Discrete Mathematics 2010-2022

Let \(D\) be an integral domain. An ideal \(I\) of \(D\) is said to be a valuation ideal of \(D\) if there exists a valuation overring \(V\) of \(D\) such that \(I=IV\cap D\). Furthermore, a nonzero nonunit \(u\in D\) is called a valuation element if \(uD\) is a valuation ideal of \(D\). It is well known that every prime ideal of \(D\) is a valuation ideal, but primary ideals (in general) need not be valuation ideals. The domain \(D\) is called a VIFD (valuation ideal factorization domain) if every ideal of \(D\) is a finite product of valuation ideals of \(D\). Moreover, \(D\) is said to be a VFD (valuation factorization domain) if every nonzero nonunit of \(D\) is a finite product of valuation elements of \(D\).

In this talk, we study VIFDs and discuss their connections to VFDs. It is shown that \(D\) is an h-local Prüfer domain if and only if every ideal of \(D\) is a finite product of pairwise comaximal valuation ideals of \(D\). Moreover, we prove that every treed VIFD is an h-local Prüfer domain. Besides that, we investigate domains for which every principal ideal is a finite product of valuation ideals and we study the \(t\)-analogs and the \(w\)-analogs of the aforementioned types of domains. This talk is based on a joint work with Gyu Whan Chang.

An integral domain \(R\) is called atomic if every nonzero nonunit factors into irreducibles, while \(R\) satisfies the Let \(S \subset \mathbb{N}_0\) be a numerical monoid and let \(\mathcal P_{\mathrm{fin}} (S)\), resp \(\mathcal P_{\mathrm{fin},0}(S)\), denote the power monoid, resp. the restricted power monoid, of \(S\), that is the set of all finite nonempty subsets of \(S\), resp. the set of all finite nonempty subsets of \(S\) containing 0, with set addition as operation. The arithmetic of power monoids received some attention in recent literature, in particular in works of Tringali. We complement these investigations by studying algebraic properties of power monoids, such as their prime spectrum. Moreover, we show that almost all elements of \(\mathcal P_{\mathrm{fin},0} (S)\) are irreducible (i.e., they are not proper sumsets). This is joint work with Alfred Geroldinger.

Persistent sheaf cohomology is a way to generalize persistent (co)homology, a method from computational topology to analyse the topological structure of data sets. We show that most of the ideas of the theory of persistent homology for topological spaces generalize to sheaves and that we can obtain the usual theory as a special case of the sheaf theoretic version.

Given an ascending chain \((I_n)_{n\in\mathbb{N}}\) of \(\mathrm{Sym}\)-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals \((I_n^\vee)_{n\in\mathbb{N}}\). We provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of \(I_n^\vee\) is given by a polynomial in \(n\) for sufficiently large \(n\). The same is true for the number of orbit generators of minimal degree, this degree being a linear function in n eventually. The former result implies that the number of \(\mathrm{Sym}\)-orbits of primary components of \(I_n\) grows polynomially in \(n\) for large \(n\). As another application, we show that, for each \(i\geq 0\), the number of \(i\)-dimensional faces of the associated Stanley-Reisner complexes of \(I_n\) is also given by a polynomial in \(n\) for large \(n\).
Even though I might not be able to cover everything mentioned above in my talk, I will try to give a gentle introduction to \(\mathrm{Sym}\)-invariant ideals. This is joint work with Ayah Almousa, Kaitlin Bruegge, Uwe Nagel and Alexandra Pevzner.

Let \(V\) be a discrete valuation domain whose quotient field \(K\) is a global field. We show that for all positive integers \(k\) and \(2\leq n_1 \leq \ldots \leq n_k\) there exists an integer-valued polynomial on \(V\), that is, an element of \(\text{Int}(V) = \{ f \in K[X] \mid f(V) \subseteq V \}\), which has precisely \(k\) essentially different factorizations into irreducible elements of \(\text{Int}(V)\) whose lenghts are exactly \(n_1,\ldots,n_k\). This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in this case.
The talk is based on joint work with Victor Fadinger and Sophie Frisch.

By a standard result, every domain (commutative or noncommutative) with ACCP is atomic. Several constructions of commutative atomic domains without the ACCP are known. Of course, no such commutative domain can be finitely generated as algebra over a field. We discuss, in some detail, a construction of a finitely presented atomic noncommutative domain without ACCP. In particular, this yields a finitely presented noncommutative domain without bounded factorizations.

I will summarize some results related to the problem of characterizing rings over which every pure projective torsion free module is a direct sum of finitely presented ones. It appears that the existence of a 'non-trivial' pure projective module can be often deduced from an 'unexpected' idempotent ideal of the endomorphism ring of a finitely presented module. It the talk we focus on commutative noetherian domains, integral group algebras of finite groups and partially also on h-local domains.

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called , for the \(\preceq\)-non-units of a (multiplicatively written) \((H, \preceq)\) (i.e., a monoid \(H\) endowed with a preorder \(\preceq\)), where an element \(u \in \allowbreak H\) is a if \(u \preceq 1_H \preceq u\) and a \(\preceq\)-non-unit otherwise. The ``building blocks'' of these factorizations are the of \(H\) (i.e., the \(\preceq\)-non-units \(a \in H\) that cannot be written as the product of two \(\preceq\)-non-units \(x \prec a\) and \(y \prec a\)); and it is interesting to look for sufficient conditions for the \(\preceq\)-factorizations of a \(\preceq\)-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the talk.

In particular, we will discuss the interaction between (i.e., a refinement of \(\preceq\)-factorizations used to counter the ``blow-up phenomena'' that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the ``local behaviour'' of the pair \((H, \preceq)\). Besides a number of examples and remarks that will guide the audience through, we will state (and outline the proof of) a number of arithmetic results, a part of which are new already in the basic case where \(\preceq\) is the divisibility preorder on \(H\) (and hence in the setup of the classical theory).

Many algebraic properties of commutative rings and monoids have been investigated concerning whether the corresponding monoid algebra has these properties, if the ring and the monoid do. In some cases, for instance being completely integrally closed, it turned out, that the monoid algebra has the property if and only if the ring and the monoid satisfy it. In other cases, the situation is more involved. For instance Chouinard proved, that a monoid algebra is a Krull domain if and only if it is constructed from a Krull domain and a torsion-free Krull monoid with appropriate group of units. In this talk, we discuss the related question concerning the property of being weakly Krull. Also, we talk about the distribution of prime divisors in Krull and affine monoid algebras. As a combination of these results, we deduce factorization theoretical statements about weakly Krull affine monoid algebras.

the stable rank \({\mathrm{s.r.}}(R)\) of a ring \(R\) is a fundamental invariant in algebraic K-theory. For instance, \({\mathrm{s.r.}}(R)\le n\) ensures that \({\mathrm{GL}_n}(R)\) acts transitively on unimodular column vectors of length \(m\) for all \(m>n\).

\({\mathrm{s.r.}}(R)\) is a concept of dimension, comparable to \(\dim(R) +1\) for global rings (\(\dim(R)\) being Krull dimension), but note that \({\mathrm{s.r.}}(R)=1\) always holds for local rings.

Let \[{\mathrm{Int}}(D) = \{ f\in K[x] \mid f(D)\subseteq D\},\] where \(K\) is the quotient field of \(D\), be the ring of integer-valued polynomials of a domain \(D\).

We determine \({\mathrm{s.r.}}({\mathrm{Int}}({\mathcal{O}}_K))\) for the ring of algebraic integers \({\mathcal{O}}_K\) in all number fields \(K\) that are not imaginary quadratic. The imaginary quadratic case is wide open (except when \({\mathcal{O}}_K\) is Euclidean, in which case \({\mathrm{Int}}({\mathcal{O}}_K)\) is amenable to our methods).

An integral domain \(R\) is called atomic if every nonzero nonunit factors into irreducibles, while \(R\) satisfies the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals of \(R\) eventually stabilizes. One can readily check that every integral domain satisfying the ACCP is atomic. The converse was asserted (without proof) by Paul Cohn back in 1968. The first counterexample for this wrong assertion was constructed by Anne Grams in 1974. Every known construction of atomic domains not satisfying the ACCP comes with a touch of sophistication. During this talk, we will take a tour through some constructions of atomic domains not satisfying the ACCP, including a recent generalization of the Grams' first example as well as some recent constructions using pullbacks of rings and monoid algebras.

• 14:15 – 15:15 Joan Bosa (University of Zaragoza, Spain)
  The realization problem for von Neumann regular rings
  

• 15:30 – 16:30 Roozbeh Hazrat (Western Sydney University, Australia)
  Sandpile models and Leavitt algebras

 
Joan Bosa:
Abstract. The realization problem for von Neumann (vN) regular rings asks whetherall conical refinement monoids arise from monoids induced by the projectivemodules over a vN regular ring. We will quickly overview this problem, andshow the last developments on it.
 
Roozbeh Hazrat:
Abstract. Sandpile models are about how things spread along a grid (think of Covid!) and Leavitt algebras are algebras associated to graphs. We relate these two subjects, via the monoids associated to sandpiles and K-groups of algebras.

Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. These arrangements are actively studied in connection to the theory of cluster algebras. In the setting of cluster algebras, the notion of a frieze pattern can be seen in a new light. A frieze is a ring homomorphism \(h\) between the cluster algebra and the ring of integers (or other domains in a more general framework). A frieze is unitary if the following implication holds: the frieze \(h\) maps all cluster variables to positive integers implies that there exists a cluster \(X\) such that \(h(x)=1\) for all cluster variables \(x\) in \(X\). We will give a brief definition of cluster algebras and friezes. We will explore which of the usual cluster types always give unitary friezes and which ones do not, and we will propose a conjecture in this direction.

Based on a study of torsion points on abelian varieties, Masser and Zannier recently showed that the set of values of t such that an algebraic function \(f(x,t)\), depending on a parameter \(t\), can be integrated (with respect to \(x\)) in elementary terms, is small, in fact, often finite, if \(f\) cannot be integrated generically (i.e., when \(t\) is considered to be transcendental). As integration can be seen as solving a very particular differential equation, it is natural to wonder about generalizations to linear differential equations. Based on a specialization theorem for the differential Galois group of a linear differential equation, we will provide a result in this spirit. Another application of our specialization theorem is a proof of Matzat's conjecture in full generality: The absolute differential Galois group of a one-variable function field is a free proalgebraic group. All of the above concepts will be introduced. Indeed, a large part of the talk will be a gentle introduction to differential Galois theory. This is joint work with Ruyong Feng.

We generalize the notion of Davenport constants to a `higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples that capture the essence of these higher degree Davenport constants are the following. 1) Suppose \(n = 2^k\), then every sequence of integers \(S\) of length \(2n\) contains a subsequence \(S'\) of length at least two such that \(\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}\) and the bound is sharp. 2) Suppose \(n \equiv1 \pmod{2}\), then every sequence of integers \(S\) of length \(2n -1\) contains a subsequence \(S'\) of length at least two such that \(\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}\). These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions have to vanish on the elements of a subsequence.

Let \(G\) be an additive finite abelian group and \(\Gamma \subset \operatorname{End} (G)\) be a subset of the endomorphism group of \(G\). A sequence \(S = g_1 \cdot \ldots \cdot g_{\ell}\) over \(G\) is a (\(\Gamma\)-)weighted zero-sum sequence if there are \(\gamma_1, \ldots, \gamma_{\ell} \in \Gamma\) such that \(\gamma_1 (g_1) + \ldots + \gamma_{\ell} (g_{\ell})=0\). We study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.

In the context of representation theory, an oriented graph is called a `quiver'. A `dimer quiver' \(Q\) on a compact surface \(\Sigma\) is a quiver that embeds in \(\Sigma\), such that each connected component of \(\Sigma \setminus Q\) is simply connected and bounded by an oriented cycle of \(Q\). I will describe noncommutative algebras constructed from such quivers, called `ghor algebras'. I will then sketch what their centers look like, as well as their simple modules. Finally, I will explain how something magical happens between the Krull dimension of the center of a ghor algebra and the projective dimension of a special simple module, using in part the Koszul complex and the topology of \(\Sigma\). This is joint work with Karin Baur.

An automorphism \(\alpha\) of a group \(G\) is called a cyclic automorphism if the subgroup \(\langle x,x\alpha \rangle\) is cyclic for every element \(x\) of \(G\). The concept of cyclic automorphism is a natural generalisation of the one of power automorphism, i.e. an automorphism which maps each subgroup of a group onto itself. It can be proved that cyclic automorphisms share a number of meaningful properties with power automorphisms and their behaviour can be described thoroughly in many cases. In this seminar, we will show some main properties of cyclic automorphisms of a group, providing a characterisation of groups in which every automorphism is cyclic and presenting some generalisations and open problems on the subject.

The Chevalley-Warning Theorem tells that for a finite field \(\mathbb{F}\) and \(f_1, \ldots, f_r \in \mathbb{F}[x_1,\ldots, x_N]\) with \(\sum_{i=1}^r \deg(f_i)\le N\), the number of solutions of \(f_1 = \ldots = f_r = 0\) is divisible by the characteristic of \(\mathbb{F}\).

With a suitable definition of the degree of a function, this theorem also holds if \(f_1, \ldots, f_r\) are mappings on an abelian \(p\)-group. Here, the notion of degree is defined using the forward difference operator \(\Delta_a\) and Frech\(\’{e}\)t’s functional equation \(\Delta_{a_1} \ldots \Delta_{a_n} f = 0\). We will report on recent results on the degree and on applications to proving bounds on the number of solutions of systems of equations.
(Joint research with J. Moosbauer; partially supported by the Austrian Science Fund FWF (P33878))

A generalization of Krull domains are so called weakly Krull domains. They have many important applications, e.g. in number theory. We will discuss when a monoid algebra is weakly Krull. Based on this and some results of the previous talk, we will deduce factorization theoretic statements for weakly Krull affine monoid algebras.
This talk is based on joint work with Gyu Whan Chang and Daniel Windisch.

In this talk I will present joint work with V. Fadinger concerning class groups and prime divisors of monoid algebras. Corollaries include new structure theorems for factorizations in monoid algebras and the existence theorem for primes in arithmetic progressions of function fields over finite fields due to Pollack.

Additive combinatorics has been described as “a marriage of number theory, harmonic analysis, combinatorics, and ideas from ergodic theory, which aims to understand very simple systems: the operations of addition and multiplication and how they interact.” The term was coined this century by Fields Medalist Terrence Tao to describe a collection of results, most appearing over the speaker’s lifetime but some dating back to Cauchy in the early 19th-century. Other results that belong to the subject area include a masterpiece theorem of Abel Prize winner Endre Szemer\(\’{e}\)di, which states that in any set of integers with positive density there are arbitrarily long arithmetic progressions, and the Green-Tao Theorem, which says that the primes contain such progressions.
The list of techniques that have been used or developed for tackling these problems include, for instance, the probabilistic method, Szemer\(\’{e}\)di’s Regularity Lemma, and Ramsey theory. However, the speaker’s favorite method for these problems is the so-called polynomial method. For instance, Noga Alon has shown that under certain conditions one can know much about the zero locus of a polynomial by strengthening a particular case of David Hilbert’s Nullstellensatz. When one is able to “capture” the set of objects being studied in the zero locus of a polynomial whose degree is “small”, then Alon’s Combinatorial Nullstellensatz has been able to give many deep arithmetical results, often with sharp bounds (and elegant, short proofs), not attainable by other methods.

In this talk we present our recent joint work on secant and tangential varieties of toric varieties. In particular, our focus will be on Cohen-Macaulay and Gorenstein classification of secant and tangential varieties of the Segre-Veronese varieties.

This talk is based on recent joint work with Mateusz Michałek, Martin Vodicka and Piotr Zwiernik.

The discriminant arises naturally in many situations in mathematics, often as a measure of size or arithmetic complexity. In this talk, we pursue discriminants in the context of pure algebra with a view towards arithmetic. Generalizing a theorem of Stickelberger, we prove that the discriminant of a free algebra of finite rank over a commutative ring is a square modulo 4. Our proof hinges on a few delightful linear algebra observations. This is joint work with Asher Auel and Owen Biesel.

The derived set of a topological space \(X\) is the set of all its non-isolated points; this construction can be iterated, obtaining a chain of closed subsets of \(X\) indexed by the ordinal numbers. In this talk, I will present two algebraic analogues of this construction.

The first one is constructed from pre-Jaffard families, a generalization of the concept of Jaffard families: while the latter have strong factorization properties (for example in studying star operation or length functions), a derived set-like construction allows to show that also the former enjoy some good properties when dealing with stable semistar operations or singular length functions.

The second analogue involves almost Dedekind domains: starting from the concept of SP-domains and critical maximal ideals, we show how, in many cases, a derived-set like construction allow to represent the group \(\mathrm{Inv}(D)\) of invertible ideals as a direct sum of groups of continuous functions, and in particular to show that this group is free.

A sequence \(x(n)\) is called \(k\)-regular, if the set of subsequences \(x(k^j n + r)\) is contained in a finitely generated module. In this talk, we will consider the asymptotic growth of \(k\)-regular sequences. When is it possible to compute it? ...and when not? If possible, how precisely can we compute it? If not, is it just a lack of methods or are the underlying decision questions recursively solvable (i.e., decidable in a computational sense)? We will discuss answers to these questions. To round off the picture, we will consider further decidability questions around \(k\)-regular sequences and the subclass of \(k\)-automatic sequences.

Let \(H\) be a commutative cancellative monoid. Then \(H\) is called a transfer Krull monoid if there exists a (commutative cancellative) Krull monoid \(B\) and a transferhomomorphism from \(H\) to \(B\). The transfer Krull property has been studied in various contexts (including noncommutative monoids and noncancellative monoids). For instance, it is well known that both Krull monoids and half-factorial monoids are transfer Krull monoids. That said, a transfer Krull monoid is (in general) neither a Krull monoid nor half-factorial. Furthermore, it was recently shown in a work of A. Geroldinger and Q. Zhong that whether \(H\) is transfer Krull is determined by the overmonoids of \(H\).

In this talk we provide several conditions that force a transfer Krull monoid to be half-factorial (or a Krull monoid). In particular, we elaborate how the root closure can affect the underlying monoid. Besides that, we study transfer Krull monoids in the context of (generalized) Cohen-Kaplansky domains and weakly factorial monoids. Finally, we complement these results by several counterexamples (involving affine monoids) and discuss their limitations. This talk is based on a joint work with Aqsa Bashir.

