David Holmes Title: Pure extension of the theta divisor on a family of abelian varieties Abstract A symmetric line bundle L on an abelian variety has the useful property that pulling it back along the `multiplication by n’ map is the same as raising it to the n-th tensor power; we say L has pure weight 2. In particular this holds for a symmetric ample line bundle representing a polarisation. Such a line bundle extends naturally to an ample line bundle on a toroidal compactification of the universal abelian variety over the moduli space, but this extension no longer satisfies the purity property. We will show how to correct this using tropical theta functions and adelic- or b-divisors, and discuss consequences for extending the theta divisor. This is joint work with Ana Maria Botero, Jose Burgos, and Robin de Jong.

Lycka Drakengren Title: The Taylor expansion of the Torelli map and applications to intersection theory Abstract: Using results of Hu, Norton '18 and Yamada '80, we can deduce the full Taylor expansions of the Torelli map and the Prym map at the boundary in terms of plumbing coordinates. We use the Taylor expansion to determine the scheme structure for the fiber product of the Torelli map $t: M_g^{ct} \rightarrow A_g$ and the product map $A_{g_1} \times ... \times A_{g_k} \rightarrow A_g$, where $g_1 + ... + g_k = g$. This allows us to compute the class $t^*[A_{g_1} \times ... \times A_{g_k}]$ using excess intersection theory in arxiv:2601.04353. The Taylor expansions can also be used for studying the scheme structure for the self-fiber product of the Torelli map, as well as for the fiber product of the product map and Beauville's extended Prym map.

Sam Molcho Title: Fourier-Mukai transforms and log geometry. Abstract: I will review progress in our understanding of the intersection theory of degenerating families of abelian schemes, namely the compactified Jacobians and Mumford's canonical partial compactification of the moduli space of principally polarized abelian varieties, parametrizing degenerations of torus rank at most one. I will state the main results and attempt to explain our approach to these problems, which combines Arinkin's perspective on how Mukai's derived autoequivalence of the derived category of a ppav degenerates with the logarithmic resolutions of the fundamental geometric structures enjoyed by an abelian scheme. The talk will be based on joint work with Bae-Pixton and joint work in progress with Bae-Feusi-Iribar Lopez.

Dhruv Ranganathan Title: Lines in the plane Abstract: I will discuss intersection theory on the moduli space of lines in the plane — the 2-dimensional generalization of stable pointed rational curves. I will present a few different paths to compact moduli spaces of these objects, touching on work of many others, including KSBA, Kapranov, Lafforgue, and Kennedy-Hunt. One path leads to a space with a virtual fundamental class, a beautiful stratification coming from matroid theory, and a good notion of tautological classes. I’ll explain how tautological integrals on these spaces can be recursively determined by certain “virtually rigid” evaluations, and a strategy to compute the virtually rigid ones. Based on work with Abramovich and Pandharipande.

Junliang Shen Title: A tale of two Fourier transforms: universal Jacobians and hyper-Kähler varieties Abstract: Fourier transforms for abelian varieties have been studied for decades since the work of Beauville in the 80s. The purpose of this talk is to explain how these ideas can be applied to the study of two different types of geometric objects: universal compactified Jacobians and hyper-Kähler varieties. For universal compactified Jacobians, Fourier transforms lead to the construction of the intrinsic cohomology ring; for hyper-Kähler varieties of K3[n]-type, Fourier transforms lead to a proof of the multiplicative Orlov conjecture for homological motives. I will explain the proofs in both settings, which rely on completely different geometric ingredients but are parallel in a certain sense. Further open questions will be discussed if time permits. This talk is based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin. Best, Junliang

Davesh Maulik Title: GW/PT correspondence for toric pairs Abstract: I will discuss work with Dhruv Ranganathan, in which we prove the primary and descendent GW/PT correspondence for snc pairs (X,D) where X is a toric threefold and D is a union of toric divisors. Even when D is empty, where everything has been known for a while, I’ll try to explain how the log approach gives a new proof and even strengthens some of the older results.

