Algebraic Geometry Seminar Fall 2014

The seminar meets on Fridays at 2:25 pm in Van Vleck B131.

The schedule for the previous semester is here.

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Fall 2014 Schedule

date	speaker	title	host(s)
September 12	Andrei Caldararu (UW)	Geometric and algebraic significance of the Bridgeland differential	(local)
September 19	Greg G. Smith (Queen's University)	Toric vector bundles	(Daniel)
October 3	Daniel Erman (UW)	Tate resolutions for products of projective spaces	(local)
October 10	Lars Winther Christensen (Texas Tech University)	Beyond Tate (co)homology	Daniel
October 17	Claudiu Raicu (Notre Dame University)	TBA	Daniel
October 31	Anatoly Libgober (UIC)	Landau-Ginzburg/Calabi-Yau and McKay correspondences for elliptic genus	Max
November 7	Vlad Matei (UW)	Moments of arithmetic functions in short intervals	Local
November 14	No seminar (room will be used for a specialty exam)
December 5	Eyal Markman (UMass Amherst)	Integral transforms from a K3 surface to a moduli space of stable sheaves on it	Andrei
December 12	DJ Bruce (UW)	Betti Tables of Graph Curves	local
March 13	Jose Rodriguez (Notre Dame)	TBD	Daniel

Abstracts

Andrei Caldararu

Several years ago Tom Bridgeland suggested that there should exist interesting chain maps C_*(M_{g,n}) -> C_{*+2}(M_{g,n+1}) and he conjectured some applications of these maps to mirror symmetry. I shall present a precise definition of these maps using techniques from the theory of ribbon graphs, and discuss a recent result (joint with Dima Arinkin) about the homology of the total complex associated to the bicomplex obtained from these maps. Then I shall speculate (wildly) about applications to mirror symmetry.

Eyal Markman

Let S be a K3 surface, v an indivisible Mukai vector, and M(v) the moduli space of stable sheaves on S with Mukai vector v. The universal sheaf gives rise to an integral functor F from the derived category of coherent sheaves on S to that on M(v). We show that the functor F is faithful (but not full). The bounded derived category of M(v) is rather mysterious at the moment. As a first step, we provide a simple conjectural description of its full subcategory whose of objects are images of objects on S via the functor F. We verify that description whenever M(v) is the Hilbert scheme of points on S. This work is joint with Sukhendu Mehrotra.

Lars W Christensen

Tate (co)homology was originally defined for modules over group algebras. The cohomological theory has a very satisfactory generalization---Tate--Vogel cohomology or stable cohomology---to the setting of associative rings. The properties of the corresponding generalization of the homological theory are, perhaps, less straightforward and have, in any event, been poorly understood. I will report on recent progress in this direction. The talk is based on joint work with Olgur Celikbas, Li Liang, and Grep Piepmeyer.

Anatoly Libgober

I will discuss elliptic genus of singular varieties and its extension to Witten's phases of N=2 theories. In particular McKay correspondence for elliptic genus will be described. As one of applications I will show how to derive relations between elliptic genera of Calabi Yau manifolds and related Witten phases using equivariant McKay correspondence for elliptic genus.

Vlad Matei

In 2012, J.P Keating and Z. Rudnick published a paper where they resolved a function field version of the Montgomery-Goldston pair correlation conjecture. Their proof relies on a recent equidistribution result of N. Katz. In joint work with Daniel Hast, we reprove their result by counting points on a certain variety using a twisted Grothendieck-Lefschetz formula and obtain also information about higher moments. Moreover our method allows us to also give a proof of the autocorrelation of the Mobius function on average in the function field setting, also known as the Chowla conjecture.

DJ Bruce

Given a graph one may obtain a reducible algebraic curve by associating a P^1 to each vertex with two P^1’s intersecting if there is an edge between the associated vertices. Such curve are called graph curves, or line arrangements, and were introduced by Bayer and Eisenbud in studying Green’s conjecture. I will discuss how the combinatorics of the graph affect the Betti table of its associated curve. In particular, I will present formulas for the Betti table for all graph curves of genus zero and one. Additionally, I will give formulas for the graded Betti numbers for a class of curves of higher genus. This talk is based on joint work with Pete Vermeire, Evan Nash, Ben Perez, and Pin-Hung Kao.