Research

I am interested in the following three areas:

• Quantum groups: their algebraic structures and representation theory.

• Derived symplectic and Poisson structures: these structures naturally appear in classical field theories (see arXiv:1311.6429), categorified geometric and deformation quantization and so on. See these notes for a non-technical introduction aimed at representation theorists and arXiv:1709.07698 for a more technical introduction. Also see arXiv:1706.02623 for a connection between derived Poisson structures and quantum groups.

• Supersymmetric field theories: many conjectures and phenomena predicted by physicists have a rigorous formulation after supersymmetric twisting (see arXiv:1805.10806 for what this means).

I am usually interested in applying ideas from the above areas to neighboring topics: skein theory (arXiv:1901.11450 and arXiv:1908.05233), operads (arXiv:1611.09668), higher categories (arXiv:1804.00879 and arXiv:1904.11312) and integrable systems (arXiv:1302.3540).

Quantum groups

• A categorical approach to dynamical quantum groups (with Artem Kalmykov), arXiv:2008.09081, Forum of Mathematics, Sigma 

• Invertible braided tensor categories (with Adrien Brochier, David Jordan and Noah Snyder), arXiv:2003.13812, Algebraic & Geometric Topology 

• The finiteness conjecture for skein modules (with Sam Gunningham and David Jordan), arXiv:1908.05233, to appear in Inventiones Math

• The quantum Frobenius for character varieties and multiplicative quiver varieties (with Iordan Ganev and David Jordan), arXiv:1901.11450, to appear in Journal of the EMS

• A categorical approach to quantum moment maps, arXiv:1901.09031, Theory and Applications of Categories 

• Poisson-Lie structures as shifted Poisson structures, arXiv:1706.02623, Advances in Math

Supersymmetric twists

• Batalin-Vilkovisky quantization from supersymmetric twists (with Brian Williams), arXiv:2107.07218

• A taxonomy of twists of supersymmetric Yang--Mills theory (with Chris Elliott and Brian Williams), arXiv:2002.10517, Selecta Math

• Topological twists of supersymmetric algebras of observables (with Chris Elliott), arXiv:1805.10806, Communications in Math Physics

Derived Poisson structures

• Shifted coisotropic correspondences (with Rune Haugseng and Valerio Melani), arXiv:1904.11312, Journal of the IMJ 

• Lectures on shifted Poisson geometry, arXiv:1709.07698

• Derived coisotropic structures II: stacks and quantization (with Valerio Melani), arXiv:1704.03201, Selecta Math

• Braces and Poisson additivity, arXiv:1611.09668, Compositio Math

• Derived coisotropic structures I: affine case (with Valerio Melani), arXiv:1608.01482, Selecta Math

• Poisson reduction as a coisotropic intersection, arXiv:1509.08081, Higher Structures

Derived symplectic structures

• Classical BV formalism for group actions (with Marco Benini and Alexander Schenkel), arXiv:2104.14886, Communications in Contemporary Math

• Shifted geometric quantization, arXiv:2011.05730

• Shifted symplectic Lie algebroids (with Brent Pym), arXiv:1612.09446, IMRN

• A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes (with Dominic Joyce), arXiv:1506.04024, Ann. Fac. sci. Toulouse

• Symplectic implosion and the Grothendieck--Springer resolution, arXiv:1411.2962, Transformation Groups

• Quasi-Hamiltonian reduction via classical Chern--Simons theory, arXiv:1311.6429, Advances Math

Other projects

• Torsion volume forms (with Florian Naef), arXiv:2308.08369

• The categorified Grothendieck--Riemann--Roch theorem (with Marc Hoyois, Sarah Scherotzke and Nicolò Sibilla), arXiv:1804.00879, Compositio Math

• Gerbal central extensions of reductive groups by K3, arXiv:1407.4113, Journal of Algebra

• Virasoro constraints in Drinfeld--Sokolov hierarchies, arXiv:1302.3540, Selecta Math