"Pure mathematics is the poetry of logical ideas."

Interests
Geometry and representation theory, more specifically, the Hitchin system and the geometric Langlands program.

Publications and preprints
On the image of the parabolic Hitchin map (joint with David Baraglia, to appear in Quarterly Journal of Math.).

Complete integrability of the parabolic Hitchin map (joint with David Baraglia and Rohith Varma, to appear in IMRN).

A new approach to Jordan decomposition for formal differential operators (joint Samuel Weatherhog, submitted).

On a geometric analogue of a conjecture of Gross and Reeder (joint with Dan Sage, submitted).

Stabilisers of eigenvectors of finite reflection groups 

Conductors in local geometric Langlands correspondence (Canadian Journal of Math., Vol 69, no. 1, 2017, 107-126) 

Preservation of depths in local geometric Langlands program (joint with Tsao-Hsien Chen, Trans. Amer. Math. Soc. (2017), 1345-1364)

Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for Jet algebras (Trans. Amer. Math. Soc. (2016), 2019-2038)

Ramified Satake Isomorphisms for strongly parabolic characters (Joint with Travis Schedler, 2014, Documenta Mathematica) 

Geometrization of principal series representation of reductive groups (Joint with Travis Schedler, 2015, 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) - Not intended for publication. 

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) - Not intended for publication. 



Brief Explanation of my research and papers
Hitchin System
 Let Bun_G denote the moduli of G-bundles on a compact Riemann surface. Let T*Bun_G denote the cotangent bundle. This is known as the moduli of Higgs bundles. For G=GL_n one can think of elements of T^*Bun_G as pairs (E,A) where E is a vector bundle and A is (roughly) a linear operator on this vector bundle. Thus, at each point on the Riemann surface one has a matrix, and one can take the characteristic polynomial of this matrix. This is what Hitchin did in his seminal paper in late 80s. The corresponding map which goes from T^*Bun_G to a vector space is called the Hitchin map. The receiving vector space is called the Hitchin base. Hitchin proved that this map is a completely integrable system; which roughly speaking means that it is proper and has Lagrangian fibres. 

I'm currently thinking about the Hitchin system on non-compact surfaces. While the Higgs bundles on non-compact spaces are well-studied, the corresponding Hitchin map is still mysterious. For instance, it is not known in general if it is a completely integrable system. In my paper with David Baraglia and Rohith Varma, we proved that this is the case if one allows for regular singularity with unipotent monodromy (the tamely ramified case). In a companion paper with Barglia, we described the Hitchin base in this case. Currently, I'm thinking about the case where one allows higher singularities (the wildly ramified case). 

Nonabelian Hodge theory tells us that moduli of Higgs bundles is diffeomorphic to the moduli of flat connections. Thus, studying flat connections is another interest of mine. In particular, in the course of our study with Dan Sage of formal flat connection, we made use of Jordan decomposition for these objects. This result, known as Levelt-Turrittin Theorem, is fundamental in the study of connection. However, its proof is rather roundabout and looks nothing like the proof of usual Jordan decomposition. In my paper with Sam Weatherhog, we set out to give a straightforward proof of this result modelled on the linear setting. 

My interest in the Hitchin system started by trying to understanding Beilinson and Drinfeld's seminal book. They proved that quantum fibres of the Hitchin map are Hecke eigensheaves, the sought after objects in the geometric Langlands program. Thus, there is an intimate relationship between geometric Langlands and Hitchin system. More specifically, if one knew that the Hitchin system with Moy-Prasad level structure is flat, then one can prove the preservation of depths conjecture mentioned below. 

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. 

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. 



Character sheaves
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.


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. 


Abandoned Projects
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 AcharClifton 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. 






Notes from some of my talks
More recent notes can be found in the Schedule/Talks page.
On the life 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)  
Introduction to local geometric Langlands (Max Planck Institute in Bonn, 2011)



Seminars I have organized
Quantum Field Theory Seminar at UQ has its own website! I established this upon my arrival at UQ in 2013. 
Nearby cycles (UBC, Semester 1, 2011) 
Geometrization and categorification (UBC, Semester 1, 2010) 
Junior algebraic geometry seminar ( The University of Chicago, Year along program, 2007)