Abstract: A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. Previously in [J. Cui, Y. Li and H. Wang. On nil clean group rings, Commun. Algebra 49(2) (2021) 790-796], the nil clean group rings over dihedral groups and generalized quaternion groups were completely determined. In this paper, we investigate group rings \(RG\) over more general metacyclic groups \(G\), where \(G=\langle a,b\mid a^n=b^m=1,b^{-1}ab=a^r\rangle\). We provide a new criterion for a general nil clean group ring based on the upper central series of \(G\) consisting of 2-groups, and whether the group ring over its respective factor group is nil clean. We also determine values of \(m\) and \(n\) (as a prime power) in order for \(RG\) to be nil clean, and verify whether their corresponding group rings are indeed nil clean.

Abstract: For a one dimensional analytically unramified local domain, the blowup algebra of the canonical ideal is a birational extension. The conductor of this birational extension always contains the conductor ideal. In this talk, I will talk about the case where there is equality and classify numerical semigroups with this property.

Abstract: I will outline a geometric framework that aims to make sense of varieties with nonnoetherian coordinate rings of finite Krull dimension. In short, such varieties may viewed as algebraic varieties with positive dimensional 'smeared-out' points. I will then describe some work in progress with Daniel Windisch on properties of these varieties in the toric setting, that is, when their coordinate rings are generated by a set of monomials in a polynomial ring. These rings correspond to certain (infinitely generated) monoids in Z^n. Finally, I will discuss an application to toric degenerations of invariant rings which are nonnoetherian.

Abstract: Hereditary Noetherian prime (HNP) rings are a noncommutative generalization of Dedekind domains. Projective modules over HNP rings are well understood, and recently, Rump and Yang developed a two-sided multiplicative ideal theory for HNP rings. For a special class of HNP rings, Dedekind prime rings, there is also a one-sided multiplicative ideal theory. In this talk, we will present a one-sided multiplicative ideal theory for general HNP rings -- understood through divisors, i.e., integer combinations of simple modules -- that generalizes both the two-sided ideal theory for HNP rings and the one-sided ideal theory for Dedekind prime rings.

Abstract: I will present some results on the arithmetic of the monoid \(\mathcal I(R)\) of nonzero ideals of a multivariate polynomial ring \(R=K[X_1,\dots,X_N]\) (where \(K\) is a field and \(N\ge 2\)), under ideal multiplication. In recent work with Laura Cossu and Azeem Khadam, we extend techniques from [A. Geroldinger and M. A. Khadam, On the arithmetic of monoids of ideals, Ark. Mat. N. 60 (2022), 67--106.] in order to construct new families of atoms in \(\mathcal I(R)\), such as a class of ideals corresponding to sum-free subsets of \(\mathbb N\), recovering some of the previously known examples. We further analyze the submonoid \(\mathcal M\rm{on}(R)\) of monomial ideals, deriving some arithmetic properties and computing sets of lengths for specific families of ideals.

Abstract: A finite Laurent Intersection Ring (FLIR) is a ring defined as intersection of finitely many Laurent polynomial rings. FLIRs over Krull domains are Krull domains themselves, and when we have an explicit description of the Laurent polynomial rings involved in the intersection, things become particularly interesting from the point of view of factorization theory. Krull domains are a well-studied class of integral domains which generalize UFDs. In particular, every nonzero nonunit element in a Krull domain can be written in finitely many different ways as a finite product of irreducible elements, but in general such factorizations need not be unique. The divisor class group of a Krull domain is a key invariant that measures the failure of unique factorization. Although the importance of the divisor class group in the study of nonunique factorization, there are very few nontrivial families of rings for which explicit computations are available. Finite Laurent Intersection Rings therefore provide a valuable supply of explicit, computable examples: from the combinatorial data of the ambient Laurent polynomial rings one can compute the class group and can also describe factorizations of elements into irreducibles concretely. We conclude the talk with Banff cluster algebras, introduced by Muller in 2013, which serve as our main motivating family of explicit FLIRs. In particular, cluster algebras arising from triangulations of marked Riemann surfaces are Banff and hence yield many concrete examples amenable to the methods described above. This is a joint work with D. Smertnig.

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Abstract: Recently, the concept of Aut-stable subspaces has played a role in the study of polynomial rings. In particular, the speaker together with Huang, Wang, and Zhang, proved that polynomial rings in more than two variables over an algebraically closed field of characteristic zero do not admit any \Aut-stable subspaces. In this talk, which is based on joint work with Mithat Konuralp Demir, we characterize all Aut-stable subspaces and Aut-stable subalgebras of Grassmann algebras.

