Abstracts - Fall 2018

Speaker:  Eva Bayer

Title: Isometries of lattices

Abstract: In a joint work with Lenny Taelman, we characterize the irreducible polynomials that occur as a characteristic polynomial of an isometry of an even, unimodular lattice with given signature, answering a question of Gross and McMullen. It turns out that the criteria are local ones, and that in the case of an irreducible polynomial, one has a local-global principle. This is no longer true for reducible polynomials. The aim of the talk is to describe these results, and give to a criterion for the local-global principle to hold.

Speaker: Daniel Bragg

Title:  Supersingular twistor spaces

Abstract: We will describe how the crystalline cohomology of a supersingular K3 surface gives rise to certain one-parameter families of K3 surface. This is a positive characteristic analog of the twistor spaces associated to K3 surfaces over the complex numbers. Our construction relies on the special behavior of p-torsion Brauer classes in characteristic p. As applications, we find new proofs of Ogus's crystalline Torelli theorem and Artin's conjecture on the unirationality of supersingular K3 surfaces. These results are new in small characteristics.

Speaker: Marc Hoyois

Title:   Moduli stacks of varieties and algebraic bordism

Abstract:  A "moduli stack of varieties" M is a functor associating to  every scheme  X some groupoid M(X) of schemes over X. An object of M(A^1) is thus a scheme over the affine line A^1, which we can interpret as a bordism between its fibers over 0 and 1. The A^1-homotopy type of such a moduli stack is thus a naive algebro-geometric analog of bordism spaces in algebraic topology, which are closely related to Thom spectra by a famous theorem of Thom. I will present a version of Thom's theorem in algebraic geometry, which states that the A^1-homotopy type of the moduli stack of proper 0-dimensional local complete intersections is the motivic Thom spectrum MGL defined by Voevodsky. This is joint work with Elden Elmanto, Adeel Khan, Vladimir Sosnilo, and Maria Yakerson.

Speaker: Junliang Shen

Title: Perverse filtrations, Higgs bundles, and hyperkahler geometry

Abstract: For a hyperkahler variety which admits a Lagrangian fibration, an increasing filtration is defined on its rational cohomology using the perverse t-structure. We will discuss the role played by this filtration in the study of the  topology and geometry of hyperkahler varieties. First, we will focus on the perverse filtration for the moduli of Higgs bundles with respect to the Hitchin fibration. We will discuss our recent proof of de Cataldo, Hausel, and Migliorini's P=W conjecture for parabolic Higgs bundles labelled by affine Dynkin diagrams. Then I will present a surprising connection between the perverse filtration for a projective hyperkahler variety and the (pure) Hodge structure on itself. Based on joint work with Qizheng Yin and Zili Zhang.