Carolina Araujo (IMPA)  On Fano foliations

Abstract: Holomorphic foliations provide a powerful tool in the study of geometric properties of complex algebraic varieties. More recently, techniques from higher dimensional algebraic geometry, specially from the minimal model program, have been successfully applied to the study of global properties of holomorphic foliations. This led, for instance, to the birational classification of foliations by curves on surfaces.

Exploring this interplay between holomorphic foliations and birational geometry, we have been carrying out a systematic study of Fano foliations. These are holomorphic foliations with ample anti-canonical class on complex projective varieties. In this talk I will present some results and techniques from this theory, developed in a series of joint works with Stéphane Druel.

Aaron Bertram
University of Utah)  Geometric Stabilities for Sheaves on Surfaces and LePotier Duality

LePotier duality for del Pezzo surfaces is a pairing between the sections of line bundles on two moduli spaces of coherent sheaves with orthogonal Chern class invariants. By tilting the category of coherent sheaves on the surface, we can see this as a pairing between sections of ample line bundles on moduli spaces of complexes. As in the curve case, an Euler characteristic computation should (in principle) tell us that the two spaces of sections have the same dimension. But this computation seems to be hard. Instead, I want to describe some work with Thomas Goller and Drew Johnson in which we introduce a Grothendieck Quot scheme (following Alina Marian and Dragos Oprea) that mediates between the two spaces of sections. I'll use this to produce some new evidence for the conjecture when one of the moduli spaces in question is the Hilbert scheme of points.

María Angélica Cueto (Columbia University)  Faithful tropicalization of the Grassmannian of planes

Abstract: Non-Archimedean analytic geometry, as developed by Berkovich, is a variation of classical complex analytic geometry for non-Archimedean fields such as p-adic numbers. Solutions to a system of polynomial equations over these fields form a totally disconnected space in their natural topology. The process of analytification adds just enough points to make them locally connected and Hausdorff. The resulting spaces are technically difficult to study but, notably, their heart is combinatorial: they can be examined through the lens of tropical and polyhedral geometry.

I will illustrate this powerful philosophy by means of the Grassmannian of planes. More concretely, we will see that the tropical projective Grassmannian of planes is faithfully tropicalized by the Pluecker embedding. Thus, it is homeomorphic to a closed subset of the analytic Grassmannian. Our proof is constructive and it relies on the combinatorial description by Speyer-Sturmfels of the real points of the tropical Grassmannian as a space of phylogenetic trees of Billera-Holmes-Vogtmann. Time permitted, we will discuss the combinatorics of the aforementioned space of trees inside tropical projective space.

This talk is based on joint works with M. Haebich, and A. Werner.

Daniel Greb (Universität Duisburg-Essen)   Variation of Gieseker-Maruyama moduli spaces on higher-dimensional base manifolds

Abstract: While the variation of moduli spaces of H-Gieseker-semistable sheaves on surfaces under change of the ample polarisation H is well-understood, research on the corresponding question in the case of higher-dimensional base manifolds revealed a number of pathologies. After presenting these, I will discuss recent joint work with Julius Ross (Cambridge) and Matei Toma (Nancy) which resolves some of these pathologies by embedding the moduli problem for sheaves into a moduli problem for quiver representations.

June Huh (IAS)  A tropical approach to a Hodge conjecture for positive currents

Abstract: Demailly showed that the Hodge conjecture is equivalent to the statement that any closed (p,p)-dimensional current with rational cohomology class can be approximated by linear combinations of algebraic subvarieties, and asked whether any positive closed (p,p)-dimensional current with rational cohomology class can be approximated by positive linear combinations of algebraic subvarieties. Using tropical geometric ideas, we construct a positive closed (p,p)-dimensional current on a smooth projective variety that does not satisfy the latter statement. This is a joint work with Farhad Babaee.

Mattias Jonsson (University of Michigan)  Degenerations of amoebae and Berkovich spaces

Abstract: A complex projective variety admits an analytification as a complex analytic variety and as a Berkovich (with respect to the trivial norm on the complex numbers). I will explain how a "hybrid" Berkovich space that contains both Archimedean and non-Archimedean data can be used to prove generalizations of a result of Mikhalkin and Rullgard about degenerations of amoebae onto tropical varieties.

Song Sun (Stony Brook University)  Algebraic geometry of limits of Kahler manifolds

Abstract: It is well-known in Riemannian geometry that a sequence of Riemannian manifolds with bounded Ricci curvature and non-collapsing volume has Gromov-Haudorff limits, which are in general only metric spaces. For projective manifolds endowed with Kahler metrics, we will show these limits are naturally projective varieties, and discuss the connection with algebraic geometry of the limits and their singularities. This talk is based on joint work with Simon Donaldson.

Melanie Wood (University of Wisconsin‑Madison)  Discriminants in the Grothendieck ring

Abstract: We will see there is a well defined limit, in the Grothendieck ring of varieties, of several different sequences of moduli spaces of "nice" objects, or equivalently of their complements the "discriminant" variety of "not so nice" objects. In flavor, these are algebro-geometric analogs of topological results about homological stability, or of arithmetic results about asymptotics of points counts on moduli spaces.  For example, we determine the limit of the motive of smooth divisors (or with s singularities) in increasing multiples of a linear system. We also determine the limit of the motive of configuration spaces of distinct points (or points that are allowed to come together to a limited extent) as the number of points increases. All of these limits in the Grothendieck ring are given by explicit formulas in terms of motivic zeta values. Our results motivate a large number of conjectures in topology and arithmetic.  This is joint work with Ravi Vakil.