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.

Coclass theory was originally invented as a new approach towards the classification of finite p-groups. It can be translated to other nilpotent algebraic objects as well, for example, it has been considered for nilpotent semigroups, for nilpotent Lie algebras and for nilpotent associative algebras. Here we discuss its translation to nilpotent associative algebras and we show that coclass theory is highly promising in this setting as well. We exhibit the state of the art of this approach and also show some interesting open problems and conjectures.

A subgroup of a group G is said to be maximal solvable if it is maximal among the solvable subgroups of G. In his 1870 Traité, Jordan gave a classification of the maximal solvable subgroups of symmetric groups. The classification reduces to the primitive case, which is equivalent to the problem of classifying maximal irreducible solvable subgroups of GL(d,p), where p is a prime. In GL(d,p), the problem is reduced to the case of primitive irreducible solvable subgroups. These subgroups are then constructed in terms of maximal irreducible solvable subgroups of general symplectic groups GSp(2k,r) (r prime) and orthogonal groups O±(2k,2). In this talk, we will discuss Jordan’s classification in modern terms. More generally, we consider the complete classification of maximal irreducible solvable subgroups of classical groups such as GL(n,q), GSp(n,q), and GO(n,q), where q is a power of a prime. From the classification we also get a recursive construction of the maximal irreducible solvable subgroups, and this works efficiently when implemented on a CAS such as Magma or GAP. We will also discuss the analogous problem for linear algebraic groups over algebraically closed fields.

TBA