The variety of Lie algebras is the central object of study in the realm of non-associative algebras. In this talk we explain, through a series of recent results, why this is no accident: working in the framework of semi-abelian categories, which every variety of non-associative algebras over a field K inherits, a number of a priori unrelated categorical and structural properties each single out Lie algebras as the unique non-trivial such variety. After an overview of these categorical-algebraic methods, we will discuss (some of) the following characterisations.

Given an associative algebra A over a field F, a polynomial identity for A is a polynomial in non-commuting variables that vanishes under all evaluations in A. An algebra satisfying a nontrivial polynomial identity is called a PI-algebra.
In this talk, we introduce some of the main results in PI-theory and several numerical invariants associated with polynomial identities, with particular emphasis on the use of combinatorial methods and representation theory techniques.
Finally, we will present some recent developments in the context of algebras with trace.

(Joint work with Ben Blum Smith, Johns Hopkins University, USA) The barycentric subdivision of a boolean complex preserves many combinatorial and topological properties, but its effect on the associated Stanley–Reisner ring is more subtle. In this talk, I will discuss the problem of comparing the face ring of a boolean complex with that of its barycentric subdivision in an equivariant setting. I will first explain why equivariant isomorphisms do not exist in general, presenting a counterexample in characteristic 2.

The main goal of this talk is to illustrate the role of quaternions in number theory. We propose to do this in two parts: first by studying the concept of a Non-Commutative Principal Ideal Domains (PID) in Quaternion Algebras, and, after, by using quaternions to prove the universality (or not) of some Diophantine equations. More precisely, this talk starts with a few results about factorization in quaternion orders. Then, we present a finite algorithm to determine if a given order is a PID, based on a criterion attributed to Dedekind and Hasse.

In this talk I will give a brief introduction to the study of conditional indepedence (CI) and context-specific conditional (CSI) independece through the use of commutative algebra. I will explain how algebra is useful to better understand the set of all CI and CSI statements that are implied by a set of such statements. The case when the ideals that arise are toric is combinatorially and algebraically very appealing. I will then present results concerning the toric ideals of certain context-specific independence models.

We construct what we call a "good" projective resolution of Weyl modules for the general linear group over an arbitrary commutative ring. From this, we obtain a resolution of the co-Specht modules for the symmetric group by permutation modules. These resolutions allow us to generalize in the corresponding Grothendieck groups the classical determinantal identity known for the representation theory of the symmetric group over fields of characteristic zero. This is joint work with Ivan Yudin.

The commuting graph of a finite non-commutative semigroup $S$ is the simple graph whose vertices are the non-central elements of $S$ and where two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$.

I will review some classical examples of groups whose strictly descending chains of subgroups are finite.  Some recent constructions of such groups (having large cardinality) will also be explained. Set-theoretic consistency will play a role in some of the stated results. This is joint work with Saharon Shelah.

Kruskal’s uniqueness theorem gives a simple criterion ensuring that a 3-way tensor admits a unique expression as a sum of r product tensors. This talk will present a geometric proof of Kruskal's theorem, and show that rank and border rank coincide for tensors which satisfy Kruskal's uniqueness condition. The talk will also serve as an introduction to the notions of tensor rank and border rank. We will briefly touch on applications of tensor rank decompositions in the sciences.

Cantor famously used two versions of Diagonalization for his fundamental results in set theory. First, he used it to prove the uncountability of the set of real numbers. Second, he used a more general version to prove that the power set of a set A has a higher cardinality than A. In this talk we will discuss how Diagonalization is the core of many of delimitation theorems, such as Gödel's (First) Incompleteness Theorem or Turing's Undecidability Result.