Loosely speaking, (shortly, ) are a unifying framework for a variety of problems that, in one way or another, involve the factorization or decomposition of certain elements of a monoid into a product (or sum, composition, etc.) of certain other elements that, in a sense, cannot be ``broken up into smaller pieces'' and can therefore be viewed as kind of ``minimal'': Examples of ``factorization problems'' along this line of thinking include additive decompositions into multiplicative units in (commutative or non-commutative, associative, unital) rings ; cyclic decompositions of permutations in the symmetric group of a finite set; idempotent factorizations of the ``singular elements'' of a monoid, with emphasis on submonoids of the ``monoid of zero divisors'' of a ring ; factorizations of the non-units of an arbitrary monoid into atoms (in the sense of ), irreducibles (in the sense of ), or ``similarity irreducibles'' (as implicitly defined in ); and so on.

More formally, an is an ordered triple consisting of a morphism \(|\cdot|\) (the of \(\Phi\)) from a multiplicatively written premonoid \(\mathcal H = (H, \preceq_H)\) to an additively written premonoid \(\mathcal K = (K, \le_K)\), a set \(A \subseteq H\) of , and a preorder \(\sqsubseteq\) on the free monoid \(\mathscr F(A)\) on \(A\), all subjected to some ``natural restrictions''(see for terminology); in particular, the norm is used to ``measure the length'' of \(\Phi\)-factorizations, where a of an element \(x \in H\) is a word \(\mathfrak a \in \mathscr F(A)\) that is \(\sqsubseteq\)-minimal in the inverse image of \(x\) under the canonical homomorphism \(\mathscr F(A) \to H\). There are at least two basic questions one may want to ask about \(\Phi\):

• Is there anything meaningful (e.g., a non-trivial characterization) that can be said about the elements of \(H\) that have at least one \(\Phi\)-factorization?

• How to qualify and quantify the departure of \(\Phi\) from a condition of ``factoriality'' (however defined)?

Sensible answers to these questions critically depend on the actual features of the ``component parts'' of \(\Phi\); but there are a few interesting things that can be proved in some generality. E.g., let the \(\varrho_\Phi(x)\) of an element \(x \in H\) be the inf, with respect to the standard ordering of \(\mathbf R_{\ge 0} \cup \{\infty\}\), of the set of all rational numbers \(m/n\) with \(m, n \in \mathbf N^+\) such that \(n\, |\mathfrak a| \le_K m \, |\mathfrak b|\) for all \(\Phi\)-factorizations \(\mathfrak a\) and \(\mathfrak b\) of \(x\), with the understanding that \(\inf \emptyset := \infty\) and the norm of an \(A\)-word is the sum of the norms of its ``letters'' (note that, if \(x\) has no \(\Phi\)-factorizations, then \(\varrho_\Phi(x) = 0\)). Next, define the \(\varrho(\Phi)\) of \(\Phi\) as the sup of the set \(\{\varrho_\Phi(x) \colon x \in H\} \subseteq \mathbf R_{\ge 0} \cup \{\infty\}\), with \(\sup \emptyset := 0\). Following , we will show

• that \(\varrho(\Phi)\) is a (non-negative) rational number (and hence finite) if \(H\) is commutative, \(A\) is finite, and \(|\cdot|\) and \(\sqsubseteq\) have ``reasonable properties'';

• how to extend the result to -spaces that are ``homomorphic'' to any of those considered in (i);

• that \(\varrho(\Phi)\) need not be finite even for the ``neoclassical -space'' of a reduced, atomic, cancellative, \(3\)-generated monoid, in which case \(A\) is the set of atoms of \(H\), the norm is \(\mathbf N\)-valued with \(|a| = 1\) for each \(a \in A\), and \(\mathfrak a \sqsubseteq \mathfrak b\) if and only if the \(A\)-word \(\mathfrak a\) ``embeds'' into the \(A\)-word \(\mathfrak b\) ``modulo associates and permutations'' (whence \(\Phi\)-factorizations are nothing but ``minimal atomic factorizations'' in the sense of , which are in turn a refinement — especially suited to non-commutative or ``highly non-cancellative'' monoids — of atomic factorizations as per the classical theory).

Incidentally, (ii) yields a generalization (from commutative unit-cancellative to arbitrary commutative monoids) of .

Let \(R\) be a commutative ring with identity and let \(I\) be a proper ideal of \(R\). Then \(I\) is called an \(OA\)-ideal of \(R\) if for all nonunits \(x,y,z\in R\) with \(xyz\in I\), we have that \(xy\in I\) or \(z\in I\). Furthermore, \(I\) is said to have an \(OA\)-factorization if \(I\) is a finite product of \(OA\)-ideals of \(R\) and \(R\) is called an \(OAF\)-ring if every proper ideal of \(R\) has an \(OA\)-factorization. In this talk, we show that \(R\) is an \(OAF\)-ring if and only if every proper \(2\)-generated ideal of \(R\) has an \(OA\)-factorization if and only if \(\dim(R)\leq 1\) and every proper principal ideal of \(R\) has an \(OA\)-factorization. We classify the rings whose proper principal ideals have an \(OA\)-factorization. Moreover, we discuss the connections between \(OAF\)-rings, general \(ZPI\)-rings and atomic pseudo-valuation domains. Finally, we apply the aforementioned results to ring theoretic constructions such as the idealization of a module. This talk is based on a joint work with Abdelhaq El Khalfi, Mohammed Issoual and Najib Mahdou.

Let \(G\) be a finite group. For a positive integer \(d\), let \(\mathsf{s}_{d\mathbb{N}}(G)\) denote the smallest integer \(\ell\) such that every sequence \(S\) over \(G\) of length \(|S|\geq \ell\) has a nonempty product-one subsequence \(T\) with \(|T|\equiv 0\) (mod \(d\)). In this talk, we mainly study this invariant for dihedral groups \(D_{2n}\).

We see that under some conditions the class of noetherian domains possessing finite partitive functions from finitely generated modules to a set of ordinal numbers are BF. In particular, we see that noetherian rings with Auslander dualizing complex are BF. Some examples of these rings are all known noetherian Hopf algebra (including the group ring \(kG,\) where \(k\) is a field and \(G\) is a polycyclic-by-finite group) and Weyl algebras \(A_n(k),\) where \(k\) is field of characteristic zero. This is an ongoing joint work with K. Brown and D. Smertnig.

Let \(F\) be a field and let \(F\langle x_1, \ldots, x_n \rangle\) denote the free algebra in (noncommuting) variables \(x_1,\ldots,x_n\); we call its elements noncommutative polynomials. For an algebra \(A\) and a noncommutative polynomial \(f \in F\langle x_1, \ldots, x_n \rangle\) we call the set \[f(A) = \{f(a_1,\ldots,a_n) \mid a_1,\ldots,a_n \in A\} \subseteq A\] the image of \(f\) in \(A\).

The L'vov-Kaplansky conjecture states that the image of a multilinear polynomial \(f\), i.e., a polynomial of the form \(f = \sum_{\sigma \in S_n} \lambda_\sigma x_{\sigma(1)} \dots x_{\sigma(n)}\) for some \(\lambda_\sigma \in F\), in the matrix algebra \(M_k(F)\) for any \(k \in \mathbb{N}\) is one of the following four vector spaces: \(\{0\}\), the space of scalar matrices \(F\), the space of trace zero matrices \({\rm sl}_k(F)\), and \(M_k(F)\). The conjecture holds for \(k=2\), but even with a lot of effort by several authors to solve the conjecture, it remains only partially solved even for \(n=3\) (see for a detailed survey article regarding this).

Various variations of the L'vov-Kaplansky conjecture were also studied by many authors. For instance, if an algebra \(A\) has a surjective inner derivation, i.e., there exists an element \(v \in A\) such that the map \(x \mapsto vx-xv\) is surjective, then any nonzero multilinear polynomial \(f\) is surjective on \(A\), i.e., \(f(A)=A\). Examples of such algebras are the \(n\)-th Weyl algebra, \(A_n\), and the algebra of endomorphisms of an infinite-dimensional vector space \(V\), \({\rm End}(V)\). Another variation of the L'vov-Kaplansky conjecture is to consider the images of multilinear polynomials in the algebra of finitary matrices, i.e., infinite matrices with only finitely many nonzero entries. In D. Vitas, Images of multilinear polynomials in the algebra of finitary matrices contain trace zero matrices it is proven that for any nonzero multilinear polynomial \(f\) its image in the algebra of finitary matrices contains all trace zero matrices.

In the presentation I will describe the L'vov-Kaplansky conjecture in detail, present some well-known results regarding the conjecture, and describe the method used to prove the two theorems stated in the previous paragraph, which is, surprisingly enough, the same in both cases.

This talk will be centered around the following question: How to prove that an ideal of a polynomial ring cannot be written as a product of two proper ideals?
We will also talk about some of applications on the arithmetic of the monoid of all non-zero ideals of a polynomial ring. This talk is based on recent joint work with Alfred Geroldinger and ongoing joint work with Laura Cossu.

A central object when studying the factorization theory of a BF-monoid \(H\) is its system of sets of lengths \(\mathcal L(H)\), which is a collection of finite sets of non-negative integers. So it is very natural to ask for a characterization of those collections of finite sets of non-negative integers occurring as systems of sets of lengths of BF-monoids, in terms of standard set operations and relations such as set addition and inclusion. This question was posed by Geroldinger and Zhong in their survey article Factorization theory in commutative monoids. We showed that such a characterization is not possible and will discuss the proof as well as the machinery it uses.

In the article “Criteria for Equality of Ordinary and Symbolic Powers of Primes,” Melvin Hochster proved several criteria for when for a prime ideal \(P\) in a commutative Noetherian ring with identity, \(P^n = P^{(n)}\) for all positive integers \(n\). I will discuss a generalization of the criteria to radical ideals.

• 16:00 – 16:45 Günter Lettl (University of Graz)
  Atoms of completely integrally closed submonoids of \(\mathbb{Z}^2\)
  

• 17:00 – 17:30 Christian Elsholtz (Graz University of Technology)
  Fermat’s Last Theorem implies Euclid’s infinitude of primes

 
Günter Lettl:
Abstract. For positive \(\alpha \in \mathbb R\) we consider the monoid \(H_\alpha = \{ (x,y) \in \mathbb N_0^2 \mid y \le \alpha x \}\) and show how the atoms of \(H_\alpha\) can be obtained from the continued fraction expansion of \(\alpha\).
 
Christian Elsholtz:
Abstract. We show that Fermat’s last theorem (for any fixed exponent \(n \geq 3\)) and a combinatorial theorem of Schur on monochromatic solutions of \(a + b = c\) implies that there exist infinitely many primes. Similarly, we discuss implications of Roth’s theorem on arithmetic progressions, Hindman’s theorem, and infinite Ramsey theory toward Euclid’s theorem.
As a consequence we see that Euclid’s theorem is a necessary condition for many interesting (seemingly unrelated) results in number theory and combinatorics.

Let \(\preceq\) be a preorder on a monoid \(H\). An element \(x\in H\) is said to be a \(\preceq\)-unit if \(x\preceq 1_{_H}\preceq x\); a \(\preceq\)-non-unit if it is not a \(\preceq\)-unit; and a \(\preceq\)-irreducible of degree \(s\ge 2\) if, for every \(1\le k\le s\), there do not exist any \(\preceq\)-non-units \(y_1, \ldots, y_k\) with \(y_1 \prec x, \ldots, y_k \prec x\) such that \(x = y_1 \cdots y_k\). We generalize one of the main results of by establishing that if \(\preceq\) is artinian, every \(\preceq\)-non-unit of \(H\) is a finite product of \(\preceq\)-irreducibles of degree \(s\). This abstract factorization theorem provides a general recipe to prove that every ``large element'' of \(H\) factors through a subset \(A\) of \(H\) whose elements ``cannot be broken up into smaller pieces'' (but need not be atoms in the sense of the classical theory of factorization).

With this in mind, we show in particular that, for certain choices of \(H\) and \(\preceq\), the \(\preceq\)-irreducibles (of degree \(3\)) are idempotent. Among other things, this allows us to recover, by virtue of the above factorization theorem, two classical results of J.A. Erdos (1967) and J. Fountaine on idempotent factorizations of singular matrices, as well as a pioneering result of J.M. Howie (1966) on idempotent factorizations in the transformation monoid of a finite set. In each of these cases, we also obtain upper bounds on the length of a shortest idempotent factorization. This is joint work with S. Tringali.

A fundamental result in algebraic geometry is that the rational points of an elliptic curve over a number field are finitely generated. In an effort to grasp the structure of this group, the notion of Selmer group and Shafarevich-Tate groups were introduced. We will restate some of the most significant theorems on the Birch and Swinnerton-Dyer conjecture, as well as construct Shafarevich-Tate groups in the setting of abelian varieties.
Using an Euler system of Heegner points, we will generalize some of the results by V. A. Kolyvagin, and W. G. McCallum on the structure of Shafarevich-Tate groups from Elliptic Curves to modular abelian varieties.

The singularity set of a noncommutative polynomial \(f=f(x_1,\dots,x_d)\) is the graded set \(Z(f)=(Z_n(f))_n\), where \(Z_n(f)=\{X \in M_n^d: \det f(X) = 0\}.\) Two main results will be presented. Firstly, irreducible factors of \(f\) are shown to be in a natural bijective correspondence with irreducible components of \(Z_n(f)\) for every sufficiently large \(n\). In particular, \(f\) is irreducible if and only if \(Z_n(f)\) is eventually irreducible. Secondly, we give Nullstellensätze for noncommutative polynomials. For instance, given two noncommutative polynomials \(f_1,f_2\), we have \(Z(f_1) \subset Z(f_2)\) if and only if each irreducible factor of \(f_1\) is (up to stable associativity) an irreducible factor of \(f_2\).
The talk is based on joint works with Jurij Volčič and Bill Helton.

A ghor algebra is type of quiver algebra whose quiver embeds in a compact surface, and whose relations come from the perfect matchings of its quiver. In this talk I will introduce such algebras with examples, and explain how the relations of a ghor algebra induce homotopy and homology relations on the paths and cycles in its quiver. I will also describe relationships between its representation theory, the genus of its surface, and certain algebraic and geometric properties of its center.

The arithmetic of monoids of zero-sum sequences over finite abelian groups is a classical subject due to their crucial role in understanding the arithmetic of (transfer) Krull monoids. In recent years, weighted zero-sum sequences received increased attention, yet the focus was on zero-sum constants. We present the begining of a systematic study of the arithmetic of these monoids, including results on unions of sets of lengths and various results on the elasticity of these monoids. More detailed results are obtained for the special case that the set of weights is \(\{+1, -1\}\). Applications to factorizations of norms of algebraic integers are also discussed. This is joint work with Safia Boukheche, Kamil Merito and Oscar Ordaz.

An atomic monoid is length-factorial if each two distinct factorizations of any element have distinct factorization lengths. We provide a characterization of length-factorial Krull monoids in terms of their class groups and the distribution of prime divisors in the classes. Main parts of the proofs are done with methods from additive combinatorics.

Let \(G\) be a multiplicatively written group and let \(H\) be a subset of \(G\). We consider the set of all finite unordered sequences over \(H\) for which there exists a permutation of the elements whose product is 1. We can endow this set with a product operation (concatenation of sequences) so that it becomes a monoid. It is called the monoid of product-one sequences over \(H\) and we will investigate its algebraic and arithmetic properties, with a special emphasis on subsets of the infinite dihedral group. This is joint work with Qinghai Zhong.

A cap is a point set in affine or projective space without any three points on any line. We will discuss the current state of the art, and give an exponential improvement for the size of caps of AG(n, p), which one can think of as (Z/pZ)^n, and PG(n,p).

For certain primes, 5,11,17,23,29 and 41, we improve the asymptotic growth of these caps, for example, when p=23 from (8.091...)^n to (9-o(1))^n, as n tends to infinity.

We discuss similar results for progression-free sets in (Z/mZ)^n.

This is joint work with Gabriel Lipnik, Benjamin Klahn and Peter Pal Pach.

Let \(f_k(m,n)\) denote the number of solutions of \(\frac{m}{n}= \frac{1}{x_1} + \cdots + \frac{1}{x_k}\) in positive integers \(x_i\).
The case \(k=2\) is essentially a question about a divisor function,
the case \(k=3\) is closely related to a sum of certain divisor functions.
For the case \(k=3, m=4\) Erdős and Straus conjectured that \(f_3(4,n)>0\), for all \(n>1\).
For general \(k\), the case \(m=n=1\) received special attention, and even has applications in discrete geometry, (e.g.: Benjamin Nill, Volume and lattice points of reflexive simplices)

We give a survey on previous results and report on new results over the last years, e.g.:
Theorem 1: There are infinitely many primes \(p\) with \(f_3(m,p)\gg\exp (c_m \frac{\log p}{\log \log p})\).

Theorem 2: For fixed \(m\) and almost all integers \(n\) one has: \(f_3(m,n)\gg (\log n)^{\log 3+o_m(1)}\).

Theorem 3: \(f_3(4,n)=O_{\varepsilon}\left(n^{3/5+\varepsilon}\right)\), for all \(\varepsilon >0\).
There are related but more complicated bounds when \(k\geq 4\).

Theorem 4: \(f_k(1,1)\ll_{\varepsilon} c^{(\frac{1+\varepsilon}{5}+\varepsilon) 2^k}\), where \(c\approx 1.264\).

There is a double exponential growth, as the following result of Konyagin shows: \(f_k(1,1,) \gg \exp(\exp(\frac{(log 2)(\log 3)+o(1)}{3} \frac{k}{\log k}))\) and this is still true (with a different constant) if one restricts to solutions with odd integers \(x_i\).

The Green-Tao theorem states that the set of primes contain arbitrarily long arithmetic progressions. It is reasonable to wonder whether the same holds for certain subsets of the primes, and also whether arithmetic progressions may be replaced by other linear configurations, i.e. solutions to systems of linear equations. I will review known (and unknown) results in that area, and talk about an ongoing work with Joni Teräväinen and Fernando Shao, where we provide a general criterion for a set of integers to contain linear configurations, and show that this criterion applies to certain interesting sets of primes such as Chen primes or almost twin primes.

The degree of a dominant rational map \(f:\mathbb{P}^n\to \mathbb{P}^n\) is the common degree of its homogeneous components. By considering iterates of \(f\), one can form a sequence \({\rm deg}(f^n)\), which is submultiplicative and hence has the property that there is some \(\lambda\ge 1\) such that \(({\rm deg}(f^n))^{1/n}\to \lambda\). The quantity \(\lambda\) is called the first dynamical degree of \(f\). We'll give an overview of the significance of the dynamical degree in complex dynamics and describe an example in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller and Mattias Jonsson.

