• All talks take place in Room 1324 East Hall (aka the Psych Auditorium in East Hall; see https://goo.gl/maps/qGpA5BijteE19aBS9 for a map).
• During the breaks, refreshments will be available in the upper atrium in the Math wing of East Hall.
• The poster session will take place during the coffee break on Monday afternoon.

Titles and abstracts

• Aise Johan de Jong: Local Picard groups

We will discuss a local Lefschetz type theorem for Picard groups. In particular, we extend a theorem of Kollar on injectivity of the restriction map to the mixed characteristic case. The proofs use only classical commutative algebra.

• Hélène Esnault: Frobenius invariant Subloci of formal Lie groups of multiplicative type over an l-adic ring, and applications.

(All joint with Moritz Kerz.) One application of the understanding of those loci is the Hard Lefschetz theorem in rank 1 in positive characteristic. Another one is on the codimension of the cohomological subloci defined by the Mellin transform in special towers.

We shall review Deligne’s purity concept and the application of L. Lafforgue’s theorem to purity, then the notions defined in the title, then the method to find torsion points of those loci, then the applications.

• Wei Ho: Hessian constructions for genus one curves

The Hessian of a plane cubic curve is classically described using partial derivatives or polars. In this talk, we will revisit the cubic Hessian and several variants for other models of genus one curves, and highlight arithmetic applications, e.g., relating them to isogenies of a modular curve and constructing families of elliptic curves with isomorphic mod p representations.

• Daniel Huybrechts: Algebraic and arithmetic aspects of twistor spaces

• Daniel Krashen: Field patching, local-global principles and rationality

This talk will present a brief survey of local-global principles for torsors for algebraic groups over higher dimensional arithmetic fields via field patching techniques. In particular, I'll discuss new work which makes a connection between obstructions to such local-global principles and obstructions to rationality of algebraic groups.

• Max Lieblich: Topological reconstruction theorems

I will report on joint work in progress with János Kollár, Martin Olsson, and Will Sawin on topological reconstruction theorems for algebraic varieties.

• Jacob Lurie: Tamagawa Numbers in the Function Field Case

Let G be a connected semisimple algebraic group over a global field K, and let A denote the ring of adeles of K. Tamagawa observed that the locally compact group G(A) is equipped with a canonical translation-invariant measure. A celebrated conjecture of Weil asserts that if G is simply connected, then the measure of the quotient space G(A)/G(K) is equal to 1. When K is a number field, this conjecture was proven Kottwitz (following earlier work of Langlands and Lai). In these talks, I'll discuss joint work with Dennis Gaitsgory about the function field case, exploiting ideas from algebraic topology.

• Davesh Maulik: Topology of Higgs moduli spaces via abelian surfaces

In this talk, we study cases of the P=W conjecture for Higgs bundles on a curve, using techniques from compact hyperkahler geometry. This is joint work in progress with Mark de Cataldo and Junliang Shen.

• Martin Olsson: Degenerations of varieties and sections of vector bundles on moduli spaces.

I will explain some observations and examples illustrating how one can use stack-theoretic techniques to construct canonical sections of vector bundles in formal neighborhoods of boundary points in moduli with large stabilizer groups. This is related to Mumford's classical theory of the theta group for abelian varieties and algebraic theta functions.

• Bjorn Poonen: The local-global principle for stacky curves

For smooth projective curves of genus g over a number field, the local-global principle holds when g=0 and can fail for g=1, as has been known since the 1940s. Stacky curves, however, can have fractional genus. We construct stacky curves of genus 1/2 that violate the local-global principle, and show that 1/2 cannot be reduced. This is joint work with Manjul Bhargava.

• Rachel Pries: Infinite clutching systems and unlikely intersections with the Newton polygon stratification

Clutching morphisms have been important for proving many results about the moduli space of curves. In this work, we study clutching systems for moduli spaces of cyclic covers of the projective line and PEL-type Shimura varieties. We focus on the Kottwitz sets and Newton polygon stratification for the moduli p reduction of these moduli spaces. We prove that the Newton polygon stratification cooperates well with the clutching morphisms under certain compatibility conditions. As an application, we find infinitely many situations when a conjecture of Oort is true and when the Newton polygon stratification of the moduli space of abelian varieties has an unlikely intersection with the Torelli locus. This is joint work with Li, Mantovan, and Tang.

