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For subsets \( A \) and \( B \) of an abelian group \( G \), their sumset \( A+B \) is the subset consisting of all elements \( g \) from \( G \) having at least one representation of the form \( g=a+b \) with \( a\in A \) and \( b\in B \). For an integer \( t\geq 1 \), the \( t \)-popular sumset \( A+_t B \) is the subset of all elements from \( G \) having at least \( t \) representations \( g=a_1+b_1=\ldots=a_t+b_t \) with \( (a_1,b_1),\ldots,(a_t,b_t)\in A\times B \) distinct tuples. In this talk, we will introduce/review two classical results regarding sumsets and \( t \)-popular sumsets. The first is Kneser's Theorem, and gives the general lower bound for the size of \( |A+B| \) in terms of \( |A| \) and \( |B| \) for a general abelian group \( G \). The second is Pollard's Theorem, which gives the general lower bound for the sum \( |A+_1 B|+\ldots +|A+_t B| \) for a cyclic group of prime order. The goal is then to find a common generalization of both these results. I will overview partial work on this problem up to now, present one newly proposed potential conjecture for the common generalization, and also talk about recent partial progress towards this conjecture done by myself joint with Runze Wang.

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Contributions by Nikola Bogdanovic, Haleh Hamdi, Jun Seok Oh, Balint Rago, Wenkai Yang, Doniyor Yazdonov

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Let \(H\) be a monoid and let \(G\) denote its quotient groupoid. We introduce the Riemann-Zariski space \(\textup{Zar}(G| H)\)associated with \(H\) and prove that it is a spectral space. We then show that, if \(H\) is an \(s\)-Prüfer monoid, the space \(\textup{Zar}(G| H)\) is homeomorphic to the prime \(s\)-spectrum of \(H\) endowed with the Zariski topology. Next, given a finitary ideal system \(r\) on \(H\), we prove that the space of \(r\)-ideals \(\mathcal{I}_r(H)\), endowed with the Zariski topology, is spectral, and that the prime \(r\)-spectrum of \(H\) is proconstructible in \(\mathcal{I}_r(H)\). We also introduce a new topology on the set \(\mathcal{X}\) of all generalized \(H\)-module systems and show that \(\mathcal{X}\) is a spectral space. As an application, we prove that the subspace of finitary generalized \(H\)-module systems \(\mathcal{X}_{\textup{fin}}\), equipped with the induced topology, is proconstructible in \(\mathcal{X}\). Finally, we provide a characterization of those subsets of the set of overmonoids of \(H\) that are quasi-compact.

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Multiplicative ideal theory explores the characterization of the multiplicative structure of an integral domain through the study of its ideals or specific systems of ideals. A significant tool in this area is the concept of star operations, introduced first by W. Krull in 1936, and used first by R. Gilmer. One well-known star operation is the \(w\)-operation which plays an important role in establishing a connection between the hereditary torsion theory and multiplicative ideal theory. One of the long-term problems in multiplicative ideal theory is to prove or disprove the assertion that ``every result in classical commutative algebra has a corresponding analogue in \(w\)-module theory". Motivated by this, we first review the \(w\)-operation analogues of some properties, and then we focus on a very recent result about the \(w\)-operation analogue of the trace property and its related variations. Recall that an integral domain \(D\) satisfies the trace property if for every ideal \(I\) of \(D\), \(II^{-1}\) is either equal to \(D\) or is a prime ideal of \(D\). This talk is based on a joint work with Gyu Whan Chang.

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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.

Abstract: A set \(X\) together with a binary operation \(*\) is a quandle, whenever the following conditions hold: \(x * x = x\) for every \(x \in X\); for every \(x, y \in X\) there exists a unique \(z \in X\) such that \(x = z * y\); for every \(x, y, z \in X\) we have \((x * y) * z = (x * z) * (y * z)\). Let \(R\) be a commutative ring with \(1 \ne 0\). In this talk, we consider quandles on subsets of the polynomial ring \(R[x_1, \ldots, x_n]\) that are stable under the action of polynomial automorphisms. We show that polynomial automorphisms restricted to these sets can be viewed as automorphisms of quandles. We also show that the natural epimorphism mapping every polynomial to its polynomial function defined via evaluation is, under certain circumstances, a quandle epimorphism. Finally, we show that every polynomial automorphism in one variable can be uniquely factorized in the polynomial ring \(R[x]\) as a product of a monic linear polynomial and an invertible polynomial. This allows us to equip the polynomial automorphism group with a quandle structure which is isomorphic to the product of a quandle on the translation group of the ring \(R\) and a quandle on \(R[x]^{\times}\).

