I'm an AIM five-year fellow, currently working at UC Berkeley. In 2005-2006, I was at MIT.
Next year I will be an assistant professor at the University of Toronto.
My office in Berkeley is 1067 Evans (phone # is 510 642 2149) and my email is .
CV Here is my CV.
Teaching In Spring 2007, I taught Math 172, an undergraduate combinatorics course.
Seminar This semester (Fall 2007), I am running a seminar with Xinwen Zhu on perverse sheaves .
Blog Some friends and I have a mathematical blog, called the secret blogging seminar.
Photos Here are a few photos online of Japan and a bike trip.
Research
Crystals, coboundary categories, and the topology of the moduli space of real curves
Andre Henriques and I began by studying the relationship between the octahedron recurrence and the category of gl(n) crystals. That lead us to defining a commuter for the category of crystals for an arbitrary reductive Lie algebra. We found that this category is a coboundary category and that the cactus group acts on tensor products in coboundary categories. The classifying space for the cactus group is the moduli space of marked genus 0 real curves and so we were lead to studying the topology of this space with Pavel Etingof and Eric Rains. More recently, Peter Tingley and I have picked up the study of the commutor. First, we showed that the commutor admits an alternate definition using Kashiwara's involution. Later, we proved that the crystal commutor arising as q=0 limit of Drinfeld's unitarized R-matrix.
Crystals and coboundary categories with A. Henriques, Duke Math. J. 132 (2006).
The octaheron recurrence and gl(n) crystals with A. Henriques, Adv. Math. 206 (2006).
The cohomology ring of the moduli space of stable curves of genus 0 with marked points with P. Etingof, A. Henriques, and E. Rains.
A definition of the crystal commutor using Kashiwara's involution with P. Tingley.
The crystal commutor and Drinfeld's unitarized R-matrix with P. Tingley.
Mirkovic-Vilonen cycles and polytopes
By the geometric Satake Isomorphism, the Mirkovic-Vilonen cycles (certain subvarieties of the affine Grassmannian) give a basis for representations of complex reductive groups. In his thesis, Jared Anderson introduced MV polytopes and explained how they could be used to get some combinatorial objects out of MV cycles. In my thesis, I gave an explicit description of MV cycles and polytopes. This description used some combinatorics which had already been developed by Berenstein-Zelevinsky to describe Lusztig's canonical basis. In particular, this gives a combinatorial link between MV cycles and the canonical basis. I also studied the crystal structure on MV polytopes proving that the crystal operators coming from MV cycles (as defined by Braverman-Gaitsgory) and the crystal operators coming from the canonical basis coincide. Finally, I used this theory to construct a natural bijection between components of fibres of the convolution morphism for the affine Grassmannian of GL_n and combinatorial objects known as hives -- both objects count GL_n tensor product multiplicities.
Mirkovic-Vilonen cycles and polytopes .
The crystal structure on the set of Mirkovic-Vilonen polytopes , Adv. Math. 215 (2007).
Hives and the fibres of the convolution morphism .
Knot homology via derived categories of coherent sheaves
Misha Khovanov has begun categorifying quantum knot invariants, such as the Jones polynomial. Sabin Cautis and I have proposed a program to accomplish this categorification using geometric representation theory. Specifically we study derived categories of coherent sheaves on varieties arising in the geometric Langlands programs. In our first paper, we study the case of sl(2) and recover the original Khovanov homology. In this case, our construction is related to that of Seidel-Smith by homological mirror symmetry.
Knot homology via derived categories of coherent sheaves I, sl(2) case with S. Cautis.
Knot homology via derived categories of coherent sheaves II, sl(m) case with S. Cautis.