About me

I am a Samelson Postdoctoral Fellow in the Department of Mathematics at Stanford University. During the 2008-2009 academic year I was on leave visiting the Max Planck Institute for Mathematics in Bonn, Germany. I study geometric and categorical constructions in representation theory.

My wife Joan Licata is also a mathematician.

Teaching

In the fall of 2009 I'm teaching Math 53 , (ordinary differential equations).
In the winter of 2008 I taught Math 51 (linear algebra and multivariable calculus).
In graduate school I taught both single variable calculus and multivariable calculus.

Research

Categorification via Geometric Representation Theory
Sabin Cautis, Joel Kamnitzer and I have developed a theory of sl(2) categorification using derived categories of coherent sheaves. The varieties involved in our main examples are cotangent bundles of Grassmanians and convolution varieties in the affine Grassmanian of PGL(n). Following ideas of Chuang-Rouquier, we use categorical sl(2) actions to produce explicit equivalences of triangulated categories; in the affine Grassmanian example, these equivalences are part of the Cautis-Kamnitzer construction of generalized Khovanov-Rozansky link homology.
Coherent Sheaves and Categorical sl(2) Actions with S. Cautis and J. Kamnitzer, to appear in Duke Mathematical Journal.
Derived Equivalences for Cotangent Bundles of Grassmanians from Categorical sl(2) Actions with S. Cautis and J. Kamnitzer, submitted.
Categorical Geometric Skew Howe Duality with S. Cautis and J. Kamnitzer, to appear in Inventiones Mathematicae.

Symplectic Duality
Tom Braden, Nick Proudfoot, Ben Webster, and I are investigating natural categorical structures associated to holomorphic symplectic manifolds and to Koszul algebras. Many Koszul algebras admit flat deformations which behave like torus-equivariant cohomology rings of a manifold with a finite torus fixed-point set. Moreover, when the Koszul algebra has geometric origin, Koszul duality is often related to some other geometric duality. For example, the Koszul algebra which governs category O for a hypertoric variety, is Koszul dual to the algebra which governs category O for the Gale dual hypertoric variety. A conjectural generalization of this duality statement to other pairs of holomorphic symplectic manifolds is current work in progress.
Gale Duality and Koszul Duality with T. Braden, N. Proudfoot and B. Webster, submitted.
Goresky-Macpherson Duality and Deformations of Koszul Algebras with T. Braden, C. Phan, N.Proudfoot, and B. Webster, submitted.

Infinite dimensional algebras and moduli spaces of framed sheaves
In my thesis I constructed actions of Heisenberg and Clifford algerbas on the equivariant cohomology of moduli spaces of framed, torsion-free sheaves on P^2. Subsequently, Alistair Savage and I gave a more natural construction of these actions by considering Chern classes of tautological bundles on these moduli spaces. The Clifford algebra action we construct is related to the Heisenberg algebra action by a "boson-fermion" correpondence, so as a corollary of our construction we obtain a geometric interpretation of the vertex operators used in that correspondence.
Vertex Operators and the Geometry of Moduli Spaces of Framed Torsion-free Sheaves with A. Savage, to appear in Selecta Mathematica.
Framed Torsion-free Sheaves, Hilbert Schemes, and Representations of Infinite Dimensional Lie Algebras , to appear in Advances in Mathematics.

Revised on February 7th, 2009