Abstract: In the course of constructing the Langlands correspondence for GL(2) over a function field, Drinfeld discovered a surprising fact about the interaction between étale fundamental groups and products of schemes in characteristic p. We state this result, describe a new approach to it involving a generalization to perfectoid spaces, and mention an application in p-adic Hodge theory (from joint work with Carter and Zabradi).

Abstract: The Sato–Tate conjecture, originally stated for elliptic curves on 1963, predicts the equidistribution of the normalized Frobenius traces with respect to the Sato–Tate measure, given by the pushforward of the Haar measure on SU(2). We would like to work on an analogous question for abelian varieties of dimension g > 1; the generalized Sato–Tate conjecture, introduced by Serre, which predicts the equidistribution on a certain compact Lie group: the Sato–Tate group. In 1966, Serre presented remarkable links between the Mumford–Tate group and the Sato–Tate group. Thus, the algebraic Sato–Tate group appears as an intermediate group between the Mumford–Tate group and the Sato–Tate group. Indeed, if the algebraic Sato–Tate conjecture holds (which is a refinement of the Mumford–Tate conjecture) for some particular abelian variety A of dimension g > 1, we can obtain the Sato–Tate group and try to deduce some new instances of the generalized Sato–Tate conjecture. The main goal of this talk is to present new results in the direction of the algebraic Sato–Tate conjecture, building on the previous work of Serre, Kedlaya and Banaszak.

Abstract: Does a smooth proper variety in positive characteristic know the Hodge number of its liftings? The answer is "of course not". However, it's not that easy to come up with a counter-example. In this talk, I will first introduce the background of this problem. Then I shall discuss some obvious constraints of constructing a counter-example. Lastly I will present such a counter-example and state a further question.

Abstract: This talk will focus on geometric realizations of vertex algebras. The Virasoro uniformization provides an incarnation of the Virasoro algebra in the tangent space of the Hodge line bundle on moduli of curves with marked points and local coordinates. This allows to assign to certain representations of the Virasoro algebra a sheaf on moduli of curves together with a projective connection. After reviewing some facts on vertex algebras and moduli of curves, I will discuss the sheaves on moduli of stable curves obtained from coinvariants of modules over conformal vertex algebras, and identify their projective connection. This is joint work with Chiara Damiolini and Angela Gibney.

Abstract: Computing volumes of moduli spaces has significance in many fields. For instance, Witten's conjecture regarding intersection numbers on the Deligne–Mumford moduli space of stable Riemann surfaces has a fascinating connection to the Weil–Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The initial two other proofs of Witten's conjecture by Kontsevich and by Okounkov–Pandharipande also used various ideas in combinatorial ribbon graphs, Gromov–Witten theory, and Hurwitz theory. In this talk I will introduce an analogue of Witten's intersection numbers on moduli spaces of Abelian differentials to compute the Masur–Veech volumes induced by the flat metric associated with Abelian differentials. This is joint work with Martin Moeller, Adrien Sauvaget, and Don Zagier (arXiv:1901.01785).

Abstract: A significant step in the Pila–Zannier approach to the André–Oort conjecture is a geometric transcendence result for the uniformization map of modular curves. I will discuss joint work with François Loeser. We prove an analogue of this result in non-archimedean geometry, namely for the uniformization of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of PGL(2,ℚₚ) contained in PGL(2,ℚ). In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.

Abstract: Kazhdan–Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids. I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Examples involving the geometry of hyperplane arrangements will be given throughout. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.

Abstract: The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin's conjecture and on its motivic analog: the latter predicts the behavior of moduli spaces of curves of large degree on some algebraic varieties. It may be formulated in terms of the generating series of the classes of these moduli spaces in the Grothendieck ring, called the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition and how this can be used to give a proof of the motivic version of Manin's conjecture for equivariant compactifications of vector groups.

Abstract: To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of "Macaulayfying" a given Noetherian scheme X. Under various assumptions Faltings, Brodmann, and Kawasaki built the sought Cohen–Macaulay modifications without preserving the locus where X is already Cohen–Macaulay. We will discuss an approach that overcomes this difficulty and hence completes Faltings' program.

Abstract: The p-curvature conjecture is an analogue of the Hasse principle for arithmetic differential equations. I will discuss this conjecture in the context of families of algebraic varieties, and among other things, demonstrate a proof of the p-curvature conjecture for rank 2 vector bundles on generic curves. This is based on joint work with Anand Patel and Junho Peter Whang.

Abstract: The Langlands and Fontaine–Mazur conjectures in number theory describe when an automorphic representation f arises geometrically, meaning that there is a smooth projective variety X, or more generally a Chow motive M in the cohomology of X, such that there is an equality of L-functions L(M,s) = L(f,s). We explicitly describe how to produce such a variety X and Chow motive M in the case of powers of certain automorphic representations, called algebraic Hecke characters. This is joint work with J. Lang.

Abstract: Achar recently introduced a “nearby cycles formalism” in the framework of chain complexes of parity sheaves. In this talk, I'll explain joint work with Achar in which we compute the output of this functor in two related settings. The first is affine space, stratified by the action of a torus, and the second is the global Schubert variety associated to the first fundamental coweight of the group PGLₙ. The latter is a parity-sheaf analogue of Gaitsgory's central sheaf construction. Much of the talk will focus on the representation theoretic context for the computation.

Abstract: Motivation. Isogenies with a given polarized abelian variety give moduli points in a Hecke orbit. In characteristic zero any Hecke orbit is dense in this moduli space. What can be said about the Zariski closure of a Hecke orbit in positive characteristic?

Result. In joint work with Ching-Li Chai we prove a conjecture (1995) that predicted the answer to this question. One of basic tools is the notion of foliations. The main focus of the talk will be on describing properties of these foliations, rather well-understood by now.