In the representation theory of finite-dimensional simple Lie algebras $\mathfrak{g}$, two categories of modules stand out due to their contrasting nature. The first is the category of weight modules, consisting of $\mathfrak{g}$-representations where a fixed Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ acts semisimply. This category has been extensively studied over the past decades, with a classification of simple modules having finite-dimensional weight spaces obtained by O. Mathieu through the introduction of a special class of modules known as coherent families.

The rational numbers have been used to measure quantities since ancient times; however, their implementation in computer languages raises a significant problem: zero has no inverse. To address this issue, J. Bergstra and J. Tucker introduced an algebraic structure called a meadow, which allows for the inversion of zero.

In this seminar we will talk about results concerning faithful representations of abelian groups as automorphisms of one-rooted trees. In particular, we show the self-similarity of the free abelian group of uncountable rank and of the Baer-Specker group.

Let $G$ be a group and consider the group $\chi(G)$ obtained from the free product $G \ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. In the last 44 years, this group has been a formidable tool for obtaining finiteness conditions in Group Theory. In this talk, we want to present some important results related to the group $\chi (G)$. Moreover, we want to establish some properties of the exponent of $\chi(G)$ when $G$ has finite exponent. 

In the context of the theory of Gelfand-Tsetlin modules, Drinfeld categories were introduced in 2017 by V. Futorny et al. to prove that every irreducible 1-singular Gelfand-Tsetlin module is isomorphic to a subquotient of the universal 1-singular Gelfand-Tsetlin module. The authors also observed that these categories could be used to generalize the classification of Gelfand-Tsetlin modules for $\mathrm{sl}(n)$, which, at that time, was only known for $\mathrm{sl}(3)$.

The non-abelian tensor product $G \otimes H$ of groups G and H was
introduced by Brown and Loday, following works of Miller and
Dennis. Ellis showed the finiteness of the
non-abelian tensor product $G \otimes H$ when both $G$ and $H$ are finite. 

I will present some related results concerning the (local) finiteness of the non-abelian tensor product $G \otimes H$.

This is a joint work with Nora\'i Rocco (Universidade de Bras\'ilia) e Irene Nakaoka (Universidade Estadual de Maring\'a). 

I shall discuss prosoluble subgroups of free profinite products towards a complete characterization of them in terms of the intersections with the free factors.

The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of $D$-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of those $D$-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman

I will discuss some recent forays into some counting problems for free objects. I will focus on free inverse semigroups and free regular $*$-semigroups. I will first discuss recent results joint with M. Kambites, N. Szakács, and R. Webb giving a precise rate of exponential growth of the free inverse monoid of arbitrary (finite) rank, which turns out to be given by a surprisingly complicated but algebraic number.