Geometric representation theory, geometric Langlands, character sheaves, higher categories
Publications and preprints
Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for Jet algebras (ArXiv 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)
A novel algorithm for decompunding Poisson random sums (joint with H. Hasheminia, and D. Gillen, 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)
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)
Conformal Field Theory (UQ, 2014), joint with Jorgen Rasmussen.
Quantum Field Theory (UQ, 2013)
Nearby cycles (UBC, 2011)
Geometrization and categorification (UBC, 2010)
Junior algebraic geometry seminar (University of Chicago, 2007)
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.
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.
Geometrization of principal series representation of reductive groups (To appear in Annals of Institute Fourier)
Motivated by the geometric consideration, we studied the analogue of Satake Isomorphisms for ramified characters. Here are our papers in this direction.
Ramified Satake Isomorphisms for strongly parabolic characters (sent for publication)
Ramified Satake Isomorphisms II (In progress)
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.
Geometrization of continuous characters of Z_p^* (Pacific Journal of Math, Vol 261 (2013), No. 1.)
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.
Stacky abelianization of algebraic groups (Transformation Groups, vol. 14, n. 4, 2009)
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 $
Base change (In progress)
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.
Compatible intertwiners for representations of nilpotent groups (Representation Theory, 15, 2011, p. 407-432)
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.
Maximal representation dimension for finite p-groups (Journal of Group Theory, 14, no. 4, 2011, p. 637-648)
The results of this paper have been used by Burt Totaro in his computation of cohomology of p-groups.
Optimization: Decomposing Poisson Sums
Poisson counting process with bulk arrivals are seen in various applications such as queuing theory, Insurance Mathematics, and quantitative risk management. In this paper, a new decompounding algorithm based on rudimentary number theory is developed. The algorithm is capable of decompounding Poisson processes when bulk sizes are eitherpairwise co-prime or composed of two elements.
A novel algorithm for decompounding Poisson randum sums
https://sites.google.com/site/masoudkomi/Poisson.pdf (joint with H. Hasheminia, and D. Gillen)