Balazs Szendroi

TL;DR: In this article, an Artin group is constructed on the coherent sheaves of a smooth quasiprojective threefold containing a configuration of ruled surfaces described by a finite type Dynkin diagram.

TL;DR: In this article, the authors identify the vector space underlying refined Donaldson-Thomas theory on the conifold geometry as the exterior space of the space of polynomial functions on the affine plane, with the (Lefschetz) SL(2)-action on the threefold side being dual to the geometric SL( 2)-action.

TL;DR: In this paper, the authors prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of ff on proper components of the critical locus of ff, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities.

Abstract: This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type A and type D, respectively arbitrary, simple singularities, confirming predictions of S-duality.