• Karen Smith: Non-Commutative Resolution of Singularities and Frobenius.

Consider a finitely generated commutative algebra R over a field K. Roughly speaking, a non-commutative resolution of singularities of Spec R is a (non-commutative) R-algebra A with finite global dimension, meaning that (like a commutative regular local ring), every module over A has a finite projective resolution. Typically, the algebra A has the form End(M) where M is some finitely generated R-module. The existence of a non-commutative resolution for a commutative ring R places strong conditions on R, such as rational singularities. In this talk, we discuss how in prime characteristic, the Frobenius can be used to construct non-commutative resolutions of nice enough rings. We conjecture that for a strongly F-regular ring R, End(F_*R) is a non-commutative resolution of R, where F_*R denotes R viewed as an R-module via restriction of scalars from Frobenius. We prove this conjecture when R is the coordinate ring of an affine toric variety. We also show that for toric rings, the ring of differential operators D(R) has finite global dimension (joint with Eleonore Faber and Greg Muller).

• Jason Starr: Symplectic invariants and rational points in positive characteristic.

Tsen's Theorem produces a rational point over a function field of a curve for every smooth complete intersection of type (d_1,...d_c) in projective n-space provided the Fano index i=n-(d_1+...+d_c) is positive. Is there more than one rational point? Zhiyu Tian, Runhong Zong and I prove "weak approximation" by rational points at all places of potentially good reduction if i>1 and if the characteristic p > max(d_1,...,d_c). This follows from a general theorem proving cohomology vanishing and separable uniruledness of Fano manifolds with cyclic Picard group whenever p is prime to certain Gromov-Witten invariants.

• Lenny Taelman: Derived equivalences of hyperkähler varieties

In this talk we consider auto-equivalences of the bounded derived category D(X) of coherent sheaves on a smooth projective complex variety X. By a result of Orlov, any such auto-equivalence induces an (ungraded) automorphism of the singular cohomology H(X,\Q). If X is a K3 surface, then work of Mukai, Orlov, Huybrechts, Macrì and Stellari completely describes the image of the map \rho_X : \Aut D(X) → Aut(H(X, \Q)). We will study the image of \rho_X for higher-dimensional hyperkähler varieties. An important tool is a certain Lie algebra acting on H(X, Q), introduced by Verbitsky, Looijenga and Lunts. We show that this Lie algebra is a derived invariant, and use this to study the image of \rho_X.

• Burt Totaro: Curves, K3 surfaces, Fano 3-folds

Which smooth projective curves are contained in some K3 surface? Which K3 surfaces are contained in some Fano 3-fold? These questions have been studied for about 40 years, with some striking advances in the past year or so.

• David Treumann: Symplectic, or mirrorical, look at the Fargues-Fontaine curve

Homological mirror symmetry describes Lagrangian Floer theory on a torus in terms of vector bundles on the Tate elliptic curve. A version of Lekili and Perutz's works "over Z[[t]]", where t is the Novikov parameter. I will review this story and describe a modified form of it, which is joint work with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on the torus. When the fiber of this sheaf of rings is perfectoid of characteristic p, and the holonomy around one of the circles in the torus is the pth power map, it is possible to specialize to t = 1, and the resulting theory there is described in terms of vector bundles on the equal-characteristic-version of the Fargues-Fontaine curve.

• Chenyang Xu: On moduli of K-stable Fano varieties

Family of Fano varieties usually doesn’t behave well unless extra conditions are posted. Inspired by the Kahler-Einstein problem, we now expect Fano varieties with K-polystability yield a good projective moduli space. In this talk, I will discuss the recent progress, using tools from higher dimensional algebraic geometry, on this problem.