Hopf braces, related structures and their associated categories by Brais Ramos Pérez (Santiago de Compostela)

Abstract: Hopf braces are recent mathematical objects introduced by I. Angiono et al. [1] and obtained through a linearisation process from skew braces,  which give rise to non-degenerate, bijective and not necessarily involutive solutions of the Quantum Yang-Baxter Equation (see [4]), whose formulation is the following

(τ ⊗ idV ) ◦ (idV ⊗ τ ) ◦ (τ ⊗ idV) = (idV ⊗ τ ) ◦ (τ ⊗ idV ) ◦ (idV ⊗ τ ), (QYBE)

Abstract: In 1892, Klein’s Erlangen program proposed that all geometric problems should ultimately be studied through the lens of group theory. In the 1950s, Jacques Tits introduced coset geometries, a structure that bridges geometries and their automorphism groups, allowing properties of groups to be studied via geometry and vice versa. Coset geometries play a central role in establishing the one-to-one correspondence between regular polytopes and a class of groups known as string C-groups. Consequently, classifying polytopes becomes equivalent to classifying these groups.

Let $F$ be a family of finite groups closed under taking subgroups, quotients and finite direct products. Given an element $g$ of a profinite group $G$, the $F$-izer of $g$ in $G$ is the set of elements $x$ in $G$ such that $\langle g,x\rangle$ is a pro-$F$-group. Let $F(G)$ be the set of elements $g$ of $G$ such that the $F$-izer of $g$ in $G$ has positive Haar measure.

Numerical semigroups are the subsemigroups of the set of natural numbers that are cofinite and contain $0$. Let $S$ be a numerical semigroup and $c$ be the smallest number such that $S$ is the union of a finite subset of $[0,c]$ and the integer interval $[c,\infty)$. Wilf's conjecture states that the density of elements of $S$ in the interval $[0,c]$ is at least equal to $1/d$, where $d$ is the dimension of the numerical semigroup $S$.

The classification of tuples of matrices up to simultaneous conjugation is a classical problem in linear algebra and representation theory. While the case of a single matrix leads to the well-known Jordan normal form, the situation becomes far more intricate for tuples, where the problem is considered "wild" and therefore not classifiable in any reasonable sense. Geometric Invariant Theory (GIT) provides a powerful framework for studying such spaces through quotient varieties, which parametrize semisimple representations of free associative algebras.

A numerical semigroup $S$ is a cofinite submonoid of the aditive monoid $(\mathbb{N},+)$. The (finite) complement of $S$ in $\mathbb{N}$ is the genus of $S$.

I plan to recall a well-known process of exploring the classical numerical semigroups tree as a means to count numerical semigroups by genus. The process is easily adapted to verify properties: one verifies the property at each explored node.

How much does the universal enveloping algebra of a Lie algebra remember about the Lie algebra itself? This is the statement of a classical problem in algebra called ``the isomorphism problem for enveloping algebras". In this talk, I'll explain how modern tools from algebraic homotopy theory and deformation theory allow us to make new progress on this problem. Along the way, we'll see that this is closely related to the following intriguing question: in the realm of homotopy theory, is being a commutative algebra simply a property, or is it an additional structure?

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.