
In this article, we construct moduli spaces of representations of quivers with multiplicities, i.e. representations of quivers over rings of truncated power series. This requires tools from non-reductive GIT, as one has to perform quotients by non-reductive groups. In doing so, we define new stability conditions for quiver representations with multiplicities. Moreover, we show that several smooth moduli spaces of quiver representations with multiplicities are cohomologically pure. We achieve this by constructing contractions of NRGIT quotients under torus actions.
Positivity for toric Kac polynomials in higher depth
arXiv:2310.02912, 2023, pp. 35
In this article, I study counts of absolutely indecomposable quiver representations over rings of truncated power series - an analogue of Kac polynomials. When the rank vector has entries at most 1, I prove that these polynomials have non-negative coefficients, as conjectured by Hausel, Letellier and Rodriguez-Villegas. I also compute the cohomology of the associated moment map fibre and show that it can be expressed in terms of generalised Kac polynomials, similarly to the preprojective cohomological Hall algebra.
Counting representations of quivers with multiplicities
Infoscience, 2024, pp. 136, Ecole Polytechnique Fédérale de Lausanne (Switzerland)