Titles and abstracts

Irene Bouw: Computing the Weil representation of superelliptic curves

Abstract: Let Y be a curve with potential good reduction to characteristic p>0. The local Galois representation of Y is determined by a Weil representation. We show how to determine this Weil representation from the stable reduction of Y. In the case of superelliptic curves, we make this explicit.   This is joint work with D.K. Do and Stefan Wewers.

Tony Yue Yu: Mirror structure constants via non-archimedean analytic disks

Abstract: For any smooth affine log Calabi-Yau variety U, we construct the structure constants of the mirror algebra to U via counts of non-archimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, I will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.

Julia Hartmann: Local-global principles for homogeneous spaces over arithmetic function fields

Abstract: Classically, local-global principles for rational points on homogeneous spaces under linear algebraic groups are studied over number fields and function fields of curves over finite fields. In this talk, we consider analogous questions for function fields of curves over complete discretely valued fields. These fields naturally admit several collections of overfields with respect to which one can study local-global principles. These are accessible using methods that come from patching and are related to formal and rigid geometry. (Joint work with J.L.-Colliot-Thélène, D. Harbater, D. Krashen, R. Parimala, and V. Suresh. )

Eric Riedl: Non-free curves and Geometric Manin's Conjecture

Abstract: One important way to study a variety is to study the curves on it. Free curves, those whose restricted tangent bundle has vanishing H^1, are particularly nice. They are smooth points of the space Hom(B,X) of morphisms from B to X, and they lie on a unique component of Hom(B,X) of the expected dimension. Nonfree curves, on the other hand, are more mysterious, and the components of Hom(B,X) consisting entirely of nonfree curves are particularly challenging to study. In this talk, we provide a geometric classification of the ways that a family of curves can be nonfree. In the more general setting of a Fano fibration, we show that these curves come from a-covers of X, and we show that the set of such a-covers up to equivalence form a bounded family. This proves the first of Batyrev's heuristics that make up Geometric Manin's Conjecture for Fano fibrations over arbitrary base curves and provides strong evidence for Lehmann, Sengupta and Tanimoto's proposed characterization of the thin set in Manin's Conjecture. This is joint work with Brian Lehmann and Sho Tanimoto.

Carlos Simpson: Nonabelian mixed Hodge structures

Abstract: With Ludmil Katzarkov and Tony Pantev in the 1990's, we defined and investigated a notion of nonabelian mixed Hodge structures. This is related to the Morgan-Hain MHS on rational homotopy and the Malcev completion of the fundamental group, and to Hain's MHS on the relative completion and the MHS on the local ring of the character variety in work with Eyssidieux. These generalize to schematic homotopy types in work of Katzarkov-Pantev-Toen and Pridham. After talking about these structures, we'll look at weights that come from noncompactness of the base variety and how these relate to parabolic structures.

John Pardon: Counting curves in Calabi--Yau threefolds

Abstract: Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety.  There are many such compactifications, hence many different enumerative invariants (Gromov--Witten, Donaldson--Thomas, Gopakumar--Vafa, Pandharipande--Thomas, . . .).  I will make the case for studying instead a "universal" enumerative invariant which takes values in a certain Grothendieck group of 1-cycles.  It is often the case with such "universal" constructions that the resulting group is essentially uncomputable.  But in this case, the cluster formalism of Ionel and Parker gives some nontrivial computations for Calabi--Yau threefolds.  As a result, we hope to reduce conjectural identities between enumerative invariants (e.g. the MNOP conjecture) to some simple special cases (of "local curves").  This is work in progress.

Will Sawin: Quantitative ell-adic sheaf theory

Abstract: Sheaf cohomology is a powerful tool both in algebraic geometry and its applications to other fields. Often, one wants to prove bounds for the dimension of sheaf cohomology groups. Katz gave bounds for the dimension of the étale cohomology groups of a variety in terms of its defining equations (degree, number of equations, number of variables). But the utility of sheaf cohomology arises less from the ability to compute the cohomology of varieties and more from the toolbox of functors that let us construct new sheaves from old, which we often apply in quite complicated sequences. In joint work with Arthur Forey, Javier Fresán, and Emmanuel Kowalski, we prove bounds for the dimensions of étale cohomology groups which are compatible with the six functors formalism (and other functors besides) in the sense that we define the "complexity" of a sheaf and control how much the complexity can grow when we apply one of these operations. The pre-talk will explain how étale cohomology groups, and bounds for their dimensions, are useful in concrete situations.