We present some basic results concerning isoartinian and isonoetherian rings and modules. These algebraic objects generalize the classical well known notions of noetherian and artinian modules. We will see that, among these, semiprime right isoartinian rings have a particularly good description. If time allows, we will talk of the isoradical of rings and modules, generated by isosimple modules.

Our motivating goal is the study of factorization in Krull Domains \(H\) with finitely generated class group \(G\). For an integer \(k\geq 1\), let \(\rho_k(H)\) denote the \(k\)-th elasticity of \(H\), which is the maximal number of atoms (irreducibles) in any re-factorization of a product of \(k\) atoms (with \(\rho_k(H)=\infty\) if there is no such finite bound). The elasticity of \(H\) is then \(\rho(H)=\sup\{\rho_k(H)/k:\; k\geq 1\}\). Having finite elasticities is a key indicator that factorization, while not unique, is not completely wild. This talk will describe, in brief, parts of our recent characterization of finite elasticity for any Krull Domain with finitely generated class group \(G\) by means of Convex Geometry. Most results are valid for the more general class of Transfer Krull Monoids (over a subset \(G_0\) of a finitely generated abelian group \(G\)). I will highlight some of the arithmetic consequences of the characterization, including the implied finiteness of the set of distances \(\Delta(H)\) and catenary degree \(\mathsf c(H)\), as well as the Structure Theorem for Unions holding, and broadly describe some of the key aspects of Convex Geometry involved in the lengthy proofs.

Let \(D\) be an integral domain with quotient field \(K\). Then \(D\) is called an order in a Dedekind domain if \(D\) is a noetherian one-dimensional domain with nontrivial conductor in its integral closure. Furthermore, an ideal \(I\) of \(D\) is called stable if \(I\) is an invertible ideal of the ring of multipliers of \(I\) (i.e., \(I\) is an invertible ideal of \((I:I)=\{x\in K\mid xI\subseteq I\}\)). The domain \(D\) is called stable if every nonzero ideal of \(I\) is stable. It is well-known that every order in a quadratic number field is a stable order in a Dedekind. In this series of talks, we will discuss some arithmetical properties of stable orders in Dedekind domains and their generalizations.
Talk 1: In the first talk, we give a brief introduction to stable domains and several types of arithmetical invariants. We discuss the most important results of the known literature and complement them by several counterexamples. We will put special emphasis on papers of V. Barucci and M. Roitman which study the connections between stable domains and Mori domains.
Talk 2: The second talk will be based on a joint work with A. Bashir and A. Geroldinger. Let \(D\) be a stable order in a Dedekind domain. We will prove that the monoid of nonzero ideals of \(D\) is transfer-Krull (resp. half-factorial, resp. of catenary degree at most two) if and only if the monoid of invertible ideals of \(D\) is transfer-Krull (resp. half-factorial, resp. of catenary degree at most two). We also show that the aforementioned characterization does not hold for arbitrary orders in Dedekind domains. Finally, we supplement these results by providing other arithmetical advantages of stability.

Talk 1: We give an overview of some classical results in the theory of Prüfer domains, including characterizations in terms of ideal arithmetic and valuation theory. Moreover, we consider various examples and state some open problems.
Talk 2: For an integer \(n \geq 1\), we consider the binomial polynomial \(\binom{x}{n}\) in the ring \(\text{Int}(\mathbb{Z}) = \{ f \in \mathbb{Q}[x] \mid f(\mathbb{Z}) \subseteq \mathbb{Z} \}\). It is well-known that \(\binom{x}{n} \in \text{Int}(\mathbb{Z})\) is irreducible. It was an open question whether \(\binom{x}{n}\) is absolutely irreducible, that is, \(\binom{x}{n}^k\) has a unique factorization into irreducible elements for all positive integers \(k\). We give a positive answer to this question and present the main ideas of the proof.
This is joint work with Roswitha Rissner.

Let \(R\) be a commutative ring with identity \(1\ne 0\). A function \(F:R\longrightarrow R\) is called a polynomial function on \(R\) if there exists a polynomial \(f\in R[x]\) such that \(F(r)=f(r)\) for each \(r\in R\). If \(F(r)\) is a unit in \(R\) for all \(r\in R\), then \(F\) is called a unit-valued polynomial function. If \(F\) is a bijection, then \(F\) is called a polynomial permutation on \(R\). When \(R\) is finite, we find the relation between the number of unit-valued polynomial functions and that of polynomial functions. Moreover, we show there is a connection between unit-valued polynomial functions and the group of those polynomial permutations on the ring \(R[x]/(x^2)\) that are represented by polynomials with coefficients in \(R\).

In 1981, Chouinard gave a characterization of when the monoid algebra \(D[S]\) is a Krull domain in terms of properties of the domain \(D\) and the monoid \(S\). In 2009, Chang investigated when a monoid algebra is weakly factorial (i.e. every non-zero non-unit is a product of primary elements) and in 2016, El Baghdadi and Kim proved a characterization of when a monoid algebra is generalized Krull. A domain resp. monoid is called weakly Krull, if the intersection of all localizations at height-one prime ideals is equal to the domain resp. monoid and every non-zero element is contained in only finitely many height-one prime ideals. We investigate the weakly Krull property of monoid algebras and use our result to characterize the weakly Krull domains among the affine monoid algebras.
This is joint work with Daniel Windisch.

Talk 1: We recall how to view direct-sum decompositions of modules in the framework of factorization theory and remind ourselves of some of the results and methods in this area. Moreover we introduce Bass rings. The talk is aimed primarily at graduate students and intended to introduce background material for the following talk.
Talk 2: We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring \(R\) through the factorization theory of the corresponding monoid \(T(R)\). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of \(T(R)\). The unit-cancellative monoid \(T(R)\) is transfer Krull of finite type. We construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of \(T(R)\).

In this talk, we discuss long sets of lengths with maximal elasticity in rings of integers \(\mathcal O_K\) (and for more general objects). In particular, we study zero-sum problems dealing with minimal zero-sum sequences of maximal length over finite abelian groups.

Let \(\mathcal L^*\) be a family of finite subsets of \(\mathbb N_0\) having the following properties.

• \(\{0\}, \{1\} \in \mathcal L^*\) and all other sets of \(\mathcal L^*\) lie in \(\mathbb N_{\ge 2}\).

• If \(L_1, L_2 \in \mathcal L^*\), then the sumset \(L_1 + L_2 \in \mathcal L^*\).

We show that there is a Dedekind domain \(D\) whose system of sets of lengths equals \(\mathcal L^*\).

We discuss sets satisfying certain conditions in finite groups, and give some applications to combinatorics and finite geometry.

Factorization in matrix rings has been investigated in a wide variety of contexts and has provided a wealth of information about how factorization works in noncommutative settings. Often, if \(S\) is a ring of matrices over a base ring \(R\), multiplicative factorization in \(R\) drives the multiplicative factorization in \(S\); that is, the additive structure of \(R\) is largely irrelevant to how elements of \(S\) factor multiplicatively. The situation is very different when \(R\) is a semiring, so that both multiplicative and additive arithmetic is nontrivial. Here we study the semigroup \(T_n(R)\) of upper-triangular matrices where \(R\) is a semiring which is both additively and multiplicatively reduced. We illustrate that factorization in such semigroups depend on both the multiplicative and additive structure of \(R\).

An irreducible element of a commutative ring is called absolutely irreducible if none of its powers has more than one (essentially different) factorizations into irreducibles. In this talk, we discuss the absolutely irreducible elements of the ring \[\text{Int}(D) = \{f \in K[x] \mid f(D) \in D \}\] of integer-valued polynomials on a principal ideal domain \(D\) with quotient field \(K\). We give a graph-theoretic sufficient condition for a polynomial \(f \in \text{Int}(D)\) to be absolutely irreducible. Furthermore, we show that our criterion is necessary and sufficient in the special case of polynomials with square-free denominator.

Let \(k\) be an algebraically closed field. Let \(f,g \in A=k[x,y]_{(x,y)}\) such that \(C=V(f)\) and \(D=V(g)\) have no component in common. Then \[e_0(f,g; A)\geq c\cdot d,\] where \(e_0(f,g; A)\) denotes the Hilbert-Samuel local multiplicity, and \(c\) and \(d\) are the initial degrees of \(f\) and \(g\) respectively. This inequality is known as local Bézout inequality in the plane \(\mathbb A^2_k\). Here equality occurs if and only if \(C\) and \(D\) intersect transversally at the origin. In this short talk, we will discuss an improvement of the above inequality when \(C\) and \(D\) do not intersect transversally. We also present local Bézout inequalities in the space \(\mathbb{A}^n_k\). To this end, we will exploit the use of techniques from commutative and homological algebra.

This is work in progress. I am establishing facts (of independent interest) that can serve as building blocks for a proof that the stable rank of the ring of integer-valued polynomials \[\mathrm{Int}(\mathbb{Z})= \{g\in \mathbb{Q}[x]\mid g(\mathbb{Z})\subseteq \mathbb{Z}\}\] is \(2\). If \(\mathrm{s.r.}(\mathrm{Int}(\mathbb{Z}))=2\), then every invertible matrix over \(\mathrm{Int}(\mathbb{Z})\) could be reduced to a \(2\times 2\) matrix by elementary row- and column-operations.

The supporting facts are:

1. (joint work with F. Halter-Koch) that the image of an algebraic number \(u\) under \(\mathrm{Int}(\mathbb{Z})\) is never contained in the algebraic integers, unless \(u\in \mathbb{Z}\), and

2. interpolation of functions \(f\colon \mathbb{Z}\rightarrow \mathbb{Z}\) at finitely many arguments and the realization of arbitrary functions on the residue class rings of \(\mathbb{Z}\) modulo finitely many powers of primes can be realized by one polynomial in \(\mathrm{Int}(\mathbb{Z})\) simultaneously, provided only that the requirements are not clearly contradictory.

We investigate ultraproducts of commutative integral domains and monoids with respect to factorizations of elements into atoms. A full description of sets of lengths in these algebraic structures is given. This allows us to construct an integral domain \(D\) such that for every multiset \(L \subseteq \mathbb{N}_{\geq 2}\) (with possibly infinite multiplicities) there exists an element in \(D\) having \(L\) as a (multi-)set of lengths.
Furthermore, as an application of the mentioned results to monoids of zero-sum sequences in joint work with Victor Fadinger a new proof of the following theorem by Geroldinger, Schmid and Zhong (2017) is given: Let \(L \subseteq \mathbb{N}_{\geq 2}\) finite non-empty. Then there are only finitely many pairwise non-isomorphic abelian groups \(G\) such that \(L\) is not the set of lengths of any element in the monoid of zero-sum sequences over \(G\).

A classical problem in ring theory is to characterize integral domains \(R\) such that every singular matrix over \(R\) is a product of idempotent matrices. As a partial answer, Ruitenburg proved that over a Bézout domain every singular matrix can be written as a product of idempotent factors if and only if every invertible matrix can be written as a product of elementary matrices. Moreover this happens if and only if the domain admits a weak version of the Euclidean algorithm. In these two seminars we first give an overview of the classical results on the idempotent factorization of matrices over domains and then we present some recent developments on the topic. In particular, we focus on products of idempotent matrices over non-Bézout integral domains.

Let \(D\) be a domain with quotient field \(K\). The ring of integer-valued polynomials on \(D\),

\[\text{Int}(D) = \{f \in D[x] \mid \forall~ a \in D, f(a) \in D \}\] in general is far from having unique factorization of elements into irreducibles. In this talk, our main focus is on the absolutely irreducible elements of \(\text{Int}(D)\) when \(D\) is a principal ideal domain. Recall that an irreducible element of a commutative ring is called absolutely irreducible if none of its powers has more than one (essentially different) factorizations into irreducibles. We give a graph-theoretic sufficient condition for a polynomial \(f \in \text{Int}(D)\) to be absolutely irreducible. Furthermore, we show that our criterion is necessary and sufficient in the special case of polynomials with square-free denominator. In addition, we construct non-absolutely irreducible elements in \(\text{Int}(\mathbb{Z})\).

Let \(k\) be an algebraically closed field. Let \(f,g \in A=k[x,y]_{(x,y)}\) such that \(C=V(f)\) and \(D=V(g)\) have no component in common. Then \[e_0(f,g; A)\geq c\cdot d,\] where \(e_0(f,g; A)\) denotes the Hilbert-Samuel local multiplicity, and \(c\) and \(d\) are the initial degrees of \(f\) and \(g\) respectively. This inequality is known as local Bézout inequality in the plane \(\mathbb A^2_k\). Here equality occurs if and only if \(C\) and \(D\) intersect transversally at the origin. In this short talk, we will discuss an improvement of the above inequality when \(C\) and \(D\) do not intersect transversally. We also present local Bézout inequalities in the space \(\mathbb{A}^n_k\). To this end, we will exploit the use of techniques from commutative and homological algebra.

Let \(R\) be a multivariate polynomial ring over a field and \(I\) a (monomial) ideal of \(R\). A prime ideal \(P\) of \(R\) is called associated to \(I\) if it is of the form \(P = \text{Ann}_R(x)\) for a non-zero element \(x\in R/I\). For a given ideal \(I\), we aim to understand the sequence \((a_n)_{n\in\mathbb{N}q}\) of numbers of associated prime ideals of the powers \(I^n\) of the ideal \(I\). While the sequence is known to stabilize for large \(n\), its initial behaviour is yet not very well understood. This talk serves as an introduction to the topic and presents the latest results of a joint research project with Irena Swanson.

Let \(G\) be a finite group. A sequence over \(G\) means a finite sequence of terms from \(G\), where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over \(G\) (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

The coordinate ring of an algebraic variety is noetherian, and so the question can be asked: is it possible to associate something like an algebraic variety to a nonnoetherian commutative algebra? The answer, surprisingly, is yes: if the algebra has finite Krull dimension, then its geometry looks just like an algebraic variety but with positive dimensional 'smeared-out' points. This new geometry has applications to the representation theory of a class of quiver algebras called dimer algebras, as well as to a theory of quantum gravity. In the first talk I will introduce nonnoetherian geometry, and discuss its role in the study of dimer algebras.

Frieze is a lattice of positive integers satisfying certain rules. Friezes of type A were first studied by Conway and Coxeter in 1970’s, but they gained fresh interest in the last decade in relation to cluster algebras, that are generated via a combinatorial rule called mutation. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras. Thus, a frieze is an array of positive integers on the Auslander-Reiten quiver of a Jacobian algebra such that entires on a mesh satisfy a unimodular rule. In this talk, we will discuss friezes of type D and their mutations. This is joint work with K. Serhiyenko (UC Berkeley).

• 28.11: Introduction to representation theory of quantum affine algebras
  

• 05.12: Quantum affine algebras and Grassmanians

 
Introduction to representation theory of quantum affine algebras
Abstract. In this talk, I will talk about finite dimensional representations of quantum affine algebras: 1. classification of finite dimensional simple modules of quantum affine algebras. 2. q-characters of finite dimensional modules. 3. cluster algebra structure of the Grothendieck ring of certain subcategory of the category of finite dimensional modules.
 
Quantum affine algebras and Grassmanians
Abstract. Let \(\mathfrak\{g\}=\mathfrak\{sl\}\_n\) and \(U\_q(\widehat\{\mathfrak\{g\}\})\) the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism \(\Phi\) from the Grothendieck ring \(\mathcal\{R\}\_\{\ell\}\) of a certain subcategory \(\mathcal\{C\}\_\{\ell\}\) of finite-dimensional \(U\_q(\widehat\{\mathfrak\{g\}\})\)-modules to a certain quotient \(\mathbb\{C\}[\{\rm Gr\}(n, n+\ell+1, \sim)]\) of a Grassmannian cluster algebra. We proved that this isomorphism induces an isomorphism \(\widetilde\{\Phi\}\) from the monoids of dominant monomials and the monoid of semi-standard Young tableaux. Using this result and the results of Qin and the results of Kashiwara, Kim, Oh, and Park, we have that every cluster monomial (resp. cluster variable) in a Grassmannian cluster algebra is of the form \(ch(T)\) for some real (resp. prime real) rectangular semi-standard Young tableau \(T\).

We translated a formula of Arakawa–Suzuki to the setting of \(q\)-characters and Grassmannians and obtained an explicit \(q\)-character formula for every finite-dimensional \(U_q(\widehat{\mathfrak{g}})\)-module and a formula for \(ch(T)\). This is joint work with Wen Chang, Bing Duan, and Chris Fraser.

Let \(D\) be an integral domain, let \(I\) be an ideal of \(D\) and let \(x\in D\) be a nonzero nonunit. Then \(I\) is called a valuation ideal of \(D\) if there exists a valuation overring \(V\) of \(D\) such that \(IV\cap D=I\). It is well-known that every prime ideal of \(D\) is a valuation ideal. Moreover, \(x\) is called a valuation element of \(D\) if \(xD\) is a valuation ideal of \(D\). We say that \(D\) is a valuation factorization domain (VFD) if every nonzero nonunit of \(D\) is a finite product of valuation elements of \(D\). Besides that, \(D\) is called a valuation ideal factorization domain (VIFD) if every ideal of \(D\) is a finite product of valuation ideals. In a series of two talks we discuss the properties of VFDs (and VIFDs).

In the first talk, we put our focus on well-known concepts and results. We investigate valuation ideals and several types of integral domains which are important in the study of VFDs, like (pre-)Schreier domains and PSP-domains. In particular, we discuss a result of Gilmer and Ohm which states that \(D\) is a Prüfer domain if and only if every ideal of \(D\) is an intersection of valuation ideals. Moreover, we deal with integral domains for which every primary ideal is a valuation ideal (or vice versa).

The main purpose of the second talk is the systematic study of VFDs. We provide characterizations for VFDs and archimedean VFDs and show that every weakly Matlis GCD-domain is a VFD. Furthermore, we present various conditions that force a VFD to be a weakly Matlis GCD-domain and investigate the connections between VFDs and homogeneous factorization domains (HoFDs). We also discuss when the polynomial ring over \(D\) is a VFD.

These talks are based on a joint work with Gyu Whan Chang.

Let \(\mu\neq0\) be integer. A Thue equation \[f(x,y)=\mu\neq0\] where \(f(x,y)\) is irreductible of the form \(a_0x^4+4a_1x^3y+6a_2x^2y^2+4a_3xy^3+a_4y^4\in\mathbb{Z}[x,y],\ a_0>0\) is called quartic Thue equation. Tzanakis, in 1993 gave a powerful algorithm under some assumptions to tranform the quartic Thue equation to a system of simultaneous pell equations. It is used by several authors to solve quartic Thue equations. For solving the system one can use results on solutions of a Pell equation, linear form in logarithms of algebraic numbers and so on.

