The intimate relationship between the homological properties of a Koszul algebra and its corresponding Koszul dual counterpart provides a powerful tool for understanding the algebraic structure of a Koszul algebra by examining its dual counterpart. In this talk we investigate the consequences of this link in the case where a group acts on the algebras. The motivating example will be the Orlik-Solomon and Varchenko-Gel'fand algebras for the braid arrangement with the action of the symmetric group,  where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements.  They also enjoy representation stability properties that follow from Koszul duality. This is joint work with Vic Reiner and Sheila Sundaram.

In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, and in joint work with Michael Brown for smooth projective toric varieties of Picard rank 2. This talk is about a splitting criterion for arbitrary smooth projective toric varieties.

Free resolutions, or syzygies, with a graded structure are algebraic objects that encode many geometric properties.  This correspondence lies at the heart of classical projective algebraic geometry.  By analogy, multigraded resolutions should also provide powerful geometric tools.  I will discuss some foundational results from the classical story and give an overview of recent work to extend these tools to the multigraded setting of toric geometry.

A consequence of a larger investigation is that for self-injective finite dimensional algebras over a field (in the commutative case: Gorenstein rings of dimension zero), of radical length at least 3, there are indecomposable perfect complexes with arbitrarily large homology in any given degree. For complicated rings it is not surprising that there should be complicated perfect complexes, so this result is perhaps most striking in that it applies to rings as small as a polynomial ring  k[t]/(t^3) . in proving this we describe in some detail properties of the Auslander-Reiten quiver of perfect complexes in the bounded derived category. An application to equivariant homotopy theory is given.

For the polynomial ring, we know that any module admits resolution no longer than the number of variables. This can be translated into a purely categorical statement: for R a polynomial ring in n variables, any object of the bounded derived category of finitely-generated modules D^b(mod R) can built using basic categorical operation (shifts, sums, and summands) using at most n mapping cones starting from R.

For general, non-regular R, this fails dramatically. Is there a natural object to substitute for R to recover finite (strong) generation? In prime characteristic, there is. As one application, using categorical invariants, we can define a hierarchy of new classes of singularities extending a familiar one. 

This is joint work with P. Lank, S. Iyengar, A. Mukhopadhyay, and J. Pollitz. 

Happy Thanksgiving!

Given a toric GIT problem A^n//T, the Bondal-Ruan-Thomsen (BRT) collection of line bundles is known to generate the derived category of [A^n//T] for all chambers. In this talk, I will discuss a window functor from the derived category of [A^n//T] into the derived category of multigraded modules which factors through the BRT generator, obtained by mirror symmetry. This talk is based on work in progress with David Favero.