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.

The Hasse principle is the idea that a Diophantine equation over the rational numbers should have a rational solution if and only if it has solutions in all of its completions, namely, the real numbers and all p-adic fields. In recent work of Lorenzo and Vullers, they give twists of the modular curve X(7) that are counterexamples to the Hasse principle. In this talk, we will discuss generalizations of their result, for example, that there are infinitely many counterexamples to the Hasse principle that are twists of the modular curve X(p) for primes p congruent to 1 mod 4.

Quasigroupoids and weak Hopf quasigroups are non-associative generalizations of groupoids and weak Hopf algebras. In this talk, we will establish their main properties and an equivalence between the category of finite quasigroupoids and that of pointed cosemisimple weak Hopf quasigroups over a field K. As an immediate consequence, we obtain a categorical equivalence between quasigroups (in the sense of Klim and Majid, i.e., loops with the inverse property) and pointed cosemisimple Hopf quasigroups over K.

Polynomial expansion concerns the heuristic expectation that, for a typical polynomial P in n variables over a field F and subsets A₁,...,A_n of F, the image P(A₁,...,A_n) is substantially larger than each of the individual sets A_k. We establish new expansion results for certain classes of polynomials over finite fields, including a classification result for ternary quadratic polynomials. Our methods rely on spectral bounds for certain graphs arising from incidence geometry. This is joint work with Sam Chow.