## Bubbling symplectic structures on moduli, a talk by Tony Pantev

I will describe a new geometric method for constructing and controlling shifted symplectic structures on the moduli of vector bundles along the fibers of a degenerating family of Calabi-Yau varieties. The method utilizes bubbling modifications of the boundaries of limiting moduli spaces to extend the symplectic structure on the general fiber to a relative symplectic structure defined on the whole family. As a proof of concept we show that this produces a universal relative symplectic structure on the moduli of Gieseker Higgs bundles along a semistable degeneration of curves. We also check that the construction works globally over the moduli stack of stable curves and show that the Hitchin map has the expected behavior in the limit. This is a joint work with Oren Ben-Bassat and Sourav Das.

## Density of Hasse failures for diagonal affine cubic surfaces, a talk by Vladimir Mitankin

In this talk we shall apply the integral version of the Brauer-Manin obstruction to construct the first examples of such failures not explained by local conditions in the setting of affine diagonal ternary cubics. We will then explore in three different natural ways how such failures are distributed across the family of affine diagonal ternary cubics.

## Tropical structures in sandpile model, talk by Mikhail Shkolnikov, IMI-BAS

I will tell how tropical curves arise in the scaling limit of the sandpile model in the vicinity of the maximal stable state and explain two major consequences inspired by this fact. The first one is that there is a continuous model for self-organized criticality, the only known model of a kind, defined in the realm of tropical geometry. The second is that the totality of recurrent states in the original sandpile model, the sandpile group, approximates a continuous group, a tropical Abelian variety, which is functorial with respect to inclusions of domains, allowing to compute its scaling limit as a space of circle-valued harmonic functions on the whole lattice.

## Invariant theory, homogeneous projective varieties, and momentum maps, course by Valdemar Tsanov

After introducing the basic notions, I will derive some properties of momentum images related to fundamental forms and osculating varieties, as well as a lower bound on the minimal positive degree of a homogeneous invariant, derived using secant varieties. At the end I will present a class of homogeneous projective varieties, characterized by a special property of their secant varieties, where the relations between the above three concepts take a particularly pristine form.

## Introduction to Projective Structures and Opers, course by Peter Dalakov

A complex projective structure on a Riemann surface is determined by an atlas, whose transition functions are Moebius (fractional-linear) transformations. There are multiple descriptions of these structures: as certain flat PGL_2-bundles, as Sturm-Liouville operators, as holomorphic connections on the (first) jet bundle of the dual of a theta-characteristic, etc. This mini-course is an introduction to the fundamentals of projective structures, accessible to students and non-specialists. We will also explore links to some classical geometric objects (such as quadratic differentials and Schwarzian derivatives), as well as some generalisations (G-opers) introduced by Beilinson and Drinfeld.

## About a generalisation of Sylvester’s law of inertia, talk by Stéphanie Cupit-Foutou

Sylvester’s law of inertia can be formulated in terms of group actions when considering real linear groups acting on real quadratic forms by base change. After reviewing this celebrated result from this perspective, I will give a generalisation of it in the setting of so-called spherical varieties (a class of complex varieties including flag varieties, toric varieties, symmetric spaces, etc.). This is a joint work with D. Timashev