Considering the Thue equation \[\label{eq} x^4-ax^3y+bx^2y^2+axy^3+a_4y^4=\mu,\] we find the condition on integers \(a\) and \(b\) for which Tzanakis's method works and study the solution when \(\mu=1\).

In this talk, after giving the description of Tzanakis's method we will give a recall on Pell equation and present the main result which is obtained by joint work with Bo He (China West Normal University) and Alain Togbe (Purdue University Northwest, Westville USA).

An integral domain \(D\) is said to be a stable domain if every nonzero ideal of \(D\) is invertible in its endomorphism ring. Since invertible ideals are stable, every Dedekind domain is stable. In this talk we give an overview on stable domains.

Ideal stability appears implicitly in the famous Ubiquity paper of Bass in 1963 where he proves that every two-generated domain is stable. This very interesting (and natural) class of domains was studied (among others) by Lipman, Sally, Vasconcelos and Olberding. There is an extensive literature on the subject. An admirable link to this literature is the two-part study of Olberding (On the Classification of Stable Domains and On the Structure of Stable Domains). Many papers in Commutative Algebra address ideal-theoretic and module-theoretic properties of prime ideals, overrings, localizations, and integral closure of stable domains. Our motivation arose from these two papers of Olberding.

Our aim is to briefly introduce you to some results on the ideal structure, local properties, finite characterness, the structure of integral closure, and the relationship of stable domains with some classical classes of domains.

• 15:15 – 16:15 Jaitra Chattopadhyay (Harish-Chandra Research Institute, Allahabad)
  Biquadratic fields having a non-principal Euclidian ideal class
  

• 16:15 – 17:15 Japhet Odjoumani (Université dAbomey-Calai, Benin)
  On Pell Factoriangular numbers

 
Jaitra Chattopadhyay: Biquadratic fields having a non-principal Euclidian ideal class
Abstract. In 1979, H. W. Lenstra defined the notion of an Euclidean ideal class and proved that if a number field \(K\) has a non-principal Euclidean ideal class then the ideal class group \(Cl_{K}\) of \(K\) is cyclic. Except for the imaginary quadratic fields, he was able to prove the converse, under the assumption of GRH. Later, H. Graves constructed an explicit bi-quadratic field having an Euclidean ideal class and after a few years C. Hsu provided a family of such fields. In this talk, we shall give a new class of bi-quadratic fields other than the ones given by Graves and Hsu. This is a joint work with Dr. M. Subramani.
 
Japhet Odjoumani: On Pell Factoriangular numbers
Abstract. Factoriangular numbers, defined as \[Ft_n=n!+\frac{n(n+1)}{2},\;n\ge0 \,,\] were introduced in 2015 by R. C. Castillo. He conjectured that the only Fibonacci Factoriangular numbers are \(2,\,5\) and \(34\). This conjecture was proved by Gomez Ruiz and F. Luca in 2017. We studied Pell numbers and showed that the only Pell Factoriangular numbers are \(2,\,5\) and \(34\).
This is joint work with Florian Luca (University of the Witwatersrand South Africa) and Alain Togbe (Purdue University Northwest, Westville USA).

A univariate (formal) power series over a field \(K\) represents a rational function if and only if its coefficients satisfy a linear recurrence relation. Such a series is called a rational series; it is called a Pólya series if its nonzero coefficients are contained in a finitely generated subgroup of \(K^\times\). An explicit characterization of such series emerges from work of Pólya (1921), Benzaghou (1970), and Bézivin (1987), who in turn treated more general fields.

Rational series can also be defined for multivariate formal power series in noncommuting variables. Again there is a combinatorial interpretation in terms of weighted finite automata (WFA). Noncommutative Pólya series were first studied by Reutenauer in the 1970s. We show that they correspond to unambiguous WFA, confirming a conjecture of Reutenauer from 1979. The proof combines methods from noncommutative algebra, automata theory, and number theory.

The aim of these talks is to be easily accessible to beginning PhD students; a significant amount of time will be devoted to background material. In the first talk, I plan to recall the univariate result and its connection to linear recurrences, introduce noncommutative rational series and WFAs, and state the main theorem.

In the second talk, I intend to introduce a novel invariant attached to a rational series and demonstrate the key steps of the proof. For this it will be necessary to recall unit equations.

This is joint work with Jason Bell. arXiv:1906.07271.

The theory of cluster algebras is a remarkable area of mathematics. The base case is cluster algebras on type \(A\), the clusters of which correspond to triangulations of convex polygons, representations of type \(A\) quivers. This case is also a restriction of the Grassmannian \(\mathrm{Gr}(k,n)\) cluster algebras to the case \(k=2\). The symmetries of cluster algebras may be generalised to higher dimensions in the guise of the higher Auslander-Reiten theory introduced by Iyama. We find a connection between higher Auslander-Reiten theory and Grassmannian \(\mathrm{Gr}(k,n)\) cluster algebras for \(k>2\) in the following sense.

Frieze patterns are a beautiful combinatorial interpretation of cluster algebras, first studied by Conway and Coxeter some thirty years preceding cluster algebras. We find that each Grassmannian cluster of Plücker coordinates defines an idempotent subquiver of the repetitive algebra of a higher Auslander algebra of type \(A\), and may therefore determine a higher frieze pattern. In addition, we establish a new combinatorial-geometric model for such frieze patterns, in the form of superimposed triangulations of convex polygons.

Motivated to determine the role of idempotents in higher Auslander-Reiten theory, we look instead to how they may be used for a rather different kind of algebra. Idempotent ideals have been heavily used to calculate the finitistic dimension of many classes of algebras such as quasi-hereditary algebras, by methodically finding idempotent ideals that are projective and reducing the problem. This is however precisely the method later employed by Chen and to calculate singularity categories, one application being the classification of singularity categories of Nakayama algebras by Chen and Ye. Generalising these methods to higher analogues, we introduce fabric idempotent ideals, and show they allow us to calculate singularity categories for higher Nakayama algebras and higher gentle algebras, the latter class needing to be first introduced.

By a sequence \(S\) over a finite group \(G\), we mean a finite unordered sequence of terms from \(G\) with repetition allowed. We say that \(S\) is a product-one sequence if its terms can be ordered such that their product equals the identity element of \(G\). The set \(\mathcal B (G)\) of all product-one sequences is a monoid (with the concatenation of sequences as the operation), and it pops up naturally in a variety of mathematical subfields including Factorization Theory, Arithmetic Combinatorics, and Invariant Theory. This publication-based thesis offers a detailed study of the monoid \(\mathcal B (G)\) from the algebraic, arithmetic, and combinatorial point of view.

To begin with algebraic aspects of the present thesis, the monoid \(\mathcal B (G)\) is a C-monoid. C-monoids are submonoids of factorial monoids with finite class semigroup. Every C-monoid is Mori (i.e., it satisfies the ACC on divisorial ideals) and its complete integral closure is a Krull monoid with finite class group. Wide classes of Mori domains (including non-principal orders in number fields) are C-domains. While the finiteness of the class semigroup is used substantially in a variety of arithmetical finiteness results for C-monoids, the algebraic structure of class semigroups was not studied so far. We establish an explicit structural description of the class semigroup of \(\mathcal B (G)\). Among others, we show that \(\mathcal B (G)\) is seminormal if and only if its class semigroup is a union of groups.

Based on algebraic results on class semigroups, we study the arithmetic of \(\mathcal B (G)\), with a focus on sets of lengths. Among others, we show that all unions of sets of lengths of \(\mathcal B (G)\) are intervals.

The Davenport constants of \(G\) and the Erdős-Ginzburg-Ziv constant are central combinatorial invariants of sequences over \(G\). While the precise values of these invariants are known for several classes of non-abelian groups (all of them have a large abelian subgroup), the associated inverse problems have not been studied so far. Using methods from Additive Combinatorics, we provide the first explicit characterizations of all minimal product-one sequences of maximal length and all sequences of maximal length having no product-one subsequence of certain length over dihedral and dicyclic groups.

• 12:00 – 13:00 Dr. Aihua Li
  Topics in Algebraic Graph Theory
  

• 13:00 – 13:30 Cheyenne Petzold
  Interlace polynomials of certain graphs

 
Dr. Aihua Li: Topics in Algebraic Graph Theory
Abstract. Let R be a commutative ring with identity 1 and T be the non-commutative ring of all nxn upper triangular matrices over R. In this presentation, Dr. Li will introduce the zero divisor graph G(T) of the noncommutative ring T. Some basic graph theory properties of G(T) are given, including determination of the girth and diameter. The structure of such a graph is discussed and bounds for the number of edges are given. The structure of the graph G(T) is fully described in the case that T is a 2x2 upper triangular matrix ring over a finite integral domain R; in this case an explicit formula for the number of edges is given.
 
Cheyenne Petzold: Interlace polynomials of certain graphs
Abstract. Cheyenne Petzold will talk about interlace polynomials of certain graphs. Properties of the interlace polynomials of graphs such as shell graphs, which are constructed by adding additional cords to cycles, are provided. An application of such polynomials in analyzing the adjacency matrixes of the ground graphs is given.
 

Dr. Aihua Li is a professor of Mathematics at Montclair State University. She received her Ph. D. in mathematics from the University of Nebraska-Lincoln. Her primary research is in commutative algebra, specialized in ring theory and ideal theory. In recent years, she has done research in number theory, matrix theory, discrete dynamical systems, and graph theory. Dr. Li is currently the chair of the Mathematical Association of America New Jersey Section (MAA-NJ). She is the recipient of a Distinguished Scholar Award by Montclair State University and a Faculty Mentoring Award by the Division of Mathematics and Computer Science of CUR.
Cheyenne Petzold is an undergraduate math major at Montclair State University doing research projects with Dr. Aihua Li.

We study sets of lengths and catenary degrees in numerical monoids. The focus of the talk will be on algorithmic aspects.

A commutative cancellative monoid \(H\) with \(H \neq H^\times\) is called primary provided that for all \(a,b \in H \setminus H^\times\) there exists \(n \in \mathbb{N}\) such that \(b^n \in aH\). Primary monoids and domains have received a significant amount of attention in the past two decades. Here we present some results related to the atomicity and factorization invariants of the class of primary monoids consisting of additive submonoids of the nonnegative cone of rational numbers. Then, given a field \(F\) of finite characteristic, we will show how to use such a class of monoids to construct atomic monoids whose monoid algebras over \(F\) are not atomic, giving a partial answer to a question posed by Gilmer in the 1980s.

A finite-complement ideal is a subsemigroup \(S\) of a free abelian monoid \(F\) such that \(|F\setminus S| < \infty\) and \(FS=S\). Examples include the multiplicative semigroup with underlying set a numerical monoid, affine semigroups with finite complement, and the monoid of all sequences over a finite abelian group that are not zero-sum free. These semigroups are never Krull, but are always \(C\)-monoids. A classification of the atoms into two distinct classes gives rise to a transfer homomorphism to an affine semigroup defined in terms of one class of atoms. Using this description we provide some preliminary information (length sets and elasticities) on the arithmetic of these semigroups.

Often in geometry, naturally occurring conditions define subsets of varieties which are either very big in size or tiny. For example, among all linear operators acting on a given finite dimensional vector space, the invertible ones form an open and dense subset. And so do, among all nilpotent operators, those which have only one Jordan block. A notable exception to this rule occurs in the variety of short exact sequences of nilpotent linear operators: The variety can be written, by means of Littlewood-Richardson (LR-) tableaux, as the union of components of equal dimension. We investigate several partial orders for LR-tableaux which reflect combinatorial, algebraic and geometric properties of the short exact sequences which they represent. We try to shed light on the following strange phenomenon. While LR-tableaux turn out to be useful and efficient tools to deal with the above mentioned properties, this is no longer true when it comes to multiplication by the variable in the Ext-group. Although the product appears to be simpler - it certainly has lower nilpotency index - the LR-tableau of the original short exact sequence does not in general determine the LR-tableau of the product.

Joint work with Justyna Kosawska (Torun)

For any positive integers \(n,m\) with \(\gcd(n,m)=1\), the rational Catalan number is defined as \[\mathsf {Cat}_{n,m}=\frac{1}{n+m}\binom{n+m}{n},\] which arises naturally in many areas of mathematics. A typical object counted by \(\mathsf {Cat}_{n,m}\) is the set of all \((n,m)\)-Dyck paths (lattice paths from \((0,0)\) to \((n,m)\) which only go east or north and stay above the diagonal line \(y=\frac{m}{n}x\)). In this talk, I will present two main results relate to rational Dyck paths. Firstly, using the symmetry of \(n\) and \(m\) in \(\mathsf {Cat}_{n,m}\), we present some symmetries on finite abelian groups. For a finite abelian group \(G\), we denote \(\mathsf M(G,m)\) the set of all zero-sum sequences over \(G\) of length \(m\). Let \(G\) and \(H\) be abelian groups with \(\gcd(|G|,|H|)=1\), then we have the following symmetry \[\mathsf M(G,|H|)=\mathsf M(H,|G|)=\mathsf {Cat}_{|G|,|H|}.\] Secondly, we consider a counting problem proposed by R. Stanley. Let \(G\) be a finite abelian group of order \(n\). Let \(a_1,\ldots,a_k\) be positive integers with \(k\le n\). Let \(N_G(a_1,\ldots,a_k)\) be the number of solutions (\((x_1,\ldots,x_k)\) with \(x_i\neq x_j\) for any \(i\neq j\)) of the following equation in \(G\): \[a_1x_1+a_2x_2+\cdots+a_kx_k=0,\] where \(0\) is the identity of \(G\). Our result is the following: if \(\gcd(a_1+\cdots+a_k,n)=1\), then \(N_G(a_1,\ldots,a_k)\) coincides with the Kreweras number which initially counts the number of some special rational Dyck paths.

We provide proofs of these results from two different perspectives. The first one is combinatorial interpretation by constructing bijections between rational Dyck paths and the objects (zero-sum sequences or solutions) we consider, the second one is algebraic which employ methods from the invariant theory which essentially follow the pioneering work of Molien in 1897.

The essence of the commutativity relation on a given magma \({\mathcal A}\) (i.e., a nonempty set equipped with an inner operation, written as product \(ab\)) is captured in its commuting graph. By definition, this is a simple graph whose vertices are all noncentral elements of \({\mathcal A}\) and where two distinct vertices \(a,b\) are connected if they commute in \({\mathcal A}\), i.e., if \(ab=ba\). In this talk, we shall consider the commuting graphs of rings and semirings, with an emphasis on matrix algebras. We shall survey the known results on the properties of the commuting graphs, mainly those concerning its connectedness and diameter. We shall also look at a few open problems in this area of research.

Let \(G\) be a finite group and \(\exp (G) = \lcm \{ \ord(g) \mid g \in G \}\). A finite unordered sequence of terms from \(G\), where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of \(G\). We denote by \(\mathsf s (G)\) (or \(\mathsf E (G)\) respectively) the smallest integer \(\ell\) such that every sequence of length at least \(\ell\) has a product-one subsequence of length \(\exp (G)\) (or \(|G|\) respectively). In this talk, we provide the exact values of the \(\mathsf s (G)\) for Dihedral and Dicyclic groups, and characterize the structure of all sequences of length \(\mathsf s (G) - 1\) (or \(\mathsf E (G) - 1\) respectively) having no product-one subsequence of length \(\exp (G)\) (or \(|G|\) respectively).

Each quiver appearing in a seed of a skew-symmetric cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in skew-symmetric cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups and thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and it is natural to ask whether the cluster group presentations possess comparable properties.

I will define a cluster group associated to a cluster quiver and explain how the theory of cluster algebras forms the basis of research into cluster groups. As for Coxeter groups, we can consider parabolic subgroups of cluster groups. I will outline a proof which shows that, in the type A case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups.

• 15:15 – 16:00 Nick Williams (University of Leicester)
  Higher analogues of Grassmannian clusters
  

• 16:15 – 17:00 Fabian Lenzen (Technical University of Munich)
  Braid group actions on category \(\mathscr{O}\)

 
Nick Williams: Higher analogues of Grassmannian clusters
Abstract. Scott described the coordinate ring of the Grassmannian as a cluster algebra. This description, along with results from Oh, Postnikov, and Speyer, puts the clusters in bijection with collections of subsets which satisfy a certain combinatorial condition. This combinatorial condition has a natural generalisation but it is not immediate that this case behaves as the classical case does. However, recent results of ours go some way towards showing that this generalised case is reasonably well-behaved. This is based on joint work-in-progress with Jordan McMahon.
 
Fabian Lenzen: Braid group actions on category \(\mathscr{O}\)
Abstract. For a complex semisimple Lie algebra \(\mathfrak{g}\), it is a long known fact that the bounded derived category of \(\mathscr{O}\) – a category comprising the most interesting representations of \(\mathfrak{g}\), including the simple representations, the indecomposable projectives and injectives as well as the Verma modules – can be endowed with an action B, the braid group associtated to the Weyl group of \(\mathfrak{g}\), in terms of Translation and Shuffling functors. Although it is technical to prove this, it is quite straightforward to see that there B acts on the Grothendieck group \(K_0(\mathscr{O})\). We will see that \(\mathscr{O}\) can be endowed with a grading which arises if we regard \(\mathscr{O}\) equivalently as a category of modules over a certain path algebra. With this grading in mind, which turns \(K_0(\mathscr{O})\) from an abelian group into a \(\mathbb{Z}[t, t^{-1}]\)-module, we see that the action of B lifts to an action of the Iwahori-Hecke algebra on \(K_0(\mathscr{O})\).

Tilting theory is a powerful descriptor of when two arbitrary rings might be considered to be the "same". We give an introductory account of tilting theory, using as an example the path algebras of type A. Then we present recent results on generalisations coming from higher Auslander-Reiten theory.

In this talk we introduce the notion of maximal Cohen-Macaulay approximations. We present two ways to construct Cohen-Macaulay approximations: the first one is due to Auslander-Buchweitz and is called the pitchfork construction, and the other one is by Martsinkovsky-Herzog, and is known as the gluing construction. We show how these constructions can be applied to the simple modules of an algebra that appears in categorification of Grassmannian cluster algebras.

Real algebra and geometry studies semialgebraic sets, i.e. solution sets to systems of polynomial inequalities. The classical theory provides Positivstellensätze, which are of the same spirit as Nullstellensätze in the case of varieties, i.e. solution sets to systems of polynomial equations. Many interesting (and hard) such results have been developed in the last decades, and surprising applications in optimization and convexity have arisen. A much more recent development is the theory of non-commutative semialgebraic sets. Triggered by questions in electrical engineering, control theory and quantum physics, several exciting results have been proven. But beyond the mentioned applications, the non-commutative theory also sheds light on classical questions.

