Skein theory, line defects, and quantum symmetric pairs with David Jordan and Iordanis Romaidis, draft updated September 17th, 2026.
Given a braided module over a ribbon tensor category, we define a vector space valued invariant of closed oriented 3-manifolds with embedded links as part of a 3d TQFT with defects. When the braided module in question is a quantum symmetric pair, the resulting invariant is reminiscent of dichromatic link invariants and we show its finite dimensionality. Conceptual links with shifted geometric quantization, the relative Langlands program, equivariant skein theory, and the representation theory of DAHAs are also discussed.
Quasi-algebraic quantization for the B-twist Langlands TQFT with Emilio Franco, draft updated September 9th, 2026.
It has been understood from the works of Gaiotto and Witten that the category of boundary conditions for the 4d QFT responsible for geometric Langlands duality is populated by Hamiltonian group actions. Under the B-twist, one expects to represent this category in hyperholomorphic sheaves on the Hitchin moduli stack M. We complete the first step of this program by constructing a representation of (a polarized enhancement of) this category, valued in quasi-algebraic sheaves over the twistor space of M.
Relative Langlands duality of the Bump--Ginzburg--Friedberg GSO6 integral with Antonio Cauchi and Armando Gutierrez Terradillos, draft updated September 1st, 2026.
The Bump--Ginzburg--Friedberg integral is an automorphic period on GSO6, distinguishing symplectic similitude parameters in its Langlands dual group GSpin6, and in the distinction case calculates the exterior square L-function. We develop the Langlands dual period formula of this integral, extending the techniques and philosophies of singular relative duality. Some notable new features include : unfolding to the Shalika model, stacky structure on the spectral side resulting in a finite sum of L-functions, and an analysis of the discrepancies in period formulae.
(BAA)-branes from higher Teichmüller theory with Enya Hsiao and Mengxue Yang, draft updated August 13th, 2025.
We provide a new perspective on the Cayley correspondence, a construction of special Teichmüller-like components in the real character variety of a Riemann surface, using holomorphic Lagrangians over the Hitchin moduli space. The conceptual framework of boundary conditions leads us to study the implication of S-duality on these holomorphic Lagrangians and morphisms between them.
Relative Langlands duality of Hitchin systems, in preparation.
An introduction to some of the ideas were presented at the ICMS conference Gauge fields in arithmetic, topology, and physics, in Edinburgh, April 2024. A recording of the presentation can be found here.
Some singular examples of relative Langlands duality with Akshay Venkatesh, Selecta Mathematica New Series (2026).
We develop the notion of automorphic periods and L-functions attached to certain singular varieties in order to restore the Langlands duality underlying Ginzburg's integral of the SL3 adjoint L-function and Garrett's triple product integral. In an appendix we record well behaved smooth examples of automorphic period/L-function duality in the sense of BZSV.
Relative Langlands duality of toric periods, Selecta Mathematica New Series (2026).
In this companion paper to Some singular examples of relative Langlands duality, we study in finer detail the BZSV automorphic period/L-function duality in the case when the group is a torus. Such details include the regularization and interpretation of nonconvergent periods, and extensions of duality statements to disconnected stabilizers / Deligne--Mumford quotients.
Generic extensions and generic polynomials for linear algebraic groups with J.T. Ferrara and Liam Mazurowski, Journal of Algebra (2016).
A generic extension of a field K with a group G is, roughly speaking, a moduli space of Galois extensions of K with Galois group G. In this article we construct such generic extensions for certain groups of Lie type in characteristic p.