Mini-courses (3 talks each)

Ignacio Barros: Cycles on moduli spaces of K3 surfaces 

I will use the moduli space of polarized K3 surfaces as a guiding example of a locally symmetric variety of orthogonal type. I will discuss aspects of the algebraic cycle structure of the moduli space and the connection with vector-valued modular forms. I will point out similarities and differences with the case of locally symmetric varieties of symplectic type (A_g), and discuss open questions.

Melody Chan: Tropicalizations of locally symmetric varieties

The theory of toroidal compactifications of locally symmetric varieties, developed by Ash--Mumford--Rapoport--Tai 50 years ago, lays the foundations for a rigorous theory of tropicalization of such varieties, which we have developed in joint work with Eran Assaf, Madeline Brandt, Juliette Bruce, and Raluca Vlad. This tropical theory has allowed for discovery of new structures in the cohomology of moduli spaces and arithmetic groups. In these lectures, I will focus on foundations and on applications to cohomology of A_g and GL_n(Z), touching on joint work with Francis Brown, Soren Galatius, and Sam Payne. Raluca Vlad will discuss some further aspects and related developments in her talk.

Research Talks

Michael Borinsky: On the Euler characteristic of the commutative graph complex

In 1993, Maxim Kontsevich introduced a family of complexes generated by graphs. These complexes compute, among other things, the rational cohomology of moduli spaces of (tropical) curves and automorphism groups of free groups. Recently, they also have been shown to provide new insights into the cohomology of the handle body group, and the general linear groups over the integers. I will discuss the commutative graph complex, the simplest example in this family. Asymptotic computations of its Euler characteristic show that its homology grows rapidly in genus. In particular, this implies the existence of large amounts of unexplained unstable homology. I review the definition of this graph complex and illustrate the asymptotic results by discussing the implications for the unstable cohomology of the moduli space curves and similar objects.

Angie Cueto: Tritangent planes to space sextic curves: a tropical viewpoint

A classical result due to Clebsch from the mid-nineteenth century confirms that every complex space sextic curve (given as an intersection of a quadric and a cubic surface in projective 3-space) has exactly 120 tritangent planes. In this talk we will show how to use combinatorial methods arising from tropical geometry to revisit this classical problem and perform the analogous count over the reals and extensions thereof. This is joint work with Yoav Len, Hannah Markwig and Yue Ren (arXiv:2512.24277).

David Holmes: Pure extension of the universal polarising line bundle on a family of abelian varieties

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.

Robin de Jong: The universal tropical theta function

This is based on joint work with Ana Botero, Jose Burgos and David Holmes. We continue the discussion started in the talk by David Holmes. We consider log compactifications of the moduli space of ppav with theta level structure (i.e., with given theta divisor), and of the universal abelian variety over it. We revisit the Delaunay and mixed Delaunay—Voronoi cone decompositions in this setting, and exhibit the tropical theta function with theta characteristics as an admissible polarization function with respect to these decompositions. If time permits we discuss applications.

Michael Joswig: 121 patchworked curves of degree seven

The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle $7\cdot\Delta_2$ for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve; this settles a question raised by Itenberg and Viro (1996). Co-authored with Zoe Geiselmann, Lars Kastner, Konrad Mundinger, Sebastian Pokutta, Christoph Spiegel, Marcel Wack, Max Zimmer.

Johannes Rau: Welschinger-Witt invariants

An important problem in enumerative geometry is counting rational curves that interpolate a configuration of points in P^2, leading to Gromov-Witten invariants (over algebraically closed fields) and Welschinger invariants (over the real numbers). Recently, Kass, Levine, Solomon, and Wickelgren constructed "quadratic" invariants that work over an (almost) arbitrary base field. The small “catch” here is that these new invariants are no longer numbers, but quadratic forms whose rank and signature recover the previously mentioned invariants. In a current work with Erwan Brugallé and Kirsten Wickelgren, we study these invariants in the framework of so-called Witt-invariants and show that, conversely, the quadratic invariants can be recovered from Gromov-Witten and Welschinger invariants. In my talk, I want to give an introduction to this topic (and its extension to rational del Pezzo surfaces).

Kris Shaw: Cohomologically tropical varieties  

This talk asks which tropicalisations of subvarieties of the torus know the cohomology of the original variety? A motivating example are linear embeddings of complements of hyperplane arrangements. We prove that the tropicalisation knows the cohomology of the variety in a strong sense if and only if it satisfies local tropical Poincaré duality and the original variety is so-called “wunderschön”. Following the work of Itenberg, Katzarkov, Mikhalkin, and Zharkov, we can obtain information about the mixed Hodge structure of a family of varieties from its tropicalisation when it locally satisfies the two conditions above. 

This is joint work with Edvard Aksnes, Omid Amini, and Matthieu Piquerez

Filippo Viviani: On universal compactified Jacobians

We explain how to classify all the universal compactified Jacobians over the moduli of stable pointed curves (joint work with M. Fava, N. Pagani). Then we will sketch how to relate universal compactified Jacobians to the universal logarithmic and tropical Picard (joint work in progress with M. Melo, S. Molcho, M. Ulirsch, J. Wise).

Raluca Vlad: Tropicalizations of locally symmetric varieties in type A

In type A, tropicalizations of locally symmetric varieties are combinatorial objects arising from polyhedral decompositions of positive definite cones of Hermitian matrices. I will discuss applications of these tropicalizations to the cohomology of moduli spaces and arithmetic groups. More precisely, I will describe a Hopf algebra structure on the cohomology of moduli of abelian varieties with CM. I will then discuss dimension growth results for this algebra, obtained via a special class of differential forms coming from the stable cohomology of general linear groups. This talk is based on joint work with Assaf, Brandt, Bruce, and Chan, as well as work in progress joint with Brown and Chan.

Jonathan Wise: Two case studies in bounded monodromy

The logarithmic Picard group compactifies the Picard group of a degenerating algebraic curve in a canonical way.  It was originally proposed by Illusie to parameterize torsors under the logarithmic multiplicative group, but this definition turned out to require amendment by an additional tropical condition, now known as bounded monodromy.  I will describe how this condition arises in the logarithmic Picard group, and how a much more mysterious analogue appears in other moduli problems associated with degenerating linear series.