David Hansen (University of Notre Dame)
Title: Families of perverse sheaves
Abstract: Given a morphism of varieties X -> Y, is there a good definition of a “family of perverse sheaves Fy on the fibers Xy” as y runs through the closed points of Y? It turns out that such a definition has been lurking in the geometric Langlands literature, albeit in an obscured form, for almost twenty years. In this talk, I’ll explain this definition and highlight the many remarkable properties enjoyed by families of perverse sheaves. I’ll also discuss some applications and some open questions.
June Huh (IAS/Princeton University)
Title: Lorentzian polynomials
Abstract: I will give a gentle overview of my work with Petter Brändén on Lorentzian polynomials (https://arxiv.org/abs/1902.03719), which link continuous convex analysis and discrete convex analysis via tropical geometry. The class contains homogeneous stable polynomials, volume polynomials of convex bodies and projective varieties, as well as some partition functions considered in statistical physics. No specific background will be needed to enjoy the talk.
Kiran Kedlaya (UCSD/IAS)
Title: Étale and crystalline companions
Abstract: Deligne’s “Weil II” paper includes a far-reaching conjecture to the effect that for a smooth variety on a finite field of characteristic p, for any prime l distinct from p, l-adic representations of the étale fundamental group do not occur in isolation: they always exist in compatible families that vary across l, including a somewhat more mysterious counterpart for l = p (the “petit camarade cristallin”). We explain in more detail what this all means, describe some key ingredients in the proof (particularly the role of the Langlands correspondence for function fields), and mention some concrete applications.
David Nadler (UC Berkeley)
Title: Traces, characters, and… loops?
Abstract: Traces of matrices, and more generally characters of representations, are indispensable tools in algebra. This will be an introductory talk about their universal nature from the perspective of topology. We will see how this perspective explains and predicts various current research streams.
Bianca Viray (University of Washington)
Title: Isolated points on modular curves
Abstract: Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that all but finitely many algebraic points on a curve arise in families parametrized by P1 or positive rank abelian varieties; we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on X1(n) push down to isolated points on a modular curve whose level is bounded by a constant that depends only on the j-invariant of the isolated point. This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.