In this talk, I will give an introduction to both the classical theory and their non-commutative extension, as well as some interesting applications.

I will give an elementary introduction to cluster algebras, and describe some of their connections with other areas of mathematics (and physics). In particular, I will describe how cluster algebras are related to Somos sequences, knot theory, continued fractions, triangulations of surfaces, and string theory. (This seminar will serve as the first class for the course "Cluster algebras".)

In this thesis, we focus on relations between different kinds of algebras on the one hand and combinatorial geometric objects, especially subdivisions of convex polygons and matchings, on the other hand.

First we introduce GL\(_m\)-dimers as subdivisions of the triangles of triangulations of convex polygons for a positive integer \(m\geq 2\). They give rise to so-called dimer models with boundary, which are quivers fulfilling further conditions. We associate boundary algebras to these dimer models. We show that for any two triangulations of an initial polygon the resulting boundary algebras are isomorphic.

We call a polygonal subdivision into subpolygons with the same number of vertices a tiling, and consider combinatorial and algebraic objects they can be associated with.

One of these combinatorial objects are frieze patterns. Recently Holm and Jórgensen introduced a \(p\)-gonal generalization of classical Conway-Coxeter friezes for \(p\geq 3\). For \(p\)-gonal tilings, these friezes are called friezes of type \(\Lambda_p\). We show a bijection between certain Conway-Coxeter friezes and friezes of type \(\Lambda_p\) for \(p=4\) and \(p=6\).

On the algebraic side, we consider Temperley-Lieb algebras, and their generalization Fuss-Catalan algebras. The generators of both kinds of algebras have a pictorial presentation as planar graphs, which we call matchings. We present the first explicit bijection between matchings and tilings. Further, we explore how local transformations in one type of these planar graphs act on the other type under this bijection.

Let \(H\) be a monoid and \(H^\times\) its group of units. We say that \(H\) is f.g.u. (short for ``finitely generated up to units'') if every non-unit of \(H\) is in the subsemigroup generated by \(H^\times A H^\times\) for some finite \(A \subseteq H\); acyclic if \(uxv = x\), for some \(u, v, x \in H\), implies that \(u\) and \(v\) are units; and positive if there exists a (partial) preorder \(\preceq\) on \(H\) such that \(1_H \prec x\) for every non-identity \(x \in H\) and the preordering of the elements is preserved under the operation of \(H\).

The talk will be an introduction to the arithmetic of f.g.u., acyclic, and positive monoids. In particular, we shall see how these objects serve as an abstract model to study factorization in a large class of monoids (and rings), and accordingly to obtain non-commutative non-cancellative generalizations (and in some cases improvements) of fundamental results from the classical theory (mostly focused on the cancellative and commutative case).

Let \(G\) be a finite group. By a sequence over \(G\), we mean a finite unordered sequence of terms from \(G\), where the repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of \(G\). The large Davenport constant \(\mathsf D (G)\) is the maximal length of a minimal product-one sequence, that is a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length \(\mathsf D (G)\) over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.

We will start with the definition of cluster algebra, examples, classification and important properties. Then we will mention results on factorization from our work "Factoriality and class groups of cluster algebras". If there is time, we will dive into ideas of some proofs and future lines of work.

We study the set of catenary degrees, the set of distances, and the unions of sets of lengths of the monoid of nonzero ideals and of the monoid of invertible ideals of orders in quadratic number fields.

In diesem Vortrag möchte ich Ihnen die Geschichte der mathematischen Knotentheorie erzählen und dann zeigen was wir in der Mathematik genau unter einem Knoten verstehen.

In 1990 gewann Vaughan F.R.Jones die Fields Medaille für seine Entdeckungen in der Knotentheorie.

Ich werde Ihnen diese neue Ideen erklären und auch zeigen wie man diese sogar im Gymnasium mit Schülern und Schulerinnen selber erarbeiten kann.

In the talk we show how one can extend the definition of rank of a square matrix to elements of more general rings. We present several neat examples to ilustrate the extended definition and discuss the properties of rank in semiprime rings. As an application we show that every element in the socle of a unital semirime ring is unit-regular.

The Grassmannian of k-planes in n-space can be equipped with a cluster structure using Cohen-Macaulay modules over an infinite dimensional algebra. We study this category and determine canonical short exact sequences and periodicity. Furthermore, we give an explicit construction of higher rank modules. Using these results, we establish a correspondence between rigid indecomposable modules and real roots for the associated Kac-Moody algebra in the tame cases.

This is joint work with D. Bogdanic and A. Garcia Elsener

Jacobian algebras are a particular class of algebras given by quivers with relations, originally introduced to connect cluster theory and representation theory. In this talk I will explain how the self-injective ones can be thought of as a "2-dimensional analogue" of preprojective algebras of Dynkin quivers. Moreover, I will show how the combinatorics of Grassmannian cluster categories constitutes a rich source of self-injective Jacobian algebras.

Philippe Caldero and Frederic Chapoton introduced the so called Caldero-Chapoton functions to give a revolutionary approach to cluster algebras of Dynkin type ADE. Indeed, they gave a bijection between indecomposable representations of the path algebra of a Dynkin quiver of type ADE and cluster variables of the cluster algebra associated to the alluded quiver. This bijection is done by defining a Laurent polynomial for every representation of the respective quiver.

On the other hand, Leonid Chekhov and Michael Shapiro introduced the notion of generalized cluster algebras from Teichmüller theory of surfaces with orbifold points. These algebras are a natural generalization of cluster algebras. The combinatorics of generalized cluster algebras is also governed by a skew-symmetrizable matrix, but the exchange polynomials may not be binomials.

In this talk we are going to follow the Caldero-Chapoton approach to describe a generalized cluster algebra by means of Caldero-Chapoton functions over an algebra naturally defined for any triangulation of a polygon with one orbifold point of order three. This is a joint work with Daniel Labardini Fragoso.

For an integer \( k\geq 2 \), let \( \{F^{(k)}_{n} \}_{n\geq 0}\) be the \( k\)–generalized Fibonacci sequence which starts with \( 0, \ldots, 0, 1 \) (\( k \) terms) and each term afterwards is the sum of the \( k \) preceding terms. In this seminar, for an integer \( d\geq 2 \) which is square free, we show that there is at most one value of the positive integer \( x \) participating in the Pell equation \( x^{2}-dy^{2}=\pm 1 \) which is a \( k\)–generalized Fibonacci number, with a few exceptions that we completely characterize. This paper extends previous work from Luca et. al 2017 for the case \(k=2\) and Luca et. al 2018 for the case \(k=3\). This is a joint work with Florian Luca.

Pipe dreams are certain combinatorial objects that play a fundamental role in the combinatorial understanding of Schubert polynomials. They encode many remarkable geometric structures such as associahedra, multiassociahedra, and certain polytopal subdivisions realizing the v-Tamari lattices of Préville-Ratelle and Viennot. In this talk, I will describe a Hopf algebra structure on a family of pipe dreams. This Hopf algebra gives rise to intriguing connections to the enumeration of certain lattice walks on the quarter plane and applications to the theory of multivariate diagonal harmonics. This is joint work with Nantel Bergeron and Vincent Pilaud.

C-monoids and C-domains are defined as suitable submonoids of factorial monoids having a finite class semigroup. Every C-domain \(R\) is Mori (i.e., it satisfies the ascending chain condition on divisorial ideals), has a nonzero conductor \(\mathfrak f = (R \negthinspace : \negthinspace \widehat R)\) to its complete integral closure \(\widehat R\) and the Krull domain \(\widehat R\) has a finite class group. Conversely, every Mori domain with the above properties and with \(R/\mathfrak f\) finite, is a C-domain.

It is well-known that a C-monoid is completely integrally closed if and only if its class semigroup is a group (and in that case the class semigroup coincides with the usual class group of a Krull monoid). We show that a C-monoid is seminormal if and only if its class semigroup is a union of groups. Moreover, we present some arithmetical finiteness results. Among others, we show that the elasticity of a C-monoid is either rational or infinite.

Numerical semigroups are additive submonoids of the natural numbers of finite complement. Though this rather simple structure, they reveal to be a useful tool in the understanding of invariants related to some generating functions in commutative algebra and algebraic geometry, in particular in the theory of singular algebraic curves. In this talk I will give an overview of these connections.

One way to understand a module is to understand its generators, the relations on its generators, the relations on the relations, the relations on the relations on the relations, and so on. In other words, so-called resolutions give quite a bit of information on a module. The first big result on resolutions was the Hilbert's Syzygy Theorem which says that for every finitely generated module over a polynomial ring in finitely many variables over a field there exists a finite-step resolution. In this talk I will discuss how commutative algebraists use resolutions (free, projective, injective) and how they use homological algebra. I will also discuss some computational aspects, the long-open homological conjectures, and recent breakthrough of Yves Andre proving the Direct Summand Conjecture.

This talk serves also as an advertisement for my course on homological algebra at University of Graz.

The classification of finite subgroups of SO(3) is well known: these are either cyclic or dihedral groups or one of the symmetry groups of the Platonic solids. On the other hand, work of H.S.M. Coxeter implies a bijection between the finite subgroups of SO(3) and finite subgroups of O(3) generated by reflections. This picture can be completed to a "square" by certain finite subgroups of SU(2) and O(2). In the 19th century, Felix Klein investigated the orbit spaces of the finite subroups of SO(3) and their double covers, the so-called binary polyhedral groups. This investigation is at the origin of singularity theory.

Quite surprisingly, in 1979, John McKay found a direct relationship between the resolution of the singularities of the orbit spaces and the representation theory of the finite group one starts from. This "classical McKay correspondence" is manifested, in particular, by the ubiquitious Coxeter-Dynkin diagrams.

In this talk I will first review the history of this fascinating result, and then give an outlook on recent joint work with Ragnar-Olaf Buchweitz and Colin Ingalls about a McKay correspondence for finite reflection groups in GL(n,C).

This work is inspired by surface cluster algebras studied by Fomin-Shapiro-Thurston, mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, string modules associated to arcs on unpunctured surfaces by Assem-Brüstle-Charbonneau-Plamondon and Quivers with potentials associated to triangulated surfaces part II by Labardini-Fragoso.

For an ideal triangulation of (\(\Sigma,M\)) and an ideal arc Labardini-Fragoso defined an arc representation of \((Q(\tau),S(\tau))\). To this end he associated a quiver with potential \((Q(\tau),S(\tau))\) for a surface with marked points (\(\Sigma,M\)).

This talk focuses on extending the definition of arc representations to a more general context by considering tagged triangulations and tagged arcs. Then I associate a representation of \((Q(\tau),S(\tau))\) and prove that the Jacobian relations are met.

The interplay between geometric and combinatorial models for cluster algebras and representation theory influenced research in various directions. Such link for the Grassmannian cluster algebra became of interest with the introduction of Grassmannian cluster categories of Jensen-King-Su and with dimer algebras of Baur-King-Marsh. In particular, using geometric and combinatorial structure of the Grassmannian, Marsh-Scott showed that certain special functions on the Grassmannian (namely twisted Pluecker coordinates) can be expressed as 'dimer partition functions', i.e. sums over perfect matchings. In joint work with Alastair King and Matt Pressland, we relate this formula to a more representation theoretic formula by introducing perfect matching modules over dimer algebras which allows us to recover combinatorial data in the Marsh-Scott formula representation theoretically and also to deduce information about the representation theory of various algebras associated to the dimer model.

See next page.

We point out a connection between Conway-Coxeter friezes of triangulations and a \(p\)-angulated generalisation of frieze patterns recently introduced by Holm and Jørgensen: the friezes of type \(\Lambda_p\) coincide with Conway-Coxeter friezes of certain triangulations for \(p=4\) and \(p=6\) in every second row. Furthermore we determine the values of the vertices of the Farey graphs corresponding to the frieze patterns of type \(\Lambda_p\) by the entries of the frieze patterns and vice versa for general \(p\).

tba.

Let \(\Lambda\) be a cluster-tilted algebra of finite type and let \(B\) be one of the associated tilted algebras. We show that the indecomposable \(B\)-modules, ordered from right to left in the Auslander-Reiten quiver of \(\Lambda\) form a maximal forward hom-orthogonal sequence of \(\Lambda\)-modules whose dimension vectors form the \(c\)-vectors of a maximal green sequence for \(\Lambda\). Thus we give a proof of Igusa-Todorov's conjecture.

Let \(\mathfrak X = (X_k)_{k \ge 1}\) be a sequence of eventually non-empty subsets of \(\mathbf Z\) such that \[X_h + X_k \subseteq X_{h+k}, \quad \text{for all } h, k \in \mathbf N^+.\] We say that \(\mathfrak X\) is strongly smooth if there exist \(M \in \mathbf N\), \(d, \mu \in \mathbf N^+\), \(X_0^\prime,X_0^{\prime\prime},\ldots,X_{\mu-1}^\prime,X_{\mu-1}^{\prime\prime} \subseteq [0, M]\), and \(x_1, x_2, \ldots \in \bf Z\) such that, for all large \(k\), \[X_k = \bigl(\inf X_k + X_{k \bmod \mu}^\prime\bigr) \uplus \mathscr P_k \uplus \bigl(\sup X_k - X_{k \bmod \mu}^{\prime\prime}\bigr) \subseteq x_k + d \cdot \mathbf Z,\] where \(\mathscr P_k := (x_k + d \cdot \mathbf Z) \cap [\inf X_k + M, \sup X_k - M]\) is an arith. progression (with diff. \(d\)). We obtain sufficient and necessary conditions for the sequence \(\mathfrak X\) to be strongly smooth, and from this we derive an all-inclusive proof of three fundamental results in additive combinatorics and factorization theory that at first have little to nothing in common with each other: a 1972 theorem of Nathanson on the (asymptotic) structure of the \(n\)-fold sumset of a finite set of integers; a theorem of Geroldinger and Halter-Koch on the structure of the sets of lengths of the powers of a fixed element in a cancellative commutative monoid that is finitely generated up to associates; and a recent theorem we have proved on the structure of unions of sets of lengths in monoids with accepted elasticity.

We give an overview on CM modules from Grassmannians following Jensen-King-Su. Starting form this, we study the Auslander-Reiten sequences of the category of Cohen-Macaulay modules of an algebra categorifying Grassmannian cluster algebra. In particular, we study infinite tame cases and construct tubes (components of the AR quiver) containing rank 1 modules, and rank 2 modules. We relate the number of rigid indecomposable modules with the number of real roots for the corresponding Lie algebra.

Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between \(1\) and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have (sufficiently many) prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.

For a multiplicative monoid H, the power monoid of H consists of all finite, nonempty subsets of H with the operation of setwise product. This is a relatively new class of non-cancellative monoids, so we will demonstrate that they are indeed a viable setting for studying factorizations. This requires the formulation of a more restrictive notion of ``factorization'', with respect to which power monoids have bouneded factorization lengths. We will discuss the motivation for the study of this class of monoids, which is twofold: (1) Factorization questions in this setting fall in line with some central themes of Additive Combinatorics and (2) they offer a class of examples wherein non-cancellative factorization can be systematically studied and better understood.

Let \(R\) be a finite commutative ring with a unity. For a natural number \(k\), a polynomial \(f(x)\in R[x]\) is said to be a null polynomial of order \(k\) if \(f,f',...,f^{(k)}\) induce the zero function over \(R\). Such polynomial induces the zero function over some homomorphic images of the ring of \(k\)th indeterminate over \(R\), \(R[x_{1},...,x_{k}]\). For a prime \(p\) and a natural number \(1\le n\le p\), we derive a formula for counting null polynomials of order \(k\) for every \(k\in \mathbb{N}\). Moreover, when \(n\le p\) we count the polynomial functions and the permutation polynomials over some homomorphic image of \(\mathbb{Z}_{p^n}[x_{1},...,x_{k}]\) which is isomorphic to \(\quad \mathbb{Z}_{p^n}[x_{1},...,x_{k}]\big/I\), where \(I\) is the ideal generated by the set \(\big\{x_j^2: j=1,2,..,k \big\}\).

The Space-filling curve (SFC) is an onto mapping from unit interval \([0,1]\) to the unit square \([0,1]^2\) which were first constructed by G. Peano in 1890. After that, it appeared many constructions, for instance, Hilbert curve(1891), Sierpinski curve(1912), Polya curve(1913), and so on. It is well-known that the constructions of SFCs depend on certain substitution rules. We extend the concept SFC from the unit square to a self-similar set which is a set being the union of small copies of itself. For a given self-similar set, how to obtain a suitable substitution rule is somehow mysterious and it is the main concern of this talk. To this end, we introduce a new notion: a skeleton of a self-similar set. And we can find a certain substitution rule through a skeleton. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.

The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of algebraic representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of \(R\)-modules to \(R\)-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimensional persistence as a special case. (joint work with Rene Corbet)

Let \(G\) be a finite group. By a sequence over \(G\), we mean a finite unordered sequence of terms from \(G\), where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of \(G\). It is well-known that the class semigroup of the monoid \(\mathcal B (G)\) of product-one sequences is a group if and only if \(G\) is abelian (equivalently, \(\mathcal B (G)\) is root closed). We show that \(\mathcal B (G)\) is seminormal if and only if its class semigroup is a Clifford semigroup.

The \(f\)-vector of a \(d\)-dimensional (convex) polytope \(P\) is defined as \[f(P) = (f_0(P), f_1(P), \ldots, f_d(P))\] where \(f_i(P)\) is the number of \(i\)-dimensional faces of \(P\). The question is whether a given vector of non-negative numbers is the \(f\)-vector of a polytope. For \(d\ge 4\), finding a complete characterization of \(f\)-vectors is an open problem. However, the so-called \(g\)-Theorem gives a description for \(f\)-vectors of simplicial and simple polytopes.

The purpose of this talk is to give a summary of the lectures of a summer school at MSRI on the \(g\)-Theorem and related topics.

For \(D\) a domain and \(E\subseteq D\), we investigate the prime spectrum of rings of functions from \(E\) to \(D\), that is, of rings contained in \(\prod_{e\in E} D\) and containing \(D\). Among other things, we characterize, when \(M\) is a maximal ideal of finite index in \(D\), those prime ideals lying above \(M\) which contain the kernel of the canonical map to \(\prod_{e\in E} (D/M)\) as being precisely the prime ideals corresponding to ultrafilters on \(E\).

