In this talk, I will present a recent result by Ascari concerning Whitehead automorphisms of subgroups of free groups. The result shows that the Stallings graph of a Whitehead image of a subgroup can be obtained by collapsing certain edges of the original graph. Building on this, I will show that the cogrowth—or entropy—of the image is strictly greater than that of the original subgroup. This gives the first direct connection between Whitehead techniques and cogrowth. If time permits, I will also present some open problems related to this topic.

The Yang-Baxter equation is a fundamental equation appearing in many different contexts of physics. In 1992, Drinfeld proposed a mathematical and combinatorial approach to the problem of classifying solutions, which was followed by a seminal article of Etingof, Schedler and Soloviev in 1999. Throughout this talk, I will present the basic notions behind the study of set-theoretical solutions to the Yang-Baxter equation, mainly from a group theoretical and Garside perspective. In this sense, I will focus on the known parallels between groups associated to solutions and the classical properties of finite Coxeter groups. In the last part, I'll talk about a new construction of Hecke algebras, inspired by the theory of Iwahori-Hecke algebras for Coxeter groups.

Tropical geometry studies a piecewise linear combinatorial shadow of classical algebraic geometry. In this talk I will explain what little we currently know about the tropical geometry of algebraic groups. Different avenues of progress towards an answer to this question lead to new insights both within and beyond tropical geometry, in particular in the context of vector bundles and the geometry of buildings. This talk will touch upon joint work with Luca Battistella, Desmond Coles, Andreas Gross, Inder Kaur, Kevin Kühn, Arne Kuhrs, Annette Werner, Alejandro Vargas, and Dmitry Zakharov.

We study the subrank of real order-three tensors and give an upper bound to the subrank of a real tensor given its complex subrank. For several small tensor formats, we investigate the typical subranks. Finally, we consider the tensor associated to componentwise complex multiplication in C^n and show that this tensor has real subrank n - informally, no more than n real scalar multiplications can be carried out using a device that does n complex scalar multiplications.

A polytrope (or alcoved polyhedron) is a tropical polyhedron that is also classically convex. We study the tropical combinatorial types of polytropes associated to weighted directed acyclic graphs (DAGs). This family of polytropes arises in algebraic statistics when describing the model class of max-linear Bayesian networks. We describe the combinatorial types of polytropes from weighted DAGs in terms of half-open decompositions of a certain polyhedral fans. These polyhedral fans fit together in a generalized cone complex that we also describe

TBA