I am a mathematician interested in geometric representation theory, a postdoc in Vivek Shende's group at the Centre for Quantum Mathematics, Syddansk Universitet. Before that I was a PhD student at Oxford under Dominic Joyce and an undergrad at Cambridge.
Research
Cohomological Hall algebras and vertex algebras
The moduli stack of coherent sheaves on a curve/representations of a quiver have two structures on their homology: an algebra, and a vertex coalgebra. This paper shows they are compatible.
Orthosymplectisation and twisted Yangians (coming ~Sep 2024)
We define orthosymplectic versions of cohomological Hall vertex bialgebras, and show that they agree with twisted Yangians. To do this, we develop a theory of invariants under a finite group action of cohomological Hall vertex bialgebras, and give examples coming from quivers with potential and perverse-coherent sheaves on surfaces. This gives a mathematical description of Chern Simons theory on Z/2 - orbifolds.
Joint with S. de Hority.
Vertex bialgebra structures on deformed CY3 moduli stacks and Drinfeld coproduct of Yangians (coming ~Sep 2024)
We contruct a vertex bialgebra structure on the critical cohomology of local CY3 categories and deformed versions thereof. We take their Drinfeld doubles. We show that this recovers Joyce-Liu's vertex algebra, and identify it with the Drinfeld coproduct on (affine) Yangians. Joint with S. Jindal and S. Kaubrys.
Quantum groups and factorisation algebras
Factorisation quantum groups
Shows how to get spectral R-matrices (of affine quantum groups, etc.) purely geometrically from braided categories "over the Ran space". The paper builds up the categorical machinery needed to do this, and recovers previous notions of "q-vertex algebra".
Chiral centres
Defining
Joint with W. Niu.
Algebraic geometry
Localised Euler classes for Artin stacks (coming ~Sep 2024)
Develops the theory of Euler classes for singular vector bundles on Artin stacks, valued in arbitrary sheaves (e.g. Borel-Moore, vanishing cycle, Joyce sheaves). Gets new a torus localisation formulas in situations necessary
Joint with D. Aranha, A. Khan, H. Park, and C. Ravi.
The stacky concentration theorem
Joint with D. Aranha, A. Khan, H. Park, and C. Ravi.
Other interests
W vertex algebras, Higgs branches and vertex quantisation of nilpotent orbits
There is a classical theorem that representations of the (finite) W-algebra classify quantisations of nilpotent orbits. We prove a "loop space" analogue of this result, and connect this to Beem et al's vertex algebras attached to Higgs branches. Joint with E. Bouaziz.
I also have an interest in Gaussian free fields and Louisville theory.
Notes
Recent Organisation and Teaching
Ordinary Differential equations. SDU course, Autumn 2024.
Topics in Calabi-Yau threefolds. Upcoming seminar, online/SDU, Autumn 2024 -
Spectral Graph Theory. Class for Swedish high school students at Mattekollo, Summer 2024.
Percolation and Phase Change Phenomena. Class for Swedish high school students at Mattekollo, Summer 2024.
Factorisation algebras. Organised, ran, spoke in online reading group, Spring-Winter 2023.
W algebras. Course for PhD students at SDU, spring-winter 2023.
For a complete list of past teaching, see CV.