Geometric representation theory, geometric Langlands, mathematical physics, conformal field theory, quantum field theory, character sheaves, higher categories
Publications and preprints
On the geometric analogue of a conjecture of Gross and Reeder (coming soon)
Hitchin's map with parabolic level structure (coming soon)
Preservation of depths in local geometric Langlands program (joint with Tsao-Hsien Chen, 2014, to appear in Transactions of AMS)
Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for Jet algebras (Transactions of AMS 2013)
Ramified Satake Isomorphisms for strongly parabolic characters (Joint with Travis Schedler, 2013, Documenta Mathematica)
Geometrization of principal series representation of reductive groups (Joint with Travis Schedler, 2013, Annals of Institute Fourier)
Geometrization of continuous characters of Z_p^* (Joint with Clifton Cunningham, Pacific Journal of Math, 2013)
Universal property of algebraic K-theory and homology (Joint with Justin Noel, 2012)
Compatible intertwiners for representations of nilpotent groups (Joint with Teru Thomas, Representation Theory, 2011)
Maximal representation dimension for finite p-groups (Joint with Zinovy Reichstein and Shane Cernele, Journal of Group Theory, 2011)
Stacky abelianization of algebraic groups (PhD Thesis, Transformation Groups, 2009)
Weil representation over finite field (Master's Thesis 2005)
Notes from some of my talks
On the line and work of Maryam Mirzakhani (University of Queensland, March 2015)
Parahoric subgroups and local geometric Langlands program (to appear in the Proceedings of the Conference "Frontiers in Mathematics" Tehran, 2014).
Symmetries in Mathematics: a personal perspective (Colloquium at the University of Sydney, 2014)
Categorical representations of the loop group (Frontiers in mathematics, Tehran, 2013)
Generalized Verma and Wakimoto modules (Queensland, Australia, 2013)
On the center of the universal enveloping algebras (Australian Math Society, Annual Meeting, Sydney 2013)
Analogies between representations of p-adic groups and affine Kac-Moody algebras (Oberwolfach, 2012)
Introduction to local geometric Langlands (Max Planck Institute in Bonn, 2011)
Seminars I have organized
Conformal Field Theory (UQ, Semester 1, 2014), joint with Jorgen Rasmussen.
Quantum Field Theory (UQ, Semester 1 and Semester 2, 2013)
Nearby cycles (UBC, Semester 1, 2011)
Geometrization and categorification (UBC, Semester 1, 2010)
Junior algebraic geometry seminar (University of Chicago, Year along program, 2007)
Brief Explanation of my research and papers
Local Geometric Langlands Program
The local Langlands program concerns relating Galois representations to smooth irreducible representations of reductive groups over local fields. Most of my time these days are spent thinking about how to geometric and categoric analogue of this correspondence.
The notion of conductor smooth irreducible representations of GL_n over a local field was defined by Casselman (for n=2), and Jacquet, Piatetski-Shapiro and Shalika for general n. Under Langlands duality, this numerical invariant matches with the Artin conductor of Galois representations. I explored the analogue of these objects in the local geometric Langlands program.
The notion of depth for smooth representations of local fields was defined by Moy and Prasad, building on fundamental work of Bruhat and Tits. The dual notion, under Langlands duality, is obtained via the upper numbering filtration of the absolute Galois group. In my joint work with Tsao-Hsien Chen, we consider the geometric analogue of this picture. The geometric analogue of smooth representations of a group over a local field, is the categorical representations of the loop group. On the other hand, the geometric analogue of Galois representations of flat irregular connections. The geometric analogue of depths of Galois representations is slope (or Katz invariant) of flat connections. We explain how to compute this via opers, and use this result to establish a relationship between depth of categorical representations and slope of underlying connections. Preservation of depths in local geometric Langlands program
One of the most fundamental results in local geometric Langlands program is the Feigin-Frenkel Theorem describing the centre of the (universal enveloping of) affine Kac-Moody algebra at the critical level. This is a loop version of the Harish-Chandra Isomorphism, describing the centre of simple Lie algebras. The latter description also has a jet-version, proved by Geoffriau, Rais and Tauvel. In this paper, I prove the jet and loop descriptions are naturally isomorphic, using generlized versions of Verma and Wakimoto modules.
