Research Statement PDF

Given a simple Lie group there is an associated Reshetikhin-Turaev invariant of tangles. This invariant is a map between representations of the given (quantum) group. Mikhail Khovanov's suggestion is to "categorify" these invariants by replacing each representation with a category and each map with a functor. Joel Kamnitzer and I propose doing this by using the (derived) category of coherent sheaves on certain flag-like varieties where the functor associated to a tangle is defined via a natural correspondence between these spaces.

In our first paper we do this for sl_2 and recover Khovanov homology. In our second paper we do this for sl_m and obtain a theory conjecturally isomorphic to Khovanov-Rozansky homology. The geometric constructions used are in part motivated by the geometric Satake correspondence as well as the symplectic constructions of Seidel-Smith and Manolescu.

Given a family of smooth curves over an open subset U of S when does it extend to a family of stable curves over S? Alternatively, when does a map from U to M_g extend to a regular map from S to the Deligne-Mumford compactification M_g? It turns out the answer is if and only if the local monodromy is virtually abelian. Formalizing this idea I say that a compactification X of X has the abelian monodromy extension (AME) property if a map U to X extends to a regular map S to X whenever the image of local fundamental groups is abelian. It turns out that if X has an AME compactification then it has a unique maximal one (which is therefore canonical).

In the first paper I show that the unique maximal AME compactification of M_g is M_g. Similarly the unique maximal AME compactification of the moduli space of abelian varieties is its Baily-Borel compactification. Subsequently, I hope is to explain how AME compactifications are related to log canonical models of varieties of log-general type.

Let G be a finite subgroup of SL_N(C). When N=2, the original McKay correspondence describes a bijection between irreducible representations of G and exceptional divisors in the minimal (crepant) resolution of the quotient C^2/G.

If N=3 such a minimal (crepant) resolution is not unique though there is a special one called G-Hilb. Bridgeland, King and Reid prove a "derived" version of McKay correspondence by showing that the (derived) category of sheaves on G-Hilb is isomorphic to the (derived) category of G-equivariant sheaves on C^3 via a natural functor &Phi: D(G-Hilb) -> D^G(C^3).

We explore to what extent the inverse of &Phi gives a natural correspondence between irreducible representations of G and exceptional loci (divisors and curves) of G-Hilb. When G is abelian and N=3 we show that the situation is similar to that when N=2. Little is known for N > 3.