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.
On the job market for 2025!
Research
Take the moduli stack of objects in a one-dimensional category, e.g. representations of a quiver or coherent sheaves on a curve. It has the structure of a vertex quantum group! The structure maps being the CoHA and Joyce's vertex coproduct.
Orthosymplectic modules of cohomological Hall algebras
Define CoHAs for orthosymplectic gauge groups, with their vertex/CoHAM structure which forms a vertex Yetter-Drinfeld module. Examples: quivers with potential, of preprojective algberas, and of dimension zero sheaves on a smooth proper surface. Give shuffle formulas. Develop a theory of factorisation algebras over ``orthosymplectic spacetimes'' which these are examples of. Joint with S. de Hority.
Vertex quantum groups for deformed CY3 completions and the Drinfeld coproduct on Yangians (coming in a week or so)
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
What is the structure on A-Mod when A is a Yangian, affine quantum group, ... ? This paper develops the axiomatics of "braided factorisation categories", "factorisation quantum groups", etc., which are analogues of the standard notions but "factorisably over the Ran space". We prove that these structures are controlled by spectral R(z)-matrices, and generalise previous work on q-vertex algebras, deformed chiral algebras and meromorphic tensor categories. This gives a mathematicisation of Costello-Witten-Yamikaze's work on 4d Chern-Simons theory.
Factorisation bosonisation (coming winter 2024/5)
We show how the geometric representation theorists' "adding in the affine Cartan", e.g. to CoHAs, is in fact an example of Majid's bosonisation-transmutation for quantum groups, applied to factorisation algebras.
Joint with S. de Hority.
Chiral centres
Give an operadic definition of a vertex algebra "Drinfeld centre", so recovering in a basis-free way the doubling constructions of Yangians, CoHAs, shuffle algebras.
Joint with W. Niu.
Algebraic geometry
Virtual Euler classes for Artin stacks (coming winter 2024/5)
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
and
The stacky concentration theorem
We generalise Atiyah-Bott's localisation formula so that it is powerful enough to deal e.g. with moduli stacks: instead of manifolds, we consider ~arbitrary Artin stacks, and instead of localising along torus-equivariant classes, we localise along chern classes of line bundles. In this setting, we show when the localised cohomology is concentrated on a closed substack, and give a generalisation of Graber-Panharipande's formula for the virtual fundamental class in terms of the class of the closed substack. Joint with D. Aranha, A. Khan, H. Park, and C. Ravi.
Other interests
Liouville theory and vertex algebras
Joint with V. Giri.
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 (available on request).