Categorical Heisenberg actions
In the first paper we define a 2-category whose Grothendieck group is isomorphic to the q-deformed Heisenberg algebra of affine type. We then construct an action of this category on derived categories of coherent sheaves on Hilbert schemes of points on ALE spaces (this categorifies the level one Fock space representation).
The second paper explains how to obtain a 2-representation of the quantum affine algebra from a 2-representation of the quantum Heisenberg algebra. Applying this construction to the Fock space 2-representation mentioned above gives a categorical action of the quantum affine algebra on derived categories of coherent sheaves on Hilbert schemes of points. This categorifies the Frenkel-Kac-Segal vertex operator construction of the basic representation.
In the third paper We show that on any 2-representation of the Heisenberg one can define a braid group action. This means that there exist certain explicit complexes of 1-morphisms which satisfy the corresponding braid relations.
Geometric categorical Lie algebra actions
We introduce the idea of a geometric categorical Lie algebra action and show that it induces an action of the corresponding braid group.
In the second paper we construct such an action on derived categories of coherent sheaves on Nakajima quiver varieties (as well as an associated affine braid group). This action lifts Nakajima's action on the K-theory of these quiver varieties.
Equivalences and categorical sl(2) actions
We study strong categorical sl(2) actions on derived categories of coherent sheaves and use them to construct derived equivalences. This idea is due to Chuang and Rouquier in the context of abelian categories. In the first paper below we explain how to obtain a strong categorical sl(2) action from an sl(2) action satisfying certain geometric criteria which can more easily be checked.
The second paper proves that the derived category of coherent sheaves on (a certain compactification of) the cotangent bundle to Grassmannians can be equipped with such a geometric sl(2) action. In the third paper we show how to obtain equivalences of triangulated categories from a strong categorical sl(2) action. Combining this with the first two papers we obtain new equivalences between derived categories of coherence sheaves on cotangent bundles to Grassmannians related by stratified Mukai flops.
The fourth paper identifies these equivalences more explicitly and uses this to construct equivalences between spaces related by stratified Atiyah flops (these are deformations of stratified Mukai flops).
Knot homologies via derived categories of coherent sheaves
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.
The third paper explains how, starting with any categorification of a specific sl(∞)-module, one can categorify all the Reshetikhin-Turaev sl(m) tangle invariants. An example of such a categorification is provided by categories of sheaves on certain varieties associated to the affine Grassmannian of PGL(m). These varieties are iterated Grassmannian bundles. Much of the work involves constructing the generalized Jones-Wenzl projectors (a.k.a. clasps).
Categorified quantum groups and representation theory
The first paper below shows that a simplified definition of categorical Lie algebra actions (which includes, for instance, an action in the sense of Rouquier) induces an action of the 2-category defined by Khovanov-Lauda in their work on categorification of quantum groups. This involves checking certain technical relations in their 2-category.
In the second paper we discuss a slightly nonstandard realization of the quantum affine algebra. This realization shows up when studying categorical vertex operators and seems suitable for categorification.
In the third paper we give a diagrammatic presentation in terms of generators mod relations of the representation category of quantum sl(n). This answers a question posed by Kuperberg and affirms conjectures of Kim and Morrison . Our main tool is an application of quantum skew Howe duality.
The abelian monodromy extension property
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.
The geometric McKay correspondence in dimension three
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.