Abstract: An integral domain R is said to be Generalized Euclidean (or a GE-ring) if every invertible matrix over R can be written as a product of elementary matrices. Characterizing GE-rings is a classical problem in ring theory. Notably, if R is a Bézout domain (i.e., a domain in which every finitely generated ideal is principal), R is a GE-ring if and only if any singular matrix over R can be decomposed into idempotent factors, and this holds true iff the domain admits a weak version of the Euclidean algorithm. In this seminar, we will survey some classical and more recent results and open questions about GE-rings, exploring in particular their connection with idempotent matrix factorization. To conclude, we will briefly introduce a novel approach to factorization of matrices rooted in factorization theory.

Abstract: Let \( K \) be a real quadratic number field and let \( \mathcal{O}_K \) be the principal order in \( K \). There is a unique squarefree integer \( d \in \mathbb{N}_{\geq 2} \) such that \( K = \mathbb{Q}(\sqrt{d}) \). It is well-known that there is a unique unit \( \varepsilon \) of \( \mathcal{O}_K \) (called the fundamental unit of \(\mathcal{O}_K\)) such that \(\varepsilon>1\) and \(\{\pm\varepsilon^k\mid k\in\mathbb{Z}\}\) is the unit group of \(\mathcal{O}_K\). Let \(\omega\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}\) be such that \(\mathcal{O}_K=\mathbb{Z}[\omega]\) and let \(x,y\in\mathbb{N}_0\) be the unique nonnegative integers such that \(\varepsilon=x+y\omega\). In this talk we present and discuss various problems that involve the real quadratic number fields \(K\) for which \(d\mid y\). Among these problems are the Ankeny-Artin-Chowla conjecture, the Erdos-Mollin-Walsh conjecture and the Pellian equation conjecture. We will also put our focus on problems that stem from module theory (involving orders in \(K\) with torsionfree/mixed cancellation), factorization theory (involving ``unusual'' orders in \(K\)) and number theory (involving orders in \(K\) with relative class number one). The main objective of our talk is to review the methods (that we recently used) to obtain and verify a counterexample to the Pellian equation conjecture.

Abstract: Krull monoids are well-studied objects in factorization theory. Their arithmetic can be combinatorially described by zero-sum sequences over their class group. As a generalization, a monoid H is said to be transfer Krull if there is a transfer homomorphism from H to a Krull monoid, meaning that H admits the same factorization behavior as a Krull monoid, while not necessarily being Krull itself. In this talk, we give an algebraic characterization of transfer Krull orders in certain Dedekind domains with torsion class group. Prominent examples of such Dedekind domains include rings of integers of number fields and holomorphy domains in algebraic function fields.

Abstract: Given a finite ring, we consider the set of polynomials inducing the constant zero function (null-polynomials). In the commutative case, this set is easily seen to be an ideal of the polynomial ring. However, whether this also holds in the non-commutative setting is not clear, due to the absence of a substitution homomorphism. For many classes of finite rings, it has been shown that indeed the null-polynomials form a two-sided ideal, and no counterexample is known. In this talk, we review existing results and give further examples of ideals of null-polynomials by considering matrix algebras that are defined by restricting elements in certain positions to be zero.

Abstract: Claborn's Realization Theorem states that every abelian group can be realized as the class group of a certain Dedekind domain. Since then, several similar existence results have been established. In this talk, we will show that for any given finitely generated abelian group, there exists an algebraic structure—specifically, a generalized cluster algebra—whose class group is isomorphic to that group. Generalized cluster algebras were introduced by Chekhov and Shapiro in 2014, as a generalization of the cluster algebras first developed by S. Fomin and A. Zelevinsky. While classical cluster algebras have binomial exchange relations, generalized cluster algebras allow multinomial exchange relations. Despite the many similarities between these two algebraic structures, our study reveals a key distinction in terms of factorization theory. Namely, the class group of a cluster algebra is always a free finitely generated abelian group, whereas the class group of a generalized cluster algebra can have torsion.

Abstract: There are only a few known constructions of Dedekind prime rings. Our goal is to give a new one. Let \(D\) be a commutative Dedekind domain with an automorphism \(\sigma \colon D \rightarrow D\). We study the ring of skew Laurent series \(R = D(\!(x; \sigma)\!) \text{.}\) We show that \(R\) is a noncommutative Dedekind domain. The natural morphism of the ideal class groups \( \pi \colon G(D) \rightarrow G(R)\) is surjective, and if \(\sigma\) acts trivially on the ideal class group of \(D\), then \(\pi\) is an isomorphism. This gives us a way to construct new examples of noncommutative Dedekind domains with some control over their projective modules.

Abstract: The following question is central to Bayesian statistics: Given a dataset, what statistical model fits it best? Traditionally, this is addressed using the Bayesian Information Criterion (BIC), which works well for regular models. However, BIC does not apply to singular models like neural networks and Gaussian mixture models, both highly relevant to work in artificial intelligence. In response, Japanese statistician Sumio Watanabe recently developed a novel information criterion that generalizes BIC and forms the foundation of a comprehensive theory for singular statistical models. Applying this criterion requires calculating two birational invariants of the variety defined by the model’s Kullback-Leibler divergence: the real log canonical threshold (rlct) and its multiplicity, with the rlct serving as the real analogue of the classical log canonical threshold from birational geometry. We provide explicit formulas for these invariants in cases where the singularities resemble a hyperplane arrangement, a problem suggested by Bernd Sturmfels. Our exploration will involve real log resolutions, wonderful compactifications, and asymptotics of high-dimensional volume integrals. This talk is based on joint work with Dimitra Kosta.

Abstract: Hopf Poisson order (HPO) algebras were introduced and studied by Brown, Nazemian, and Zhang (preprint, 2024). We investigate the class of Poisson modules over HPO algebras and show that it forms a monoidal category. Moreover, we prove that the left homological integral of an HPO algebra \(H\), denoted \(\int_H^l\), is a left Poisson module. It is also a right Poisson module if and only if \(\int_H^l = \int_H^r \).

Abstract: I will provide an overview of power semigroups, a topic that has gained increasing attention in recent years due to its intriguing connections with other areas of mathematics (especially, algebra and additive combinatorics). We will begin by introducing the fundamental concepts and the key properties that define power semigroups, making sure to highlight examples that illustrate their significance. After laying this groundwork, we will shift the focus to some of the most compelling open problems in the field. These include questions related to the ideal structure of power semigroups, their classification up to (global) isomorphisms, and combinatorial and arithmetic aspects that remain elusive despite substantial progress.

Abstract: A numerical semigroup is irreducible if it cannot be expressed as the intersection of two numerical semigroups that contain it properly. Every numerical semigroup can be expressed as an intersection of (finitely many) irreducible numerical semigroups. We review some of the known results on bounds for the minimum and maximum length of factorization into irreducibles. We show that the unions of sets of lengths of factorizations of numerical semigroups into irreducible numerical semigroups are all equal to \(\mathbb{N}_{\ge 2}\).