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.

Abstract: Hardy fields form a part of asymptotic analysis which is amenable to algebraic methods. In this talk I will explain a theorem which permits the transfer of statements concerning algebraic differential equations between Hardy fields and related structures, akin to the “Tarski Principle” in semi-algebraic geometry, and sketch some applications, including to linear differential equations. (Joint work with L. van den Dries and J. van der Hoeven.)

Abstract: Let \(E\) be a quadratic vector space equipped with a non-degenerate symmetric bilinear form, over a field of characteristic not 2. An \(n\)-frame is an element of \(E^n\), that is, an ordered \(n\)-tuple of vectors. An \(n\)-frame is orthogonal if the n vectors are pairwise orthogonal. Let \(S\) be the polynomial ring in the coordinates of \(E^n\), and let \(I\) be the ideal generated by the \((n \choose 2)\) quadratic polynomials expressing the orthogonality condition for \(n\)-frames. The vanishing locus of the ideal \(I\) is precisely the set of all orthogonal \(n\)-frames. In this work, we study the ring \(R = S/I\), which we call the ring of orthogonal frames. We determine the irreducible components, and study when the ring is a domain, a complete intersection, when it is normal or factorial. We give applications to the theory of Lovasz-Saks-Schrijver ideals of simple graphs, and solve problems proposed by Aldo Conca and Volkmar Welker. If time permits, we will discuss some interesting open problems. This is a joint work with Laura Casabella.

Abstract: The divisor class group traces back to Kummer and was introduced by Krull to quantify how far a ring is from being a unique factorization domain. In algebraic geometry, it measures how far a variety is from being locally factorial (i.e. the variety is covered by UFDs). In this talk, I introduce an opposite extremal notion, which I call maximal nonfactoriality. I mention why this notion is relevant in geometry and I give several examples. I then show how maximal nonfactoriality can be reformulated in terms of negative K-groups, a perspective that allows our main results to be extended to higher dimensions.

Abstract: Let \(G\) be a finite group. A finite collection of elements from \(G\), where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in \(G\) equals the identity element of \(G\). Then, the Gao constant \({\sf E}(G)\) of \(G\) is the smallest integer $\ell$ such that every sequence of length at least \(\ell\) has a product-one subsequence of length \(|G|\). For a positive integer \(n\), we denote by \(C_n\) a cyclic group of order \(n\). Let \(G = C_n \rtimes_s C_2\) with \(s^2 \equiv 1 \pmod n\) be a metacyclic group. The direct problem of \({\sf E}(G)\) in the case that \(s \equiv -1 \pmod n\) (that is, in the case of dihedral groups) was settled in [Bass, J. Number Theory 2007], while the respective inverse problem was settled in [Oh, Zhong, Isr. J. Math. 2020]. Later (see [Avelar, Brochero Martínez and Ribas, J. Combin. Theory Ser. A 2023]), this result was further generalized to \(C_n \rtimes_s C_2\), \(s \not\equiv \pm1 \pmod n\), except for the case that \(G = C_{3n_2} \rtimes_s C_2\) with \(n_2 \neq 1\), \(\gcd(n_2, 6) = 1\), \(s \equiv -1 \pmod 3\), and \(s \equiv 1 \pmod{n_2}\). In [Oh, Ribas, Zhao and Zhong, preprint 2025], we complete the remaining case and hence for all metacyclic groups of the form \(G = C_n \rtimes_s C_2\), the Gao constant and the associated inverse problem are now fully settled. This is a joint work with D.V. Avelar (Universidade Federal Fluminense), F.E. Brochero Martínez (Universidade Federal de Minas Gerais), J.S. Oh (Jeju National University), K. Zhao (Nanning Normal University) and Q. Zhong (University of Graz).

Abstract: We introduce the category \(\mathsf{AtoMon}\), a non-full subcategory of the category of monoids consisting of atomic monoids and homomorphisms that preserve atoms. By computing all limits and colimits, we show that \(\mathsf{AtoMon}\) is both complete and cocomplete. We also study the arithmetic behavior of products and coproducts, giving explicit formulas for important invariants related to factorization lengths. The goal is to connect category theory with factorization theory by providing a categorical setting for studying atomic factorizations. The seminar is based on joint work with Federico Campanini and Salvatore Tringali.