# Abstracts

#### Jon Pål Hamre: Schubert coefficients of matroids (9:15-10:15, D34)

We introduce a set of matroid invariants called Schubert coefficients. To define and understand the Schubert coefficients we use tools from algebraic geometry such as Schubert calculus and toric geometry. The main goal of the talk is to prove the non-negativity of the Schubert coefficients of sparse paving matroids, and hopefully convince the audience that these are interesting matroid invariants.

#### Signe Lundqvist: A brief introduction to combinatorial rigidity theory (10:50-11:20, D34)

Rigidity theory is a branch of combinatorial geometry that concerns problems where one is given a geometric realisation of a combinatorial structure, and a notion of admissible motionsof the geometric realisation.

Graphs drawn in d-dimensional Euclidean space, where the admissible motions are those that preserve the distance between all vertices that are connected by an edge, is a classical example. Another example is polyhedral scenes in d-dimensional space, where the admissible motions are those that preserve the projection to (d-1)-dimensional space.

The goal is typically to characterise when geometric realisations of a combinatorial structure have admissible motions. As I will outline in my talk, such characterisations are often in terms of independent sets of a matroid.

#### Klara Stokes: Lifts of polytopes and other motions of incidence geometries (11:30-12:30, D34)

Given a drawing in the plane, how can we tell if it is a projection of a polyhedron? This classical problem has been investigated from many different viewpoints in the past. In this talk I will show how to model the problem as the space of motions of a certain graph of groups, and how the same model can be used to model a variety of problems in constraint geometry, including the Euler-Cauchy problem regarding the rigidity of polytopes with the facets realised in terms of rigid panels.

#### Louis Hainaut: Introduction to the moduli spaces of Riemann surfaces and their Deligne-Mumford compactification (14:00-14:30, D34)

This talk serves as an introduction to the talk of Lionel Lang. I will introduce the moduli spaces \(M_{g,n}\), which classify compact Riemann surfaces with a hyperbolic metric. I will then discuss their Deligne-Mumford compactification. In the remaining time I will introduce a tropical version of these moduli spaces, the tropical moduli spaces \(M_{g,n}^{trop}\).## Lionel Lang: Towards tropical compactifications of moduli spaces (14:40-15:40, D34)

In this talk, I want to promote the construction of tropical compactifications of moduli spaces, for instance the moduli space of projective hypersurfaces of a given degree, or the moduli space of abstract varieties (e.g. the moduli space of curves \(M_{g,n}\)). The many applications of tropical geometry to algebraic geometry justify the construction of such compactifications where algebraic objects and their tropical counterparts coexist. I intend to present a tropical compactification of \(M_{g,n}\) and, as much as time permits, describe one approach for varieties of arbitrary dimension. This is partially based on a joint work in progress with M. Melo, J. Rau and F. Viviani.