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Abstract: The separating Noether number \(\beta_{\mathrm{sep}}(G)\) of a finite group \(G\) is the minimal positive integer \(d\) such that for every \(G\)-module \(V\) there is a separating set of degree \(\leq d\). In this manuscript, we investigate the separating Noether number \(\beta_{\mathrm{sep}}(G)\). Among others, we obtain the exact value of \(\beta_{\mathrm{sep}}(G)\) for finite abelian groups \(G\), when either \(G\) is a \(p\)-group or \(\mathsf r(G)\in \{3,5\}\).

Abstract: For a finite abelian group \( G \) and a positive integer \( k \), let \( \mathsf{D}_k(G) \) denote the smallest integer \( \ell \) such that each sequence over \( G \) of length at least \( \ell \) has \( k \) disjoint nontrivial zero-sum subsequences. It is known that \( \mathsf{D}_k(G) = n_1 + kn_2 - 1 \) if \( G \cong C_{n_1} \oplus C_{n_2} \) is a rank \( 2 \) group, where \( 1 < n_1 \mid n_2 \). We investigate the associated inverse problem for rank \( 2 \) groups, that is, characterizing the structure of zero-sum sequences of length \( \mathsf{D}_k(G) \) that cannot be partitioned into \( k+1 \) nontrivial zero-sum subsequences.

Abstract: Let \( S \) be a multiplicatively written semigroup. The family \( \mathcal{P}(S) \) of non-empty subsets of \( S \) endowed with the binary operation of setwise multiplication \[ (X,Y)\mapsto \{xy:x\in X, y\in Y\}\] induced by \( S \), is called the \textit{large power semigroup} of \( S \). Although power semigroups were already introduced in the 1950s, the study of arithmetical and algebraic properties of these objects received a lot of attention in the last decade. In this talk, we give a survey of recent developments in this area and present some new results involving the \textit{reduced finitary power monoid} \( \mathcal{P}_{\text{fin},1}(H) \) of a monoid \( H \), which consists of all finite subsets of \( H \) containing the identity element. To give an example, a central question, called the \textit{isomorphism problem}, is whether an isomorphism between \( \mathcal{P}_{\text{fin},1}(H_1) \) and \( \mathcal{P}_{\text{fin},1}(H_2) \) for two monoids \( H_1 \) and \( H_2 \) implies that \( H_1 \) and \( H_2 \) are isomorphic. This question was answered in the negative for arbitrary monoids, yet remained open for \textit{cancellative} monoids. We give sufficient conditions for commutative, cancellative monoids to satisfy the property above and show that there are "well-behaved" monoids for which the isomorphism problem has a negative answer.

Abstract: An innocent question of Erdős and Turán (1936) asked about the maximal size of sets in \(\{1,2,..., N\}\) without 3 integers in arithmetic progression. The upper bound on the size of such sets, when \(N\) is large, has seen influential ideas, and the problem became one of the corner stones of combinatorial number theory/additive combinatorics. (The list of contributors includes Roth, Szemeredi, Furstenberg, Bourgain, Gowers, Sanders, Bloom and Sisask, and most recently Kelley and Meka.) On lower bounds the situation remained unclear: Szekeres (1936) conjectured (based on small examples) that the set of integers avoiding the digit 2 in the ternary system might be best possible. Salem and Spencer (1942) and Behrend (1946) gave better constructions. These constructions required a new idea, but are quite simple otherwise. Behrend's construction was named "fairly unimprovable" and the status was described as "the lower bound has remained virtually stationary". In this talk we describe a novel idea. This new method allows to surpass a natural threshold for progression-free sets in \((Z_m)^n\), and also eventually improves Behrend's construction. This is joint work with Zach Hunter, Laura Proske, Lisa Sauermann

Abstract: Let \(d\) be a squarefree integer with \(d\geq 2\), let \(K=\mathbb{Q}(\sqrt{d})\) and let \(\mathcal{O}_K\) be the principal order in \(K\). It is well-known that if \(\omega=\begin{cases}\sqrt{d} & \text{ if } d\equiv 2,3\mod 4\\\frac{1+\sqrt{d}}{2} & \text{ if }d\equiv 1\mod 4\end{cases}\), then \(\mathcal{O}_K=\mathbb{Z}[\omega]\). Let \(\varepsilon\in\mathcal{O}_K\) with \(\varepsilon>1\) be the fundamental unit of \(\mathcal{O}_K\) and let \(x,y\in\mathbb{N}_0\) be the unique nonnegative integers such that \(\varepsilon=x+y\omega\). The Ankeny-Artin-Chowla conjecture states that if \(d\) is a prime number with \(d\equiv 1\mod 4\), then \(d\nmid y\) and Mordell's Pellian equation conjecture states that if \(d\) is a prime number with \(d\equiv 3\mod 4\), then \(d\nmid y\). Mordell's Pellian equation conjecture was refuted recently and very recently we were able to disprove the Ankeny-Artin-Chowla conjecture. In this talk, we discuss the key problems that motivated these conjectures and we will briefly outline their (long) history. We present and study some of the attempts (from the literature) that were made to solve them. Finally, we put our focus on the methods that were actually used to disprove these conjectures, namely the small step algorithm and the large step algorithm. Subsequently, we will touch on future research that is related to the aforementioned conjectures.

