In this talk, we will present the bijective correspondence between Young diagrams and proper numerical sets. Then we will use this correspondence to introduce Young diagram decompositions for symmetric numerical sets (and semigroups). We will extend these decompositions to almost symmetric numerical semigroups.

There will be a coffee break after the seminar. 

References

In this talk we present some properties of generalized torsion elements in groups. We describe the set of generalized torsion elements in finitely generated abelian-by-finite groups. In addition, we present a family of Bieberbach groups in which every element is a generalized torsion element. This presentation is mainly based in the following papers [1,2,3]. 

There will be a coffee break after the seminar.

References

[1] R. Bastos and L. Mendonça. Generalized torsion elements in infinite groups. arXiv:2411.17918 [math.GR]. 

This presentation is based on joint work with P. Shumyatsky. We study the class of all finite groups in which every commutator has prime power order. A group in this class is called a CPPO-group. Our interest in this class arose from the observation that it includes, as a subclass, all finite groups in which every element has prime power order - commonly known as EPPO-groups. These were the subject of foundational work by G. Higman and M.

Quasi-Frobenius rings were introduced by Nakayama  in the study of representations of algebras. Afterwards, Quasi-Frobenius rings played a central role in ring theory, and numerous characterizations were given by various authors. In particular, Ikeda characterized these rings as two sided self-injective and two sided Artinian.

The main objective of multiplicative ideal theory is to investigate the multiplicative structure of integral domains by means of ideals or certain systems of ideals of that domain. An essential tool in multiplicative ideal theory is the concept of  ``star operation"  which was introduced by Krull in 1936 and then was used by Gilmer in his book in 1972. In this talk, we first introduce some concepts related to multiplicative ideal theory. The emphasis will be given to the ``$w$-operation", one of the most important star operations.

In the previous talk, we revisited some general aspects of the representation theory of the Lie algebra $\mathrm{gl}(n)$ and used them as motivation to study Gelfand-Tsetlin modules via Drinfeld categories. In this seminar, we will take a step back to build a solid foundation before diving into Gelfand-Tsetlin modules. We will start by recalling key results from the classical representation theory of $\mathrm{sl}(2)$, ensuring that we have the necessary tools and intuition. With this groundwork in place, we will then explore how Gelfand-Tsetlin modules appear in this setting.

Let $\mathcal{C}\subseteq\mathbb{N}^p$ (for a non-zero natural number $p$) be a non-negative integer cone. A monoid $S \subseteq \mathcal{C}$ is called a $\mathcal{C}$-semigroup if its complement in the cone is finite. This structure naturally extends the classical notion of numerical semigroups, which are submonoids of the natural numbers with a finite complement in the set of natural numbers.

Nos últimos anos, houve um grande interesse em detectar propriedades do grupo fundamental $\pi_1(M)$ de uma $3$-variedade por meio de seus quocientes finitos ou, mais conceitualmente, pelo seu completamento profinito. Isso motiva o estudo do completamento profinito $\widehat{\pi_1(M)}$ do grupo fundamental de uma $3$-variedade. Um trabalho recente de 2017 de H. Wilton e P. Zalesskii mostra que as decomposições típicas de grupos de $3$-variedades, como produtos livres com amalgamação, extensões HNN e grafos de grupos, são preservadas sob o completamento profinito.

In this talk, extensions of solvable Lie algebras are considered. The method of central extension is mostly used to obtain a classification of nilpotent algebras. This method was generalized for the solvable Lie algebras by T. Sund in 1979. We investigate extensions of solvable Lie algebras with naturally graded filiform nilradicals. Moreover, we generalize this method for the solvable Leibniz algebras and find extensions of solvable Lie algebras with null-filiform and filiform nilradicals.

In the 1960's. W. Leavitt studied universal algebras which do not satisfy the \emph{Invariant Basis Number Property (IBN)}. These are algebras that do not have a well-defined rank, that is, algebras for which $R^m\cong R^n$ ($m\neq n$) as $R$-modules which are later called the \emph{Leavitt algebras of module type} $(m,n)$. In 2005, the Leavitt algebra of type $(1,n)$ was found to be the so-called \emph{Leavitt path algebra} of a certain directed graph.