We give a sufficient condition for when all primes above \(M\) are of this form and thus establish a correspondence to the prime spectra of ultraproducts of residue class rings of \(D\). As a corollary, we obtain a description using ultrafilters, differing from Chabert's original one which uses elements of the \(M\)-adic completion, of the prime ideals in the ring of integer-valued polynomials \(\mathrm{Int}(D)\) lying above a maximal ideal of finite index.

https://www.math.tugraz.at/discrete/events/DK-Day-2017.pdf

Let \(H\) be a multiplicatively written monoid. Given \(k \in \mathbf N^+\), we denote by \(\mathscr U_k\) the set of all \(\ell \in \mathbf N^+\) such that \(a_1 \cdots a_k = b_1 \cdots b_\ell\) for some irreducible elements \(a_1, \ldots, a_k, b_1, \ldots, b_\ell \in H\).

The sets \(\mathscr U_k\) are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large \(k\), which is usually expressed by saying that \(H\) satisfies the Structure Theorem for Unions.

The talk is about a refinement of the Structure Theorem for Unions in a wealth of situations, with an emphasis on the role played in this respect by weakly directed families, a ``purely additive model'' that, in a way, captures the deepest (combinatorial) nature of the kind of questions we are discussing.

(We recall that a weakly directed family is simply a collection \(\mathscr L\) of subsets of \(\mathbf N\) such that, for all \(L_1, L_2 \in \mathscr L\), there exists \(L \in \mathscr L\) with \(L_1 + L_2 \subseteq L\).)

In its original form, an algebraic variety is the common zero locus of a set of polynomials over the real or complex numbers. More generally, an algebraic variety \(X\) may be associated to any commutative finitely generated \(k\)-algebra \(R\), by identifying points of the variety with algebra homomorphisms from \(R\) to the field \(k\). The algebra \(R\) is then the ring of functions on \(X\). In this talk, I will introduce a way to associate geometric spaces to subalgebras of \(R\) which are infinitely generated; these new spaces look like \(X\), but have the strange property that they contain positive dimensional 'smeared-out' points. I will also describe noncommutative resolutions of these spaces, and indicate how their homological properties capture the dimensions of these smeared-out points.

In this talk, we discuss non-unique factorizations in the ring \(\text{Int}(\mathcal{O}_K)\) of integer-valued polynomials on the ring \(\mathcal{O}_K\) of algebraic integers of a number field \(K\), that is,

\[\text{Int}(\mathcal{O}_K) = \{f \in K[x] \mid f(\mathcal{O}_K) \subseteq \mathcal{O}_K\}.\]

Given a finite multiset \(N\) of natural numbers greater than 1, we explicitly construct a polynomial \(f \in \text{Int}(\mathcal{O}_K)\) which has exactly \(|N|\) essentially different factorizations of the prescribed lengths. In particular, this implies that every finite non-empty set \(N\) of natural numbers greater than 1 occurs as a set of lengths of a polynomial \(f \in \text{Int}(\mathcal{O}_K)\).  In addition, we show that there is no transfer homomorphism from the multiplicative monoid of \(\text{Int}(\mathcal{O}_K)\) to a block monoid.

We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid \(H\), let \(\Delta_{\rho} (H)\) be the set of all positive integers \(d\) which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity \(\rho (H)\). We study \(\Delta_{\rho} (H)\) for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

The distribution of pair correlations of an infinite sequence of reals in [0,1] can be seen as a test for pseudorandomness of the sequence. If the asymptotic distribution of pair correlations coincides with that of a truly random sequence, then the behavior of pair correlations is said to be "Poissonian". Typical systems which one likes to study such questions for are for example (\(n \alpha\)) and (\(n^2 \alpha\)) modulo one, for some irrational \(\alpha\). The study of pair correlations originated in mathematical physics, since the pair correlation statistics describes the distribution of energy levels of certain quantum integrable systems in the context of the Berry-Tabor conjecture. However, the problem is also very attractive from a purely mathematical point of view, since it brings together phenomena from number theory, Fourier analysis, combinatorics, Diophantine approximation, and other disciplines. In our talk we describe recent results in this field. In particular we explain the connection between the metric theory of pair correlations and additive combinatorics, and the relation between Poissonian pair correlations and the notion of equidistribution. For the latter part we will require some (simple linear) algebra.

https://www.math.tugraz.at/Tichy60/Programm-Festkolloquium_Internet_Version.pdf

One of the most influential problems in contemporary mathematics is a question of Erdős and Turán: Let \(r_k(n)\) denote the maximal number of integers in \([1,2,3, \ldots , n]\) without any arithmetic progression of length \(k\). Erdős conjectured that \(r_k(n)=o(n)\), where \(k\) is fixed and \(n\) tends to infinity. This was proved for \(k=3\) by Roth, and for general \(k\) by Szemerédi, with contributions by Bourgain, Gowers, Green, Sanders, Tao and many others.

It is expected that a good understanding of the question of maximal sets without progressions in the finite field case \(\mathbb{F}_q^n\) also leads to improvement in the integer case.

The cap set problem is to find good estimates for \(r_3(\mathbb{F}_3^n)\). Very recently there has been a breakthrough due to Croot, Lev and Pach, and soon later by Ellenberg and Gijswijt, proving that \(r_3(\mathbb{F}_3^n)\leq o((2.756)^n)\).

In this talk we discuss lower and upper bounds on \(r_k(\mathbb{F}_q^n)\).

Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. We wish to describe two variations. On the one hand, we compute determinants of matrices assembled from the squares of the cluster variables in Baur–Marsh’s matrix. One such determinant admits an interpretation as a Cayley–Menger determinant. On the other hand, we wish to present a formula for the determinant of a matrix of cluster variables in a cluster algebra of type D.

Using the injective stabilization of the univariate tensor product and the projective stabilization of the contravariant Hom functor, we define new notions of torsion and, respectively, cotorsion, which apply to arbitrary modules over arbitrary rings. For commutative domains the new torsion coincides with the classical torsion and, for finitely presented modules over arbitrary rings, it coincides with the 1-torsion (this is the kernel of the canonical map from a module to its bidual). The definition of the cotorsion does not seem to have a classical prototype.

The new definition of torsion is remarkably simple and will first be given in an elementary way, without appealing to functors. To define cotorsion, we shall use a functorial approach from the very beginning to compensate for the lack of a prototype. The resulting definition is also remarkably simple and can be given in an elementary way.

Time permitting, we shall see that the Auslander-Gruson-Jensen functor sends cotorsion to torsion. If the injective envelope of the ring is finitely presented, then the right adjoint of the AGJ functor sends torsion back to cotorsion. In particular, we obtain a duality between torsion and cotorsion over artin algebras.

This will be an expository talk. No prior knowledge of functor categories is assumed. This is joint work with Jeremy Russell.

• 15:15-16:15 Zhi-Wei Sun (Nanjing University, P.R. China)
  Restricted sums of three or four squares
  

• 16:15-16:45 Coffee Break
  

• 16:45 – 17:45 David J. Grynkiewicz (University of Memphis, USA)
  On the Degree of Regularity of \((x_1-y_1)+\ldots+(x_1-y_k)=c\)

Restricted sums of three or four squares
Abstract. The classical Gauss-Legendre theorem states that each \(n=0,1,2,\ldots\) not of the form \(4^k(8m+1)\ (k,m=0,1,2,\ldots)\) can be written as the sum of three squares. Lagrange's four-square theorem asserts that any natural number can be written as the sum of four squares. In this talk we introduce various conjectures refining these two theorems; for example, our 24-conjecture states that any nonnegative integers can be written as \(x^2+y^2+z^2+w^2\) with \(x,y,z,w\) nonnegative integers such that both \(x\) and \(x+24y\) are squares. We will also mention some recent results as well as related techniques in this direction.
 
On the Degree of Regularity of \((x_1-y_1)+\ldots+(x_1-y_k)=c\)
Abstract. It is conjecture of Fox and Kleitman that, for every integer \(k\geq 1\), there is an integer \(c_k>0\) such that the equation \((x_1-y_1)+\ldots+(x_1-y_k)=c_k\) is \((2k-2)\)-regular, meaning that coloring the positive integers using only \(2k-2\) integers always leads to a monochromatic solution. If true, this would be best possible. Previous bounds could show that the equation is \(C\log(k)\) regular. Using a multi-summand version of the Freiman \(3k-4\) Theorem due to Lev, we can improve this to the equation being \((k-1)\)-regular. We also verify the conjecture in the case \(k=3\) without the need to employ exhaustive computer search. Joint work with Adhikari and Shalom.

Organizers: K. Baur and A. Geroldinger

We shall give a brief introduction to the basic definitions and interesting problems in classical invariant theory. We will focus on the problem of determining the Noether number and we shall determine the exact value of Noether number for a new class of non-abelian groups. Meanwhile, we will show the strong connection between classical invariant theory and zero-sum theory. Then we will employ the method from classical invariant theory (Poincare series) to investigate zero-sum theory.

Gentle algebras are a particularly nice class of algebra for which the indecomposable complexes in the bounded derived category can be completely described in terms of string and band com- binatorics. This means that gentle algebras provide a natural laboratory in which to study the homological properties of finite-dimensional algebras concretely. In this talk, we shall describe the classification of indecomposable complexes in the bounded derived catgeory, a basis of morphisms between indecomposable complexes and describe a graphical calculus that computes the mapping cones of these morphisms. As an application, we shall give a complete description of the middle terms of extensions for a basis of the Ext space between any two string or band modules over a gentle algebra. The talk will be based on joint works with Kristin Arnesen and Rosanna Laking, and Ilke Canakci and Sibylle Schroll.

Iyama and Yoshino introduced a tool, known as Iyama-Yoshino reduction, which is very useful in studying the generators and decompositions of positive Calabi-Yau triangulated categories. However, this technique does not preserve the required properties for negative Calabi-Yau triangulated categories. In this talk, we establish a Calabi-Yau reduction theorem for this class of categories. This will be a report on joint work with David Pauksztello.

We give a brief review on gentle algebras, 2-Calabi-Yau categories and 2-Calabi-Yau tilted algebras. We characterize Cohen-Macaulay modules over 2-Calabi-Yau tilted algebras, and use this characterization to find all the possible gentle 2-Calabi-Yau tilted algebras.

I will discuss a Hamiltonian formalism for cluster mutations using canonical (Darboux) coordinates and piecewise-Hamiltonian flows with Euler dilogarithm playing the role of the Hamiltonian. The Rogers dilogarithm then appears naturally in the dual Lagrangian picture. I will show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from Hamiltonian/Lagrangian point of view. (Based on the joint project with T. Nakanishi and D. Rupel.)

Let \(\mathscr{L}\) be a collection of non-empty sets of non-negative integers. We denote by \(\mathscr{U}_k\), for every \(k \in \mathbf N\), the union of all \(L \in \mathscr{L}\) with \(k \in L\), and by \(\rho\) the supremum of \(\sup L/\inf L^+\) as \(L\) ranges over \(\mathscr{L}\), where \(L^+\) is the positive part of \(L\). We call \(\mathscr{L}\) a weakly directed family if for all \(L_1, L_2 \in \mathscr{L}\) there exists \(L \in \mathscr{L}\) with \(L_1 + L_2 \subseteq L\).

We show that, if \(\mathscr{L}\) is a weakly directed family and \(\rho = \sup L/\inf L^+ < \infty\) for some \(L \in \mathscr{L}\), there is \(m \in \mathbf N^+\) for which the following holds: Given \(M \in \mathbf N\), we have, for all large \(k \in \mathbf N\), \[(\mathscr{U}_{k+m} - \sup \mathscr{U}_{k+m}) \cap [\![-M,0 ]\!] = (\mathscr{U}_k - \sup \mathscr{U}_k) \cap [\![ -M, 0]\!]\] and \[(\mathscr{U}_{k+m} - \inf \mathscr{U}_{k+m}) \cap [\![ 0, M ]\!] = (\mathscr{U}_k - \inf \mathscr{U}_k) \cap [\![ 0, M ]\!].\] The result applies, in the first place, to the unions of sets of lengths of a BF-monoid with finite delta set and accepted elasticity: Most notably (and among many others), this covers transfer Krull monoids of finite type (including all commutative Krull domains with finite class group and certain maximal orders in central simple algebras over global fields), and some wide classes of weakly Krull commutative domains (including all orders in algebraic number fields with finite elasticity).

In this talk we study extensions between Cohen–Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We will prove that rank 1 modules are periodic, and we will give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine the \(i\)th extension group between arbitrary rank 1 modules. An explicit combinatorial algorithm is given for the computation of the \(i\)th extension group between rank 1 modules when i is odd, and when i even, we show that the \(i\)th extension group between rank 1 modules is cyclic over the centre, and we give an explicit formula for its computation.

Arithmetical invariants of hereditary orders (and in particular, of maximal orders) in central simple algebras over global fields have recently been studied by means of transfer homomorphisms to monoids of zero-sum sequences. In particular, in this case, invariants such as the elasticity, set of distances, and catenary degrees are finite under a certain module-theoretic condition.

The non-hereditary case has so far not been studied. As a first step towards this, we consider quaternion orders over discrete valuation rings. In this setting, we characterize finiteness of the elasticity, and show that the set of distances and all catenary degrees are finite. The proof, which splits into several cases, relies on a classification of quaternion orders in terms of ternary quadratic forms.

Joint work with Nicholas R. Baeth.

Joint work with Joseph Grant.

The braid group is a classical object in mathematics: the elements are the ways of twisting a fixed number of strands, up to isotopy, with the multiplication given by concatenation. Although it is defined topologically, the braid group has a beautiful presentation as an abstract group, given by Artin. This presentation can be associated to a Dynkin diagram of type A. In this way, a generalised braid group, or Artin braid group, can be associated to every Dynkin diagram.

A quiver is a directed graph, and an orientation of a Dynkin diagram is referred to as a Dynkin quiver. As part of the definition of a cluster algebra, Fomin and Zelevinsky introduced the notion of quiver mutation, where a quiver is changed only locally. A quiver which is mutation-equivalent to a Dynkin quiver is said to be mutation-Dynkin. Our main result is a presentation of an Artin braid group for any mutation-Dynkin quiver. We show how these presentations can be understood topologically in types A and D using a disk and a disk with a cone point of order two (i.e. an orbifold) respectively.

We introduce mutation along infinite admissible sequences for surfaces with infinitely many marked points on the boundary. This allows us to define mutation equivalence classes for the (completed) infinity-gon. For the completed infinity-gon, we then extending this notion to transfinite mutations and show that under this notion, the exchange graph is connected. This is joint work with S. Gratz (Oxford).

A key example of a geometric cluster algebra is Scott's cluster structure on the homogeneous coordinate ring of the Grassmannian. This cluster algebra admits a categorification, due to Jensen, King and Su, and it has been shown by Baur, King and Marsh that the endomorphism algebras of some special objects in this category may be described combinatorially in terms of bipartite graphs on a disc, called dimer models. Marsh and Scott give a formula computing some of the cluster monomials (the twisted Plücker coordinates) combinatorially from these dimer models. These cluster monomials may also be computed using the cluster character formula, provided in this level of generality by Fu and Keller. Understanding how these two apparently different computations arrive at the same answer reveals further connections between dimer models and homological algebra in the Grassmannian cluster category. This is joint work with İlke Çanakçı and Alastair King.

We are interested in constructing minimal linear representations of elements in the universal field of fractions (free field) of the free associative algebra (over a commutative field). I will present a recent result which serves as a first step, namely solving the word problem with linear techniques and discuss some open problems. Since the factorization of non-commutative polynomials seems to be closely related, I try to sketch a possible setup.

Let \(G\) be a finite group, \(\mathcal F (G)\) the free abelian monoid with basis \(G\), and \(\mathcal B (G) \subset \mathcal F (G)\) the monoid of product-one sequences over \(G\) (in other words, this is the semigroup of all finite, unordered sequences of terms from \(G\), where the terms can be ordered so that their product equals the identity of the group). Then \(\mathcal B (G)\) is a C-monoid and its class semigroup \(\mathcal C \big( \mathcal B (G), \mathcal F (G) \big)\) is a finite commutative semigroup. It is a Krull monoid if and only if \(G\) is abelian, and in that case the class semigroup coincides with the class group which is isomorphic to \(G\). We study the class semigroup and the arithmetic of \(\mathcal B (G)\).

Let \(\mathcal P_{\rm fin}(\mathbf Z)\) be the collection of all non-empty finite subsets of \(\mathbf Z\), which we turn into a commutative (non-cancellative) monoid by endowing it with the operation of set addition \[(X,Y) \mapsto X+Y := \{x+y: x \in X \text{ and } y \in Y\}.\] We say that a set \(A \in \mathcal P_{\rm fin}(\mathbf Z)\) is irreducible (or an atom) if there do not exist \(X, Y \in \mathcal P_{\rm fin}(\mathbf Z)\) with \(A = X + Y\) and \(|X|, |Y| \ge 2\). We prove that, for every \(r \in \mathbf N\), there exist \(X \in \mathcal P_{\rm fin}(\mathbf Z)\) and atoms \(A_0, A_1, \ldots, A_{r+2} \in \mathcal P_{\rm fin}(\mathbf Z)\) such that \[X = A_0 + A_1 = A_1 + \cdots + A_{r+2},\] and we use this result to draw a few conclusions about the arithmetic of \((\mathcal P_{\rm fin}(\mathbf Z), +)\) and, more in general (through suitable transfer principles), of an entire class of new monoids we refer to as power monoids.

Let \(H\) be a commutative cancellative monoid. We say that \(H\) has finite monotone catenary degree if there is some \(N\in\mathbb{N}_0\) such that for all \(a\in H\) it follows that each two factorizations of \(a\) can be concatenated by a monotone \(N\)-chain. A theorem of W. Hassler states that every strongly ring-like finitely primary monoid of rank at most two has finite monotone catenary degree. In this talk we extend this result to finite coproducts of strongly ring-like finitely primary monoids of rank at most two. Furthermore, we show that the monotone catenary degree of the monoid of \(v\)-invertible \(v\)-ideals of orders in quadratic number fields is finite.

Frieze patterns have an interesting combinatorial structure, which has proven very useful in the study of cluster algebras. We introduce \((k,n)\)-frieze patterns, a natural generalisation of the classical notion. A generalisation of the bijective correspondence between frieze patterns of width \(n\) and clusters of minors in the cluster structure of Grassmannian \(\mathrm{Gr}(2,n+3)\) is obtained, and we outline two inductive methods to expand and contract frieze patterns.