Abstract: Let \(\mathcal{H}\) be a class of monoids or rings, and let \(\mathsf{a}(H) \in \mathbb{R}\) be an arithmetic invariant defined for every \(H \in \mathcal{H}\). A natural question is to find a formula for \(\mathsf{a}(H)\) in function of a suitable set of indeterminates, depending on the structure of \(H\), which is valid for all \(H \in \mathcal{H}\). Assume that there is an injective function \(\varphi\colon \mathcal{H} \rightarrow \mathbb{Z}^n\), that is, every \(H \in \mathcal{H}\) can be represented uniquely by a set \(\{h_1,\ldots,h_n\}\) of integers. We say that there is a polynomial formula for \(\mathsf{a}(H)\) for every \(H \in \mathcal{H}\) if there exist a finite set of polynomials \(f_1,\ldots,f_\sigma \in \mathbb{C}[x_1,\ldots,x_n]\) such that for every \(H \in \mathcal{H}\) such that \(\varphi(H)=\{h_1,\ldots,h_n\}\) there is an \(i \in [1,\sigma]\) such that \(f_i(\varphi_H(h_1),\ldots,\varphi_H(h_r))=\mathsf{a}(H)\). There are multiple examples (across different fields) of invariants which are very hard to compute in general; however, at the same time, simple polynomial formulas can be obtained for specific classes. In this talk, we address the problem of determining whether a polynomial formula exists for a given class \(\mathcal{H}\). More specifically, we adapt a method used by Curtis to study the Frobenius number of a numerical monoid to give some conditions ensuring the non-existance of a polynomial formula for \(\mathsf{a}(H)\), and then we apply this result to two very different invariants, one involving the arithmetic of finitely generated monoids (the catenary degree) and another related to singularities for hypersurfaces in characteristic \(p\) (the \(F\)-threshold).

Abstract: The minimal generating set \(\mathsf{G}(I^k)\) of the power \(I^k\) of a monomial ideal \(I\) is typically a lot smaller than the set of \(k\)-fold products of generators of \(I\). For large \(k\), it becomes increasingly complex to identify all relevant cancellations among these products. However, it is well known that the number (\mu(I^k)\) of minimal generators of \(I^k\) is eventually the value of a polynomial function---the Hilbert polynomial of the fiber cone of \(I\). This gives reason to suspect that the actual set \(\mathsf{G}(I^k)\) eventually becomes predictable in the sense that no ``unexpected cancellations'' occur anymore. In a recent collaboration with Jutta Rath we confirmed the predictability of \(\mathsf{G}(I^k)\) as \(k\) becomes large for bivariate monomial ideals. This yields a full description of \(\mathsf{G}(I^k)\) and a quantifiable threshold for \(k\). Roughly speaking, we show that the \emph{staircase} of \(I^k\) is a ``stretch'' of the staircase of \(I^s\) where \(s\) is given explicitly. This talk aims to explain this ``staircase stretch'' in pictures..

Abstract: This talk is about the commutative ring \(\mathbb{C}[x,y]/(x^2)\), which has been studied from various points of view: in the 1980s as a ring with countable representation type in commutative algebra and its spectrum as singularity of type \(A_\infty\) in singularity theory, and more recently, in the 2020s, it has been shown that it yields a categorification of certain Grassmannian cluster algebras of infinite rank in representation theory. The key object in these studies is the category of Maximal-Cohen Macaulay modules over this ring, which has a rich combinatorial structure. In this talk I will comment on the results above and also explain how to get from them to nonperiodic infinite frieze patterns and Penrose tilings (this is joint work in progress with Ö. Esentepe).

Abstract: Let \(D\) be an integral domain with quotient field \(K\). The following subring \[ {\rm Int}(D):=\{f\in K[X]\mid f(D)\subseteq D\} \] of \(K[X]\) is usually called the ring of integer-valued polynomials over $D$. The ideal theory of this type of rings is quite fascinating and mysterious. We will concentrate our attention to the unitary ideals of \({\rm Int}(D)\), that is, ideals containing some nonzero constant. More precisely, when \(D\) is a Dedekind domain with finite residue fields, we will fully describe in topological terms certain classes of radical unitary ideals of \({\rm Int}(D)\). Joint work with A. Loper.