Texas Algebraic Geometry Symposium 2019

Angela Gibney, Classes on the moduli space of curves from affine Lie algebras and Gromov Witten theory: identities and generalizations

In this talk I will introduce positive classes on the moduli space of stable pointed rational curves that arise in two different constructions; on the one hand as Chern classes of Verlinde bundles, constructed from integrable modules over affine Lie algebras, and on the other hand, as the Gromov-Witten loci of smooth homogeneous varieties. We'll see that in the simplest cases these classes are equivalent. Examples, conjectures, and generalizations via conformal vertex algebras will be discussed.

Nicholas Proudfoot, The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials

I will give a general recipe for interpreting your favorite combinatorially defined polynomials as intersection cohomology Poincare polynomials. The main focus will be on uniting three families of examples:

The Kazhdan-Lusztig polynomials for a Weyl group can be interpreted as Poincare polynomials for certain intersection cohomology groups associated with the flag variety.
The g-polynomial of a polytope can be interpreted as the Poincare polynomial for the intersection cohomology of the associated affine toric variety.
The Kazhdan-Lusztig polynomial of a hyperplane arrangement can be interpreted as the Poincare polynomial of a certain variety associated with the arrangement.

Harold Williams, Kasteleyn operators from mirror symmetry

The goal of the talk is to explain how an explicit combinatorial description of mirror symmetry for toric surfaces appears implicitly in the statistical mechanics of dimer models. The main result is that, given a bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$, the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a curve in $(\mathbb{C}^\times)^2$ -- the defining equation of this spectral curve encodes the enumeration of perfect matchings on $\Gamma$ as well as the asymptotic behavior of perfect matchings on its covering graph in $\mathbb{R}$. The second construction is to consider a certain noncompact Lagrangian $L \subset T^* T^2$ canonically associated to $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its homological mirror coherent sheaf. This lives on a toric compactification of $(\mathbb{C}^\times)^2$ determined by the asymptotics of $L$. We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry.

This is joint work with David Treumann and Eric Zaslow.

Paul Hacking, Mirror symmetry for Fano varieties.

The mirror of a Fano n-fold is a family of Calabi--Yau (n-1)-folds over the affine line with maximally unipotent monodromy at infinity. We describe this mirror correspondence in terms of birational geometry, deformation theory, and Hodge theory. This is joint work with Corti and Petracci, and builds on work of Coates, Corti, Galkin, Golyshev, and Kasprzyk.

This talk will be about joint work with Prakash Belkale, Chiara Damiolini, and Nicola Tarasca.

Danny Krashen, The complexity of Brauer classes

Brauer classes, which classify division algebras contain a great deal of information about the arithmetic of fields and the behavior of various types of algebraic structures, such as quadratic forms. While various techniques have been developed which have informed our understanding of Brauer classes on $p$-adic curves and function fields of more general arithmetic surfaces, relatively little progress has been made in the case of function fields of higher dimensional schemes.

In this talk I will discuss some conjectures concerning the complexity of Brauer classes, and describe some recent joint work with Antieau, Auel, Ingalls and Lieblich in the case of p-adic surfaces.

Piotr Achinger, Serre-Tate theory for Calabi-Yau varieties

Classical Serre-Tate theory concerns the deformation theory of ordinary abelian varieties. It implies that their deformation spaces can be equipped with a group structure and a lifting of the Frobenius morphism, and consequently such varieties admit a canonical lifting to characteristic zero. In the talk, I will show how to obtain similar results for ordinary Calabi-Yau varieties of arbitrary dimension. The main tools will be Frobenius splittings and a new construction of relative Witt vectors of length two. This is joint work with Maciej Zdanowicz (EPFL).

Michael Groechenig, p-adic integration for Hitchin systems and the fundamental lemma

The fundamental lemma is an identity of integrals central to the Langlands programme. Despite its ostensibly combinatorial nature, it resisted all direct efforts to verify it, until Ngô finally proved it in 2008. One of the unexpected features of Ngô’s argument, was the important role played by the moduli space of Higgs bundles. His proof infers the fundamental lemma from a statement about the cohomology of moduli spaces of Higgs bundles, called geometric stabilisation.

In this talk I’ll discuss a new perspective on geometric stabilisation, provided by p-adic integration. We will see that there exists a close philosophical link between mirror symmetry for moduli spaces of Higgs bundles (à la Hausel-Thaddeus) and the fundamental lemma. This is joint work with Wyss and Ziegler.