We define GL\(_m\)-dimers of triangulations of regular \(n\)-gons. These are bipartite graphs similar to GL\(_m\)-webs defined by A.B. Goncharov in a recently published paper. The GL\(_m\)-dimers give rise to a dimer model with boundary \(Q\) (introduced by Baur, King and Marsh) and a dimer algebra \(\Lambda_Q\), a path algebra with relations arising from a potential. Let \(e_b\) be the sum of all idempotent boundary components, then the dimer algebra leads to a boundary algebra \(\mathcal{B}:= e_b \Lambda_Q e_b\). It turns out that given two different triangulations \(T_1\) and \(T_2\) of the \(n\)-gon, then the boundary algebras are isomorphic, i.e. \(e_b \Lambda_{Q_{T_1}} e_b \cong e_b \Lambda_{Q_{T_2}} e_b\).

In this talk we will present some research on skew polynomial rings. We have studied minimal prime ideals of skew polynomial rings over Dedekind domains, skew polynomial rings over generalized Asano prime rings and skew polynomial rings over Morita rings of Morita context. Moreover, we will also present skew polynomial rings in coding theory.

Subword complexes are simplicial complexes introduced by A. Knutson and E. Miller as a tool to study Gröbner geometry of Schubert polynomials. In this talk, I will present some relevant results about of these objects in algebra, combinatorics, and discrete geometry. In particular, I will focus on:

• combinatorics of triangulations and multi-triangulations of convex polygons,

• two applications in cluster algebras and Hopf algebras, and

• geometric constructions of multi-associahedra.

This talk is based on joint works with Nantel Bergeron, Jean-Philippe Labbé, Vincent Pilaud, and Christian Stump.

Let \(H\) be a Krull monoid with finite class group \(G\) such that every class contains a prime divisor. Then every non-unit \(a \in H\) can be written as a finite product of atoms, say \(a=u_1 \cdot \ldots \cdot u_k\). The set \(\mathsf L (a)\) of all possible factorization lengths \(k\) is called the set of lengths of \(a\). There is a constant \(M \in \mathbb N\) such that all sets of lengths are almost arithmetical multiprogressions with bound \(M\) and with difference \(d \in \Delta^* (H)\), where \(\Delta^* (H)\) denotes the set of minimal distances of \(H\). We study the structure of \(\Delta^* (H)\) and establish a characterization when \(\Delta^*(H)\) is an interval.

The system \(\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}\) of all sets of lengths depends only on the class group \(G\), and a standing conjecture states that conversely the system \(\mathcal L (H)\) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to \(C_n^r\) with \(r,n \in \mathbb N\) and \(\Delta^*(H)\) is not an interval.

Let \(H\) be a Krull monoid with class group \(G\). Then \(H\) is factorial if and only if \(G\) is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of \(H\) in the non-factorial case. We focus on the set \(Ca(H)\) of catenary degrees of \(H\) and on the set \(\mathcal{R}(H)\) of distances in minimal relations. We show that every finite nonempty subset of \(\mathbb{N}_{\geq 2}\) can be realized as the set of catenary degrees of a Krull monoid with finite class group. Suppose in addition that every class of \(G\) contains a prime divisor. Then \(Ca(H)\subset \mathcal{R}(H)\) and \(\mathcal{R}(H)\) contains a long interval. Under a reasonable condition on the Davenport constant of \(G\), \(\mathcal{R}(H)\) coincides with this interval and the maximum equals the catenary degree of \(H\).

Scientific Program:

• 10:00-10:30 Coffee

• 10:30-11:15 Weidong Gao (Nankai University, PR China)
  Zero-sum subsequences of distinct lengths

• 11:30-12:15 Wolfgang A. Schmid (University of Paris 8 & 13, France)
  Transfer Krull monoids and weakly Krull monoids: a comparison of their sets of lengths

• 14:00-14:45 Pedro A. García-Sánchez (University of Granada, Spain)
  Numerical Semigroups

• 15:00-15:45 Salvatore Tringali (University of Graz, Austria)
  Unions of Sets of Lengths

Organizers: K. Baur and A. Geroldinger

Detailed program: as pdf. file here

Cluster algebras were introduced by Fomin and Zelevinsky in early 2000 in the context of Lie theory. Cluster algebras are commutative rings with a set of distinguished generators. Due to their rich combinatorial structure, the theory of cluster algebras has spread to many other areas of mathematics, from triangulations of surfaces, to Teichmueller theory, to quiver representations. Fock and Gonchorov, Fomin, Shapiro, and Thurston, and Gekhtman, Shapiro, and Vahnstein established a relation between cluster algebras and hyperbolic geometry. Explicit combinatorial formulas for cluster variables in terms of perfect matchings of snake graphs were given by Musiker, Schiffler, and Williams. Thus the curves in the surface completely determine the combinatorial and algebraic structure of the cluster algebra.

In this thesis, we consider asymptotic triangulations, and describe their cluster algebra structure. We start with asymptotic triangulations of the annulus, and consider the flips of arcs in the triangulations, their associated quivers and the mutation rules of these quivers, and then look at their associated algebra. Using lambda lengths and laminations, we get a cluster algebra-like structure with principle coefficients for asymptotic triangulations. Triangulations and asymptotic triangulations of the annulus were used to characterize infinite frieze patterns of integers by Baur, Parsons, and Tschabold. Then we give a cluster-algebraic interpretation of these infinite frieze patterns, by constructing an infinite frieze where the entries are Laurent polynomials of generalized arcs in the triangulated surface. Using snake graphs and skein relations, we achieve some algebraic and combinatorial results involving the relationships between entries in the frieze, and give geometric interpretations of known results for infinite friezes with integer entries.

Both the representation theories of affine Kac-Moody groups and quivers present a tripartite structure. Representations of a Kac-Moody group \(G\) come naturally in three classes (positive, zero, and negative level representations) according to how the center of \(G\) acts. Indecomposable representation of a quiver \(Q\) are either preprojective, postinjective, or regular depending on where they sit in the associated Auslander-Reiten quiver.

We connect these two trialities using cluster algebras. By identifying the ring of coordinates of an appropriate double Bruhat cell of \(G\) as a cluster algebra we show how cluster variables coming from preprojective (resp. postinjective and regular) representations of \(Q\) can be interpreted as generalized minors of \(G\) arising from positive level (resp. negative level and 0 level) representations.

Let \(\Lambda\) be a representation-finite \(\mathbb C\)-algebra which has Hall polynomials. In this talk I will describe the \(\mathbb Z\)-Lie algebras \(L(\Lambda)\) and \(K(\Lambda)\) associated with \(\Lambda\) which are defined by Riedtmann and Ringel. Then I will show that if \(\Lambda\) has a universal cover \(\Lambda'\) which is a locally bounded directed \(\mathbb C\)-algebra, then \(K(\Lambda)\) and \(L(\Lambda)\) are isomorphic. As an application we will see that if \(\Lambda\) is a representation-finite (generalized) cluster-tilted algebra or representation-finite trivial extension algebra, then \(K(\Lambda)\) and \(L(\Lambda\)) are isomorphic.

A diagram category is, more or less, a subcategory of the partition category. We recall the definition of the partition category, observe various subcategories within it, and discuss connections to representation theory, low-dimensional geometry, and aspects of combinatorics.

We consider the generalized concept of order relations on \(B(H)\), the algebra of all bounded linear operators on a Hilbert space \(H\), which was proposed by Šemrl and which covers the star partial order, the left-star partial order, the right-star partial order, and the minus partial order. The notion of these and some other partial orders is extended to Rickart rings or Rickart *-rings and some well-known results are generalized.

The class of multiserial algebras contains many well-studied examples of algebras such as the intensely-studied biserial and special biserial algebras. These, in turn, contain many of the algebras of tame representation type arising in the modular representation theory of finite groups such as tame blocks of finite groups and all tame blocks of Hecke algebras or in another direction they contain the finite dimensional algebras arising out of cluster theory such as gentle algebras associated to angulations of surfaces. However, unlike biserial algebras which are of tame representation type, multiserial algebras are generally of wild representation type. Roughly speaking this means that their representation theory is at least as complicated as that of the free associative algebras in two generators. We will show that despite this fact, we retain some control over the representation theory of multiserial algebras thus giving a new class of algebras that allow some insight into the wild representation type.

The null ideal of a matrix \(A\) (with entries in a commutative ring \(R\)) is the ideal of \(R[X]\) consisting of those polynomials \(f\) with \(f(A) =0\). Better understanding of the null ideals of matrices over \(R=D/J\) (for a domain \(D\) and an ideal \(J\)) has applications in the theory of integer-valued polynomials and polynomial mappings on non-commutative rings. Introducing the \(J\)-ideal of a matrix, a generalization of the null ideal, allows us to work over \(D\) instead of \(D/J\). We present certain generating sets of \(J\)-ideals and discuss computational aspects.

An important recent development in singularity theory is the appearance of non-commutative resolutions. The idea, which first occurred in physics, is similar to the idea of classical (commutative) resolutions. A space is by the geometry/algebra duality replaced by a ring, and thus one looks for a non-singular (possibly non-commutative) ring that replaces the original (singular) ring.

The notion of a non-commutative crepant resolution (NCCR) was introduced by Van den Bergh in 2004. In the talk we will first give an introduction to non-commutative resolutions. Then we will present some recent joint work with Michel Van den Bergh on non-commutative (crepant) resolutions of quotient singularities for reductive groups.

Let \(B/A\) be a ring extension and let \(X\) be the collection of all the \(A\)-submodules of \(B\). The aim of this talk, based on paper in collaboration with D. Spirito, is to present a result that describes a relation between a new natural topology on \(X\) and the algebraic properties of a flat module \(M\) in \(X\). Namely, we give topological conditions on a subspace \(Y\) of \(X\) under which multiplication by \(M\) commutes with the intersection of the members of \(Y\). Among the applications of this result, we will provide a very short proof of the Glaz-Vasconcelos conjecture.

Let \(H\) be a Krull monoid with finite class group \(G\) such that every class contains a prime divisor. For \(k\in \mathbb{N}\), let \(\mathcal{U}_k(H)\) denote the set of all \(m\in \mathbb{N}\) with the following property: There exist atoms \(u_1, \ldots, u_k, v_1, \ldots, v_m \in H\) such that \(u_1\cdot \ldots \cdot u_k=v_1\cdot \ldots \cdot v_m\). It is well-known that the sets \(\mathcal{U}_k(H)\) are finite intervals and their minima \(\lambda_k(H)\) can be expressed in terms of \(\rho_k(H)\). Furthermore, the invariants \(\rho_k(H)\) depend only on the class group \(G\). If \(|G|\leq 2\), then \(\rho_k(H)=k\) for every \(k\in \mathbb{N}\). Suppose that \(|G|\geq 3\). An elementary counting argument shows that \(\rho_{2k}(H)=kD(G)\) and \(kD(G)+1 \leq \rho_{2k+1}(H)\leq kD(G)+\lfloor \frac{D(G)}{2}\rfloor\) where \(D(G)\) is the Davenport constant. It is known that for cyclic groups we have \(kD(G)+1= \rho_{2k+1}(H)\) for every \(k\in \mathbb{N}\). We show that (under a reasonable condition on D(G)) for every noncyclic group there exists a \(k^*\in \mathbb{N}\) such that \(\rho_{2k+1}(H)= kD(G)+\lfloor \frac{D(G)}{2}\rfloor\) for every \(k\geq k^*\).

The set of distances of a monoid or of a domain is the set of all \(d \in \mathbb N\) with the following property: there are irreducible elements \(u_1, \ldots, u_k, v_1, \ldots, v_{k+d}\) such that \(u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{k+d}\), but \(u_1 \cdot \ldots \cdot u_k\) cannot be written as a product of \(l\) irreducible elements for any \(l\) with \(k < l < k+d\). We show that the set of distances is an interval for certain seminormal weakly Krull monoids which include seminormal orders in holomorphy rings of global fields.

Scientific Program:

• 15:00 Opening

• 15:00-15:15 Robert F.Tichy (Graz University of Technology)
  Laudatio

• 15:15-16:00 Nigel Byott (University of Exeter)
  Scaffolds and Local Galois Module Structure

• 16:00-16:30 Coffee Break

• 16:30-17:15 Ernst-Ulrich Gekeler (Saarland University)
  Stochastic Structures in Number Theory

• 17:20-18:05 Attila Pethő (University of Debrecen)
  Parametrized Families of Thue Equations a'la Günter Lettl

Organizers: K. Baur and A. Geroldinger

The non-commutative analogue of a Dedekind ring is a hereditary noetherian prime ring. The ideal theory of such rings has been ``well known'' for a while, but - as will be demonstrated - not quite fully understood. As in the commutative case, divisors form a free abelian group containing the essence of unique prime factorization, but in a non-obvious way: in general, the divior group is decorated with a 1-cocycle, every ideal is determined by its divisor, but a divisor need not always come from an ideal. Nevertheless, every ideal is a unique product of primes, up to commutation rules which account for the grain of salt. In the local case, ideals are not just powers of primes, they are given by monotonic periodic functions, composition of functions standing for multiplication of ideals. The structures arising here follow a pattern which has been observed in other topics like differential geometry, Lie theory, and quantum groups.

We survey results on factorizations of non-zero-divisors into atoms in noncommutative rings. Two approaches to unique factorization are covered: First, unique factorization of elements into products of atoms up to order and similarity, in particular in \(2\)-firs, and generalizations thereof; secondly, UFRs and UFDs in the sense of Chatters and Jordan, which take Kaplansky's characterization of commutative UFDs as a starting point for a definition in the noncommutative setting. As far as rings with non-unique factorizations are concerned, we survey transfer results for matrix rings, rings of triangular matrices, and hereditary noetherian prime rings (which include classical hereditary orders in central simple algebras over global fields).

The Ziegler spectrum is a topological space that was first introduced by M. Ziegler in the context of the model theory of modules. Surprisingly, the definition of the space has an entirely algebraic description, and its structure enables us to study interactions between finite-dimensional and infinite-dimensional objects. We will introduce the Ziegler spectrum of a (compactly generated) triangulated category, as well as two dimensions on the space. We will then use the class of derived-discrete algebras to illustrate how the complexity measured by these dimensions is reflected in the structure of the bounded derived category.

This is a report on joint work with K. Arnesen, D. Pauksztello and M. Prest.

Frieze patterns were introduced in 1971 by Coxeter who subsequently studied them also in collaboration with Conway. They are arrays composed of finitely many shifted rows of numbers, bounded from above by a row of zeros, followed by a row of ones, and bounded from below by a row of ones, followed by a row of zeros, and satisfying everywhere a local determinant rule. An interesting feature of these frieze patterns is that they are invariant under a glide reflection and hence periodic. A key classical result of Conway and Coxeter establish a one-to-one correspondence between friezes of positive integers and triangulations of polygons.

In the first part of this talk, we introduce the notion of infinite frieze patterns of positive integers in the plane as a modification of the classical notion of friezes patterns and characterize them via triangulations. They differ from classical frieze patterns in that they are not bounded from below, and consist instead of infinitely many rows. In addition, they need not be periodic. One of our main results shows that triangulations of once-punctured discs and annuli give rise to periodic infinite friezes of positive integers, with all such friezes arising in this way. More generally, we will see that all infinite friezes of positive can be obtained from triangulations of an infinite strip in the plane. Furthermore, we provide a geometric interpretation of all entries of an infinite frieze via matching numbers, extending an important classical result. Part of this is joint work with Karin Baur and Mark Parsons.

In the second part, we then focus on periodic infinite friezes of positive integers. We first establish a key feature of friezes arising from triangulations of a once-punctured discs. More precisely, each diagonal of such a frieze is made up of a collection of arithmetic sequences. This motivates introducing the notion of growth coefficients for periodic infinite friezes of positive integers, which enables us to see that friezes associated to triangulations of once-punctured discs exhibit the only friezes of linear growth, while friezes arising from triangulations of annuli grow exponentially. Furthermore, we give some recent results on this new subject. This is work in progress with Karin Baur, Klemens Fellner and Mark Parsons.

Let \(L/K\) be a normal extension of number fields. The Hasse norm principle is a local-global principle for norms. It is satisfied if any element \(x\) of \(K\) is a norm from \(L\) whenever it is a norm locally at every place. For any fixed abelian Galois group \(G\), we investigate the density of \(G\)-extensions violating the Hasse norm principle, when \(G\)-extensions are counted in order of their discriminant. This is joint work with Dan Loughran and Rachel Newton.

Orbit categories of derived categories have been the subject of significant interest in recent years, particularly cluster categories. We consider a different orbit category C, which can be realised as a negative Calabi-Yau cluster category. In particular, we study the endomorphism algebras of maximal rigid objects of C, including an explicit description of these algebras in terms of quivers with relations.

As an interesting byproduct, we are led to consider a more general class of algebras, which also include the "classical" cluster-tilted algebras, called tiling algebras. This is a report on joint work with Mark Parsons.

In this talk, we consider infinite frieze patterns of positive integers in the plane. In the periodic case, we have seen a characterization via triangulations of once-punctured discs and annuli. We briefly recall how to obtain a periodic infinite frieze from triangulations of once-punctured discs and annuli. In particular, a triangulation of an annulus provides two infinite friezes, namely the pair of outer and inner friezes. We then introduce the notion of growth coefficients for periodic infinite friezes and give some recent results on this new subject, especially on the growth behaviour of periodic infinite friezes. We will see that for every infinite frieze of period \(n\), the difference between the \(n\)th and the \((n-2)\)th non-trivial rows is constant. Moreover, there is a linear recursion formula for the entries in a diagonal of a periodic infinite frieze that depends on this value. Given a periodic infinite frieze, we may ad or remove triangles in the associated triangulation. We will show that the growth coefficient stays invariant under these operations. Another nice result is that the growth coefficients are the same for the pair of outer and inner friezes arising from a triangulation of an annulus. Finally, we will conclude that triangulations of once-punctured discs provide the only friezes of linear growth, while friezes arising from triangulations of annuli have exponential growth. This is work in progress with Karin Baur, Klemens Fellner and Mark Parsons.

In this talk, we consider a lifting of Joseph ideals for minimal nilpotent orbit closures to the setting of affine Kac-Moody algebras and find new examples of affine vertex algebras whose associated varieties is a minimal nilpotent orbit closure. As an application, we obtain a new family of lisse W-algebras. This is a joint work with Tomoyuki Arakawa.

Frieze patterns were introduced by Coxeter who subsequently studied them in collaboration with Conway. Together, they gave a characterisation of frieze patterns in terms of triangulations of polygons, establishing that every frieze pattern has an associated triangulation of a polygon and vice versa. This was later extended by Broline, Crowe and Isaacs who showed that all of the entries of a frieze pattern can in fact be obtained from its associated triangulation of a polygon via matchings of triangles to the vertices of that polygon.

