Robert Lazarsfeld:
Title: Measures of association between algebraic varieties
Abstract: I will talk about joint work with Olivier Martin that attempts to measure "how far from birationally isomorphic" two given varieties X and Y of the same dimension may be. The idea is to study the minimal complexity of correspondences between them. Besides presenting a few results, I will discuss some open problems and directions for further work.
Farbod Shokrieh:
Title: Arakelov invariants and non-archimedean geometry
Abstract: I will describe some connections between arithmetic/Arakelov geometry and non-archimedean/tropical geometry. The interplay arises from the study of analytic invariants on degenerating families of curves and abelian varieties, as well as the theory of heights of abelian varieties. (Based on recent and ongoing projects with Robin de Jong and Robert Wilms.)
Chiara Damiolini:
Title: Parahoric Bruhat-Tits from Galois covers
Abstract: Moduli spaces of vector bundles over an algebraic curve X has been a central focus in algebraic geometry, with numerous avenues of research also in adjacent fields. A natural generalization can be obtained by replacing vector bundles with G-torsors, for G an algebraic group. In today's talk I will focus on the case when G takes the form of a parahoric Bruhat-Tits group. These are groups defined over X which are generically reductive, and that display specific "parahoric" behaviors at finitely many points on X. In particular I will discuss how, under appropriate conditions, these groups arise from decorated principal bundles on Galois coverings of X. This description was first developed for generically split parahoric Bruhat--Tits groups by Balaji and Seshadri. I will present an extension of their result to a much more general class of groups. This is based on joint work with J. Hong.
Alexander Perry:
Title: The period-index conjecture for abelian threefolds
Abstract: The period-index conjecture asks for a precise bound on one measure of complexity of a Brauer class (its index) in terms of another (its period). I will discuss joint work with James Hotchkiss which proves this conjecture for Brauer classes on abelian threefolds.
Aaron Landesman:
Title: Canonical representations of surface groups
Abstract: For $\Sigma_{g,n}$ a genus $g$ surface with $n$ punctures, we study the character variety parameterizing representations of $\pi_1(\Sigma_{g,n})$. This character variety has a natural action of the fundamental group of the moduli space of curves. In joint work with Josh Lam and Daniel Litt, we aim to describe the points with finite orbit under this action, which we call canonical representations.
Emelie Arvidsson:
Title: The singularities of the minimal model program in positive characteristic
Abstract: My talk will be about the cohomological properties of the singularities that appear in the context of the Minimal Model Program. In particular, we will see that these singularities behave differently over the complex numbers and over an algebraically closed field of positive characteristic. I will discuss some consequences this pathological characteristic p behavior has, in particular when trying to formulate a moduli theory for varieties of general type in positive characteristics. In the end of my talk, we will focus on dimension three. We will see some positive results and discuss how these results can be applied to the moduli theory of stable surfaces in characteristic $p>5$. This talk will be based on joint work with F. Bernasconi and Zs. Patakfalvi.
Brendan Hassett:
Title: Rationality criteria for cubic hypersurfaces
Abstract: An algebraic variety is rational if it admits a birational parametrization by affine space. We still do not know which cubic fourfolds are rational or how to prove that any are not rational! We survey recent results - using classical geometry, Hodge theory, and newer techniques like derived categories - as well as concrete open questions.