(Ellenberg talk) The Ceresa cycle is an algebraic cycle canonically attached to a curve C, which appears in an intriguing variety of contexts; its height can sometimes be interpreted as a special value, the vanishing of its cycle class is related to the Galois action on the nilpotent fundamental group, it vanishes on hyperelliptic curves, etc. In practice it is not easy to compute, and we do not in fact know an explicit non-hyperelliptic curve whose Ceresa class vanishes. We will discuss a definition of the Ceresa class for a tropical curve, explain how to compute it in certain simple cases, and describe progress towards understanding whether it is possible for the Ceresa class of a non-hyperelliptic tropical curve to vanish. The tropical Ceresa class sits at the interface of algebraic geometry, graph theory (because a tropical curve is more or less a metric graph), and topology: for we can also frame the tropical Ceresa class as an entity governed by the mapping class group, and in particular by the question of when a product of commuting Dehn twists can commute with a hyperelliptic involution in the quotient of the mapping class group by the Johnson kernel. (Joint work with Daniel Corey and Wanlin Li.)

(Florence talk) In this talk, I will discuss a recent notion of smoothness for profinite groups, introduced by Charles De Clercq and myself. I will try to avoid technicalities, and present the ideas as simply as possible, in terms of equivariant representations and affine spaces, over Witt vectors. We will see how usual Kummer theory, appropriately axiomatized, implies very general lifting statements for these objects.

(Bakker talk) The cohomology groups of complex algebraic varieties come equipped with a powerful but intrinsically analytic invariant called a Hodge structure. Hodge structures of certain very special algebraic varieties are nonetheless parametrized by algebraic varieties, and while this leads to many important applications in algebraic and arithmetic geometry it fails badly in general. Joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman remedies this failure by showing that parameter spaces of Hodge structures always admit "tame" analytic structures in a sense made precise using ideas from model theory. A salient feature of the resulting tame analytic geometry is that it allows for the local flexibility of the full analytic category while preserving the global behavior of the algebraic category. In this talk I will explain this perspective as well as some important applications, including an easy proof of a celebrated theorem of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and the resolution of a longstanding conjecture of Griffiths on the quasiprojectivity of the images of period maps.

(Corwin talk) If a variety has no p-adic points for some prime p, then it has no rational points, and this is a standard way to prove non-existence of rational points. The Hasse or Local-Global Principle asks the converse, whether a variety with p-adic points for every p must have a rational point. Lind, Reichardt, and later Selmer found counterexamples to the Hasse principle in the 1940's, which they showed using reciprocity laws. In 1971, Manin unified their methods into a single "obstruction" to the local-global principle using class field theory, via the Brauer group, and Skorobogatov extended this further in the 90's into what is known as the etale-Brauer obstruction to the local-global principle. In 2009, Poonen constructed a smooth proper variety with no rational points, whose lack of rational points is not explained even by the etale-Brauer obstruction, and since then, Harpaz-Schlank reinterpreted the etale-Brauer obstruction in terms of etale homotopy theory. In this talk, we review the above and then discuss recent work of Schlank and the speaker using the topological perspective of Harpaz-Schlank to better understand Poonen's counterexample.

(Bhaskar talk) Any central simple algebra A over a field K is a form of a matrix algebra. Further A/K comes equipped with a reduced norm map which is obtained by twisting the determinant function. Every element in the commutator subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A), the group of reduced norm one elements of A. Whether the reverse inclusion holds was formulated as a question in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*]. Platonov negatively settled the Tannaka-Artin question by giving a counter example over a cohomological dimension (cd) 4 base field. In the same paper however, the triviality of SK_1(A) was shown for all algebras over cd at most 2 fields. In this talk, we investigate the situation for l-torsion algebras over a class of cd 3 fields of some arithmetic flavour, namely function fields of p-adic curves where l is any prime not equal to p. We partially answer a question of Suslin by proving the triviality of the reduced Whitehead group for these algebras. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields.

(Burgos talk) The height pairing between algebraic cycles over global fields is an important arithmetic invariant. It can be written as a sum of local contributions, one for each place of the ground field. Following Hain, the Archimedean components of the height pairing can be interpreted in terms of biextensions of mixed Hodge structures. In this talk we will explore how to extend the Archimedean contribution of the height pairing to higher cycles in the Bloch complex and interpret it as an invariant associated to a mixed Hodge structure. This is joint work with S. Goswami and G. Pearlstein.

(Florence talk) Let X be a smooth projective variety over a field F. Automorphisms of X are then represented by a group scheme over F, denoted by Aut(X). Conversely, given an algebraic group G over F, it is natural to ask whether G=Aut(X), for some smooth projective F-variety X. This question is the topic of active current investigation. Using analytical techniques, Lombardo and Maffei recently showed the following. Assume that G=A is an abelian variety, and that F=C is the field of complex numbers. Then, A=Aut(X) for some smooth projective X/F, if and only if A/F has finitely many group automorphisms. Their result was extended to all algebraically closed fields F by Blanc and Brion. In this talk, I will extend it further, to the case of an arbitrary field F. Note that the same question, when G/F is a linear algebraic group, is wide open.

(Besser talk) Quadratic Chabauty is a recent method for finding integral or rational points on certain curves. In particular, the version of the theory developed by Balakrishnan and Dogra was recently used successfully to find the rational points on a certain modular curve related with a conjecture of Serre. In this talk I will report on oint work in progress with Steffen Mueller and Padmavarthi Srinivasan, which simplifies the method significantly using the theory of p-adic height pairing on abelian varieties. The classical theory of real valued heights was recast, starting with the work of Zhang in the 90's, as a theory of "adelic" metrics on line bundles. Using techniques from the theories of iterated integrals developed by Coleman and Vologodsky and the theory of line bundles on abelian varieties we can give an analogous theory of p-adic adelic metrics, leading to p-adic heights. Quadratic Chabauty is shown to come from a situation where a non-trivial line bundle on the Jacobian of a curve X pulls back to a trivial line bundle on X. The resulting computations for finding rational points are far simpler than previously known. Perhaps the most surprising aspect of the theory is the way that the recent understanding of Vologodsky integration by Katz and Litt gives a totally new way of understanding the contributions to the height at primes of bad reduction. In this talk I will largely treat p-adic interated integrals as a black box so that relevant background consists mostly of algebraic geometry and a bit of combinatorics.

(Yi talk) Hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. In my joint work with Masoud Kamgarpour, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric sheaves, thus confirm the geometric Langlands conjecture for hypergeometrics. In this talk, I will introduce the key method we use called rigid automorphic data, due to Zhiwei Yun. The definition of automorphic data for tame hypergeometrics involves the mirabolic subgroup, while in the wild case, Moy-Prasad subgroups of principal parahorics intervene. I would also mention the generalization to classical groups in the wild case, which is part of my recent work with Masoud Kamgarpour and Daxin Xu.

(Chen talk) It is well-known that each cohomology group of compact K\"ahler manifold carries a Hodge structure. If we consider a degeneration of compact K\”ahler manifolds over a disk then it is natural to ask how the Hodge structures of smooth fibers degenerate. When the degeneration only allows a reduced singular fiber with simple normal crossings (i.e. semistable), Steenbrink constructed the limit of Hodge structure algebraically. A consequence of the existence of the limit of Hodge structure is the local invariant cycle theorem: the cohomology classes invariant under monodromy action come from the cohomology classes of the total space. In this talk, I will try to explain a method using D-modules to construct the limit of Hodge structure even when the degeneration is not semistable.