Papers

A derived approach to geometric McKay correspondence in dimension three. PDF J. Reine Angew. Math. (to appear) (on the arXiv)

We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail in abelian case. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb(C^3). This correspondence appears to be related to Reid's recipe.

Knot homology via derived categories of coherent sheaves II, sl(m) case. PDF Inventiones Math. 174 (2008) no. 1, 165--232. (on the arXiv)

Using derived categories of equivariant coherent sheaves we construct a tangle invariant which categorifies the sl(m) knot invariant for the standard representation. It naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and related to Manolescu's by homological mirror symmetry.

The Abelian Monodromy Extension property for families of curves. PDF Submitted. (on the arXiv)

Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U of S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily-Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.

Knot homology via derived categories of coherent sheaves I, sl(2) case. PDF Duke Math Journal 142 (2008), no. 3, 511--588. (on the arXiv)

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel-Smith by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology.

On Tutte's chromatic invariant PDF Trans. AMS. (to appear)

Periodicity, morphisms, and matrices. PDF Theor. Comput. Sci. 1-3 (2003); 107--121.

The matrix of chromatic joins and the Temperley-Lieb algebra. PDF J. Combin. Theory (Ser. B) 89 (2003); 109--155.