In the geometric Langlands program, one wants replace representations of reductive groups by the action of corresponding loop group on a category. These categories are usually obtained by geometrizing the usual representation. The latter should be constructed via geometrizing/categorifying the usual smooth irreducible representations of G. How should one geometrize the principal series representations of a reductive group over a local field? Geometric Satake Isomorphism provides an answer to this question for unramified representations. In a recent preprint with Travis Schedler, we consider this problem for not necessarily unramified principal series, and prove Drinfeld's conjectures regarding geometrizing regular families.
Motivated by the geometric consideration, we studied the analogue of Satake Isomorphisms for ramified characters. Here are our papers in this direction.
Together with Tanmay Deshpande, I am currently investigating defining a notion of character sheaf for solvable groups, following Boyarchenko-Drinfeld's approach to the nilpotent groups.
Character sheaves were defined by Lusztig for studying characters of finite groups of Lie type. In a recent preprint with Clifton Cunningham, we defined the notion of trace for character sheaves on the multiplicative group over the p-adic numbers and showed that we obtain all smooth characters of p-adic integers in this manner.
It is clear that the restriction of one-dimensional characters to the commutator subgroup is trivial. The corresponding statement is false for one-dimensional character sheaves. This led Drinfeld to conjecture the existence of a "true commutator and abelianization" of an algebraic group. His conjecture is proved in this paper which is based on my thesis.
The results of this paper have been used in the theory character sheaves for unipotent groups developed by Boyarchenko and Drinfeld.
Local Langlands Program
Jointly with Bhama Srinivasan and Ramin Takloo-Bigash, I am investigating the relationship between Langlands base change and Shintani's base change for finite groups. Here is some computations I did for the case of two by two matrices
In another direction works of Vogan, Soergel, Ben-Zvi and Nadler suggest that there should be a categorical framework for the local Langlands Conjecture, which combines Langlands duality with Koszul duality. The aformentioned people, and others, have carried out this program for algebraic groups over real numbers. Together with Pramod Achar, Clifton Cunningham, and Hadi Salmasian, and David Roe, I am thinking about the analogue of these categorical constructions for p-adic fields. Here is our report on this subject:
Higher categories and algebraic K-theory
Algebraic K-theory and higher category theory has intervened in several of my projects, in particular, in the stacky abelianization paper and also in the compatible intertwiners paper. Motivated by a talk given by Kapranov at MPI, Justin Noel and I starting investigating in what sense is K-theory and homology an abelianization for a group or, more generaly, for a space. Through discussions with Peter Teichner and Chris Shommer-Preis, we have come to the realization that this material known to the experts, though it doesn't appear in the literature in the form we have in mind. The following text is a summary of our thoughts on this subject.
Representations of nilpotent groups and Weil representations
Representation of Heisenberg group is important because of its relationship with quantum mechanics and because it arises naturally in many problems related to representation theory. Moreover, they give rise to a remarkable projective representation of the symplectic group known as the Weil representation. In my Master's thesis, I gave two proofs that the projective Weil representations over the finite field can be linearized (i.e., be made into a true representation).
One of the central theorems in study of representations of the Hesienberg group is the Stone-von Neumann Theorem. Kirillov's orbit method can be considered as a genralization of this theorem to higher dimensional nilpotent groups. In a joint work with Teru Thomas, we refine the orbit method by considering all polarizations at once.
The representation dimensional of a finite group G is the smallest n for which there is an embedding of G in the group of invertible n by n matrices. In a joint paper with Shane Cernele and Zinovy Reichstein, we introduce the "generalized Heisenberg group" and prove that their representations satisfy a Stone-von Neumann property. We show, moreover, that their representation dimension is maximum amongst all groups of the same size.
The results of this paper have been used by Burt Totaro in his computation of cohomology of p-groups.