Aitor Iribar Lopez Title: Weight calculations and the tautological ring Abstract: Most questions about the intersection theory of the toroidal compactifications of A_g are unanswered. In this talk, I will explain how we can solve most of these for the simplest compactification given by torus rank 1 degenerations. The main technique is an extended theory of weights. We can compute these explicitly in the tautological ring, and then use Fourier transform to extend the results to the whole Chow ring. This is a continuation of S. Molcho's talk, and is based on joint work with Y. Bae, J. Feusi and S. Molcho

Short Talks

Irene Spelta Title: Families of G-curves and maximal monodromy Abstract: Maximality results for the monodromy of families of algebraic varieties are widely studied in algebraic geometry. In this talk, we study the monodromy of families of Galois coverings of curves. The natural action of the Galois group G on cohomology induces a decomposition of the associated variation of Hodge structures, and under a certain assumption on this decomposition, we prove that the algebraic monodromy group is maximal. As an application, we analyse positive-dimensional totally geodesic subvarieties of Ag arising from such families.

Olivier Taïbi Title: The low degree cohomology of compactifications of $A_g$ Abstract: I will report on joint work with Samir Canning and Dan Petersen determining which irreducible Galois representations can occur in the low degree cohomology of any smooth toroidal compactification of $A_g$ or $X_{g,s}$, the $s$-fold fiber product of the universal abelian variety.

Filippo Viviani TITLE: On the classification of modular compactification of the universal Jacobian ABSTRACT: I will first report on a joint work with M. Fava and N. Pagani in which we give a complete classification of all the modular compactifications of the universal Jacobian over the moduli stack of pointed stable curves. Then, I will discuss the relationship between these modular compactifications and the universal logarithmic and tropical Picard stacks (joint work, partially in progress, with M. Melo, S. Molcho, M. Ulirsch, J. Wise).

Dragos Oprea Title: On the elliptic genera of Quot schemes of zero dimensional quotients on curves Abstract: We study Quot schemes of rank zero quotients on smooth projective curves. For quotients of a vector bundle of even rank, we present a formula for the level 2 elliptic genus. We also propose a conjectural generalization to higher level elliptic genera and provide supporting evidence.

Orsola Tommasi Title: Perspectives on the stable cohomology of compactifications of A_g Abstract: By classical work of Borel, it is well known that the rational cohomology of the moduli space A_g of principally polarized abelian varieties of dimension g stabilizes in degree k<g, In this talk, I would like to review stability results on compactifications of A_g, such as the Satake compactification and toroidal compactifications.

Younghan Bae Title: Curve-Jacobian correspondence Abstract: By the Riemann-Roch theorem, when m> 2g-2, the m-th symmetric product of a smooth projective curve is a projective bundle over its Jacobian. It is natural to ask whether this relationship extends to families of curves that may have singularities. In this talk, I will describe a correspondence between algebraic cycles on the universal compactified Jacobians over the moduli space of stable curves and those on the moduli space of stable curves with additional markings. This correspondence allows one to transport structures between two sides. It brings together Fourier transforms, the P=C phenomenon, ring structures, logarithmic Abel-Jacobi theory, tautological relations, and many other aspects. This is a joint work in progress with A. Pixton.

Lunch

Samuel Grushevsky Title: Non-trivial cohomology of strata of differentials Abstract: A stratum of differentials is the moduli space of complex curves together with a meromorphic form with prescribed multiplicities of zeros and poles. The strata are phase spaces of an action of SL(2, R) and thus the central object of study in Teichmuller dynamics. On the other hand, they give natural high codimension subvarieties of the moduli of curves with marked points. We construct various non-trivial and non-tautological cohomology classes on stata of differentials and their compactifications. Joint work with Prabhat Devkota

Phil Engel Title: Matroids and the integral Hodge conjecture Abstract: Associated to any regular matroid of rank g on k elements, one can associate a multivariable semistable degeneration of principally polarized abelian g-folds over a k-dimensional base. I will discuss joint work with de Gaay Fortman and Schreieder, proving that a combinatorial invariant of the matroid obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. Corollaries include the failure of the integral Hodge conjecture for abelian varieties of dimension ≥ 4 and the stably irrationality of very general cubic threefolds.