Abstract: For a complex projective variety, Serre's classical GAGA theorem asserts that the analytification functor from algebraic coherent sheaves to analytic coherent sheaves is an equivalence of categories. A particularly nice consequence is that a compact analytic variety admits at most one algebraic structure. In joint work with B. Bakker and Y. Brunebarbe, we show that a GAGA theorem holds even in the non-proper case if one instead works with "definable analytic spaces": analytic structures that are "tame" in a sense made precise by the notion of o-minimality. This result has particularly important applications to Hodge theory, where it provides a natural framework for discussing period spaces, which are rarely algebraic but always carry a definable analytic structure. We will explain how it can be used to prove a conjecture of Griffiths on the quasi-projectivity of the images of period maps, as well as a natural generalization to mixed period maps. Time permitting, we will also discuss some applications of these results to moduli theory. Hélène Esnault (Freie Universitaet Berlin / IAS): Special loci of Betti moduli
Abstract: We define special loci in Betti moduli, prove in rank one a structure theorem in arithmetic geometry out of which we draw various consequences, such as Hard Lefschetz in rank one and positive characteristic. Joint with Moritz Kerz. Tony Pantev (University of Pennsylvania): Symplectic structures on moduli of Stokes data Abstract: I will discuss the notion of shifted symplectic structures along the stalks of constructible sheaves of derived stacks on stratified spaces. I will describe a general pushforward theorem producing relative symplectic forms and will explain explicit techniques for computing such forms. As an application I will describe a universal construction of Poisson structures on derived moduli of local systems on smooth varieties and will explain how symplectic leaves arise from fixing irregular types and local formal monodromies at infinity. This is a joint work with Dima Arinkin and Bertrand Toën. Simon Donaldson (Stony Brook / Imperial College): Moment maps in complex differential geometry Abstract: The talk will be a survey of results connecting existence theorems in complex differential geometry with stability conditions in algebraic geometry. One key idea is the interpretation of tensors arising in the first area (derived from curvature) as "moment maps" for the action of infinite dimensional groups. This fits the discussion into a general notion of "equality of symplectic and complex quotients". We will review this background and discuss various examples and problems of current research interest. Jennifer Balakrishnan (Boston University): Chabauty-Coleman experiments for genus 3 curves Abstract: We describe a Chabauty--Coleman computation of rational points on genus 3 hyperelliptic curves defined over the rationals whose Jacobians have Mordell--Weil rank 1, which was carried out on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves. We discuss a few surprising examples where the zero locus includes global points not found in the set of rational points. This is joint work with Francesca Bianchi, Victoria Cantoral-Farfán, Mirela Çiperiani and Anastassia Etropolski. We also examine tools needed to implement a Chabauty--Coleman solver and survey recent work of Sachi Hashimoto and Travis Morrison on a rational points computation for Picard curves of rank 1. Junliang Shen (MIT): Hitchin systems, hyper-Kaehler geometry, and the P=W conjecture Abstract: The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In this talk, we will focus on interactions between compact and noncompact hyper-Kaehler geometries. Such connections, together with symmetries coming from the moduli of compact hyper-Kaehler manifolds, lead to new progress on the P=W conjecture. I will discuss the results we obtained and the difficulties we met. Based on joint work with Mark de Cataldo and Davesh Maulik. J. Maurice Rojas (Texas A&M University / NSF): Complexity Chasms Over Local Fields Abstract: When can one solve a system of polynomial equations in time polynomial in the input size? This classical question is closely related to the P vs. NP problem and remains unsolved over the real numbers, despite important advances over the complex numbers. We discuss some definitive complexity limits in one variable, and how they bring together diophantine approximation, hypergeometric functions, and computational complexity. We then see how geometry of discriminant varieties leads us to an important speed-up in the multivariate case: Under certain restrictions, we can solve sparse polynomial systems (over the reals and the p-adic rationals) in polynomial-time on average. Sebastian Casalaina-Martin (University of Colorado, Boulder): Geometry and topology of the moduli space of cubic threefolds and its smooth models Abstract: Cubic hypersurfaces are of special interest in geometry. The fact that every smooth cubic surface contains precisely 27 lines is one of the most classical results of algebraic geometry. Cubic hypersurfaces of dimension 3 became famous for another reason: they were the first known example of a variety X which admits a finite dominant map from a projective space onto X (X is unirational), but is itself not birational to projective space (X is not rational). In this talk I will discuss some new results on the geometry and topology of moduli spaces of cubic hypersurfaces. This is joint work with Samuel Grushevsky, Klaus Hulek, and Radu Laza. |
Stony Brook 2020 >