Abstract: The main goal of this talk is to explore the role of quaternions in number theory. For that purpose, we start with the concept of a Non-Commutative Principal Ideal Domains in Quaternion Algebras. After, we discuss factorization in non-commutative domains, and give an algorithm that checks if an arbitrary quaternion order is a Non-Commutative PID. Finally, we analyse some examples of orders, and see how one can use the concepts addressed in the first part of the talk to show the universality (or not) of some diophantine equations.

Abstract: In this talk I will show how to extend the notion of holonomy of principal bundle connections to orbifolds. To do that, first we will talk about principal orbibundles and give a characterization of them in terms of group actions. After that, we will talk about connections, horizontal lifts and their relation with the orbifold singularities. To conclude this talk, we will talk about the holonomy group. I will show some ideas of how to proof that it is a Lie group, the reduction theorem and the Ambrose-Singer theorem.

Abstract: The “tone net” (Tonnetz in german) that Leonhard Euler already described in 1739 is an early example of using geometric objects in music theory. In fact, such spaces appear after identifying certain simple examples of Lie groupoids that one can interpret “musically”. Although surprisingly simple groupoid structures are used, advanced concepts in music theory appear as features of the geometry that such groupoids describe. In the talk, we will develop some (higher) Lie theory and some music theory, and we do so side by side. Concepts from Lie theory are introduced in so far as we interpret them musically, but we do introduce such concepts in a greater generality than necessary. This way we introduce two topics (Lie theory and music theory) at once, hopefully creating interest for both.

Abstract:  The classification of irreducible representations of noncommutative algebras, such as universal enveloping algebras U(g) of finite dimensional Lie algebras g, is a central but challenging problem in representation theory. The Dixmier–Moeglin equivalence, established in the 1980s, provides a deep connection among primitive ideals, rational ideals, and locally closed points in the prime spectrum of such algebras, yielding a powerful tool to characterize primitive ideals. Over the past two decades, the Poisson analogues of these concepts arising naturally from semiclassical limits have attracted considerable interest and motivated the study of the Poisson Dixmier–Moeglin equivalence (PDME).

In this talk, we explore the notion of a strong Poisson Dixmier-Moeglin equivalence, which extends PDME by introducing degrees related to rationality, primitivity, and local closure in Poisson prime ideals. We review both the classical and strong versions of PDME, discuss classical examples including symmetric algebras of sl 2 and abelian Lie algebras, and highlight known counterexamples, such as certain Poisson Ore extensions related to Hermite polynomials that satisfy the classical PDME but fail the strong PDME.

Moreover, we outline recent progress on the validity of the strong PDME for larger classes of symmetric algebras of Lie algebras, and pose open questions about whether the strong PDME holds in general for symmetric algebras of semisimple Lie algebras.