After briefly recalling the definition of frieze patterns and illustrating the above results, we will focus on joint work with Baur and Tschabold on 'infinite friezes' (which differ from Conway-Coxeter frieze patterns in that they have infinitely many rows), in which results analogous to the classical theory are proved. In particular, we will see that the periodic infinite friezes have a characterisation in terms of triangulations of punctured discs and annuli. Moreover, the entries of such a frieze can be obtained from any associated triangulation via matchings.

Let \(H\) be a Krull monoid with class group \(G\) such that every class contains a prime divisor. Then every nonunit \(a \in H\) can be written as a finite product of irreducible elements. If \(a=u_1 \cdot \ldots \cdot u_k\), with irreducibles \(u_1, \ldots u_k \in H\), then \(k\) is called the length of the factorization and the set \(\mathsf L (a)\) of all possible \(k\) is called the set of lengths of \(a\). It is well-known that the system \(\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}\) depends only on the class group \(G\). In this talk we study the inverse question asking whether or not the system \(\mathcal L (H)\) is characteristic for the class group. Consider a further Krull monoid \(H'\) with class group \(G'\) such that every class contains a prime divisor and suppose that \(\mathcal L (H) = \mathcal L (H')\). We show that, if one of the groups \(G\) and \(G'\) is finite and has rank at most two, then \(G\) and \(G'\) are isomorphic (apart from two well-known pairings).

This is joint work with Wolfgang A. Schmid.

Let \(H\) be a Krull monoid with finite class group \(G\) and suppose that every class contains a prime divisor. If an element \(a \in H\) has a factorization \(a=u_1 \cdot \ldots \cdot u_k\) into irreducible elements \(u_1, \ldots, u_k \in H\), then \(k\) is called the length of the factorization and the set \(\mathsf L (a)\) of all possible factorization lengths is the set of lengths of \(a\). It is classical that the system \(\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}\) of all sets of lengths depends only on the class group \(G\), and a standing conjecture states that conversely the system \(\mathcal L (H)\) is characteristic for the class group. We verify the conjecture if the class group is isomorphic to \(C_n^r\) with \(r,n \ge 2\) and \(r \le \max \{2, (n+2)/6\}\). Indeed, let \(H'\) be a further Krull monoid with class group \(G'\) such that every class contains a prime divisor and suppose that \(\mathcal L (H)= \mathcal L (H')\). We show that, if one of the groups \(G\) and \(G'\) is isomorphic to \(C_n^r\) with \(r,n\) as above, then \(G\) and \(G'\) are isomorphic (apart from two well-known pairings). The proof is based on methods from Additive Combinatorics.

Chekhov and Shapiro (2011) have introduced the notion of "generalised cluster algebra”; we will focus on an example in type \(C_n\). On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the specialisation of a quantum affine algebra's spectral parameter \(q\) at a root of unity. Our main result draws these two notions together: we state that for the Lie algebra \(\mathfrak g=\mathfrak{sl}_2\), the Grothendieck ring of a certain tensor subcategory of representations of \(U_q(L\mathfrak{sl}_2)\) specialised at a root of unity of order \(l\), is a generalised cluster algebra of type \(C_{l-1}\). We will also state similar results and conjectures for \(\mathfrak g=\mathfrak{sl}_3\) and \(\mathfrak{sl}_4\).

It is well-known that the arithmetic of a Krull domain can be described by its divisor-class group. However, there are many integral domains which fail to be Krull (e.g. ${\rm Int}(\mathbb{Z})$). The arithmetic of ${\rm Int}(\mathbb{Z})$ has been investigated by using other methods. For instance, there is a result of S. Frisch (see [1]), which states that every non-empty finite subset of $\mathbb{N}_{\geq 2}$ is the set of lengths of some nonzero element of ${\rm Int}(\mathbb{Z})$. On the other hand, F. Kainrath (see [3]) proved that whenever $H$ is a Krull monoid with infinite divisor-class group $\mathcal{C}_v(H)$ such that every element of $\mathcal{C}_v(H)$ contains a height-one prime ideal, then every non-empty finite subset of $\mathbb{N}_{\geq 2}$ is the set of lengths of some element of $H$. Recently (see [2,4]), it was shown that every monadic submonoid of ${\rm Int}(R)$ is a Krull monoid if $R$ is a factorial domain (or a Krull domain). This raises the question whether the result of Frisch can be proved by using the result of Kainrath, and it motivates the investigation of divisor-class groups of monadic submonoids of ${\rm Int}(R)$. We study these groups in a series of talks.

In the first talk we recall basic concepts which are crucial for studying Krull monoids (e.g. divisorial ideals, height-one prime ideals, and various types of submonoids, like saturated, divisor-closed or monadic submonoids). We prove several important facts about saturated submonoids of Krull monoids. Finally, we provide a few preparatory results for the second talk.

In our second talk we intensify the investigation of monadic submonoids of rings of integer-valued polynomials over $R$ (where $R$ is a factorial domain). We describe the structure of their atoms and their height-one prime ideals. If $f\in R[X]$, then we show that the divisor-class group of $[\![f]\!]$ (i.e. the monadic submonoid generated by $f$) is torsion-free. We determine the divisor-class group of various monadic submonoids of ${\rm Int}(\mathbb{Z})$ (e.g. of $[\![X(X-1)(X-2)]\!]$ and its divisor-closed submonoids).

[1]

S. Frisch, A construction of integer-valued polynomials with prescribed sets of lengths of factorizations, Monatsh. Math. 171 (2013), no. 3 – 4, 341 – 350.

[2]

S. Frisch, Relative polynomial closure and monadically Krull monoids of integer-valued polynomials, arXiv:1409.1111v2.

[3]

F. Kainrath, Factorization in Krull monoids with infinite class group, Colloq. Math. 80 (1999), no. 1, 23 – 30.

[4]

A. Reinhart, On monoids and domains whose monadic submonoids are Krull, Commutative Algebra, 307 – 330, Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions (Marco Fontana, Sophie Frisch and Sarah Glaz, eds.), Springer, 2014.

Given a finite abelian group \(G\) and zero-sum sequence \(S\) with terms from \(G\), there may be multiple ways to factor \(S\) into disjoint minimal zero-sum subsequences, also called atoms: \(S=U_1{\boldsymbol{\cdot}}\ldots{\boldsymbol{\cdot}}U_k\). Thus each atom \(U_i\) is a zero-sum sequence over \(G\) having no proper, nontrivial zero-sum sequence. If we know that \(S\) has such a factorization using \(k\) atoms, we can ask for what other integers \(\ell\) does \(S\) have a factorization into \(\ell\) atoms. If we ask the same question, not for a fixed \(S\), but instead over all possible \(S\) having some factorization using \(k\) atoms, we obtain the well-studied invariant \[\mathcal U_k (G)=\{\ell\in \mathbb N:\; \text{ there exist atoms } U_i \text{ and } V_j \text{ such that } U_1{\boldsymbol{\cdot}}\ldots {\boldsymbol{\cdot}}U_k=V_1{\boldsymbol{\cdot}}\ldots{\boldsymbol{\cdot}}V_\ell \}.\] The structure of \(\mathcal U_k(G)\) is known to be completely determined by its maximal value \(\rho_k(G)=\max \mathcal U_k(G)\). However, apart from the trivial bounds \(\rho_{2k} (G) = k \mathsf D (G)\)  and \[\label{crucialinequality} 1 + k \mathsf D (G) \le \rho_{2k+1} (G) \le k \mathsf D (G) + \Big\lfloor \frac{\mathsf D (G)}{2} \Big\rfloor,\] where \(\mathsf D(G)\) denotes the Davenport constant of \(G\), little is known about \(\rho_{2k+1}(G)\). In this talk, we will discuss some inductive lower bounds for \(\rho_{2k+1}(G)\) as well as a characterization of when a rank two group attains the upper bound for \(\rho_3(G)\), both in relation to conjectured behaviour for \(\rho_{2k+1}(G)\).

The results presented in this talk stem from joint work with Bertran Steinsky.

Labyrinth fractals are a special case of Sierpinski carpets. They were introduced and studied in two papers by Cristea and Steinsky published a few years ago (1,2) and are self-similar dendrites in the unit square.

These fractals are constructed iteratively by using a pattern (or labyrinth set) that is the generator of the fractal. Under certain conditions on the labyrinth pattern that generates the self-similar fractal, the length of the path in the fractal between any two points of the fractal is infinite.

In recent work (3), we studied mixed labyrinth fractals, which are one possible generalisation of labyrinth fractals. In general, mixed labyrinth fractals are not self-similar. Among other, we have shown that they are dendrites, and we have studied properties of the paths. Moreover, further generalisations are considered.

Acknowledgement: Ligia L. Cristea founds her research by means of the stand-alone FWF-project P27050-N26.

(1)

L.L. Cristea, B. Steinsky, Curves of Infinite Length in 4x4-Labyrinth Fractals, Geometriae Dedicata, Vol. 141, Issue 1 (2009), 1--17.

(2)

L.L. Cristea, B. Steinsky, Curves of Infinite Length in Labyrinth-Fractals, Proceedings of the Edinburgh Mathematical Society Volume 54, Issue 02 (2011), 329--344.

(3)

L.L. Cristea, B. Steinsky, Mixed labyrinth fractals, submitted for publication.

We generalize results of C. Prabpayak, which describe all orders of a pure cubic number field, to arbitrary cubic number fields.

Calabi-Yau (CY) triangulated categories are those satisfying a useful and important duality, characterised by a number called the CY dimension. Much work has been carried out on understanding positive CY triangulated categories, especially in the context of cluster-tilting theory. However, not much is known about negative CY triangulated categories. In order to understand the structure of such categories, one can consider their generating objects. Mutations of such objects play a key role in the theory developed in the positive CY case.

In this talk, we will consider a certain class of generating objects of negative CY triangulated categories, namely Hom-configurations. We will see how the behaviour of their mutations is very reminiscent to that of cluster-tilting objects in the positive CY case.

This is a report on work in progress.

Functions on non-commutative rings arising from substitution of the variable (either to the right or to the left of the coefficients) in polynomials satisfy some conditons that one would expect to fail in the absence of substitution homomorphism.

For certain kinds of non-commutative subrings of matrix algebras we show that the set of left-null polynomials (a priori a right ideal) is a two-sided ideal. An open conjecture of N. Werner states that this is true for all finite non-commutative rings. Also for certain kinds of non-commutative subrings of matrix algebras the set of integer-valued polynomials forms a ring, like in the commutative case. The methods used are combinatorial, involving the interpretation of entries in powers of matrices as sums of weighted paths in a graph.

A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings.

In this talk we deal with the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra \(A\) does not have a non-trivial grading if and only if \(A\) is basic, its quiver has one vertex, and its group of outer automorphisms \({\rm Out}(A)\) is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist non-trivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.

Let \(R\) be a bounded Dedekind prime ring. Then the multiplicative semigroup of non zero divisors, \(R^\bullet\), is an arithmetical maximal order. Hence, under certain conditions on the abstract norm, there exists a transfer homomorphism from \(R^\bullet\) to a monoid of zero-sum sequences over a subset of an abelian group \(C\). We show that, in fact, this group \(C\) is isomorphic to the projective class group of \(R\), and that the condition is that every stably free right \(R\)-ideal is free. This generalizes an analogous earlier result, in which \(R\) had to be a classical maximal order in a central simple algebra over a global field.

Infinite friezes are a variation of Coxeter-Conway frieze patterns introduced and studied by Conway and Coxeter. Triangulations of punctured discs give rise to periodic infinite friezes having special properties. There is a combinatorial interpretation of the entries of periodic infinite friezes associated to triangulations of punctured discs via matchings for certain combinatorial objects, namely periodic triangulations of strips. In this talk we will provide an alternative description of the entries of periodic infinite friezes associated to triangulations of punctured discs that provides the diagonal sequences of the periodic infinite friezes. We use a similar method of assigning new labels to the vertices of a periodic triangulation of a strip as used for Coxeter-Conway frieze patterns by Conway and Coxeter. Furthermore, we will give a connection between periodic infinite friezes associated to triangulations of punctured discs and Farey triangulations.

Elliptic curve cryptography has gained increasing interest over the last two decades. Besides the possibility of shorter key lengths, this is also due to its ability to create very efficient and multifunctional cryptographic schemes by means of bilinear pairings. A bilinear pairing is a non-degenerate bilinear map \(e : \mathbb{G}_1 \times \mathbb{G}_2 \to \mathbb{G}_T\) where \(\mathbb{G}_1\) and \(\mathbb{G}_2\) are suitable groups of elliptic curves and \(\mathbb{G}_T\) is a subgroup of the multiplicative group of a finite field. We will give an overview of necessary results to construct pairings and give examples and possible applications. After that, we discuss Miller's algorithm and other techniques that make the computation of pairings feasible. We also present algorithms to find suitable finite fields and elliptic curve parameters to obtain pairing-friendly elliptic curves from the family of Barreto-Naehrig curves.

Asymptotic triangulations can be viewed as limits of triangulations under the action of the mapping class group. We construct an alternative method of obtaining these asymptotic triangulations using Coxeter transformations. This provides us with an algebraic and combinatorial framework for studying these limits via the associated quivers and root systems.

The quivers "mutation equivalent" to an orientation of a Dynkin diagram are precisely those appearing in the seeds of the corresponding cluster algebra. A companion basis for such a quiver \(\Gamma\) is a \(\mathbb{Z}\)-basis of roots for the integral root lattice of the associated root system with the property that non-zero inner products of pairs of its elements correspond to edges in the underlying graph of \(\Gamma\). We first briefly recall some of the key results on companion bases, and then present a method for explicitly constructing companion bases which makes use of a companion basis mutation procedure.

We consider a quiver of type \(A_n\), the number of tilting modules over the path algebra is just the Catalan number \(C_n\). Moreover, the number of cluster tilting modules over this path algebra is the next Catalan number \(C_{n+1}\). In this talk we give a new interpretation of these numbers in terms of the volume of a polytope, with vertices the positive roots of type \(A\). Those polytopes are related to root polytopes, the convex hull of roots of a finite root system. We present the results explicitely for type \(A\) and relate them to combinatorial descriptions in Stanleys book about 'Algebraic Combinatorics'. In the last part, we consider various generalisations for the other Dynkin quivers and even algebras with relations.

Dividing out by the action of a group on some algebraic structure is a ubiquitous construction. In topological quantum field theory, where it appears in "orbifolding a symmetry", this leads to a natural generalisation called "equivariant completion". Equivariant completion is a simple, purely categorical construction that can uncover unexpected new relations. We shall introduce the basic ideas of this theory, and then apply it to produce equivalences between derived categories of path algebras of Dynkin quivers.

The question, which sets of integers can be written as a sumset \(S=A+B\), possibly with some exceptions, is for most given sets \(S\) open. Ostmann asked it for the set of primes, Sárközy for the set of smooth numbers, and also for quadratic squares modulo \(p\). We will give a survey of some recent results in this area.

Let \(S\) be a commutative ring with identity, \(R\) a subring of \(S\) and \(I\) an ideal of \(S\). We say that \(I\) is an \(R\)-conductor ideal of \(S\) if \(I=\{x\in S\mid xS\subseteq V\}\) for some intermediate ring \(V\) of \(R\) and \(S\). In 1919, P. Furtwängler investigated \(\mathbb{Z}\)-conductor ideals of principal orders in algebraic number fields. He elaborated a characterization of these ideals. In this talk we provide sufficient criteria for being an \(R\)-conductor ideal of \(S\). Among others we show that if \(R/I\cap R\) is a principal ideal ring and every \(P\in {\rm spec}(S)\) with \(I\subseteq P\) satisfies \(R+P\subsetneqq S\), then \(I\) is an \(R\)-conductor ideal of \(S\). Furthermore, we present a more general version of Furtwängler's characterization. We use it to rediscover the following variant of his result:
Let \(R\) and \(S\) be Dedekind domains and \(I\cap R\not=\{0\}\). For \(P\in\max(S)\) with \(P\cap R\not=\{0\}\) set \(e_P=v_P((P\cap R)S)\) and \(f_P=\dim_{R/P\cap R}(S/P)\). Then \(I\) is an \(R\)-conductor ideal of \(S\) if and only if every \(P\in\max(S)\) with \(I\subseteq P\) satisfies at least one of the following conditions:

• \(f_P\geq 2\).

• \(e_P\nmid (v_P(I)-1)\).

• \(\frac{v_P(I)-1}{e_P}<\frac{v_Q(I)}{e_Q}\) for some \(Q\in\max(S)\setminus\{P\}\) such that \(I\subseteq Q\) and \(Q\cap R=P\cap R\).

Finally, we supplement our results by a few counterexamples.

We study class semigroups in the setting of seminormal weakly Krull monoids. The main results apply to the monoid of invertible ideals in seminormal orders of algebraic number fields and to seminormal orders with trivial Picard group.

Let \(H\) be a Krull monoid, let \(G\) be its divisor class group, and let \(G_0\subset G\) be the classes containing prime divisors. For simplicity, assume \(H\) is reduced. It is well known that each nonunit \(x\in H\) has only finitely many factorizations into irreducibles. If \(x=a_1\cdots a_n\) is a factorization of \(x\) into irreducibles, the length of this factorization is \(n\). One can collect all lengths of factorizations of \(x\) into irreducibles and obtain the well-studied set of lengths of \(x\), \(\mathcal{L}(x)\). One can also take a more detailed account of factorizations by considering how often factorization lengths occur. The length multiplicity set of \(x\), denoted \(\mathcal{LM}(x)\) is a subset of \(\mathbb{N}_0^2\) where \((n, k)\in\mathcal{LM}(x)\) if and only if \(x\) factors as a product of \(n\) irreducibles in exactly \(k\) different ways.

Kainrath has shown that if the Krull monoid \(H\) has infinite class group \(G\) and \(G_0=G\), then for any finite subset \(S\) of \(\mathbb{N}_{\geq 2}\), there is an \(x\in H\) with \(\mathcal{L}(x)=S\). Moreover, excluding a small class of groups \(G\), one can realize every possible finite length multiplicity set \(S\subset \mathbb{N}_{\geq 2}\times \mathbb{N}_{\geq 1}\). Kainrath's proof was nonconstructive. In this talk we will give the background on Kainrath's result and illustrate a constructive proof for \(G=\mathbb{Z}\). We will also discuss recent work to extending Kainrath's result to Krull monoids with \(G=\mathbb{Z}\) but \(G_0\) a proper symmetric subset of \(\mathbb{Z}\). We finish with a brief discussion of the analogous problem over finite class groups.

Let \(n\geq 2\) be an integer. For \(x\) an integer or residue class modulo \(n\), we use \((x)_n\in [0,n-1]\) to denote the least non-negative integer representative for \(x\) modulo \(n\). Then, given \(x_1,\ldots,x_\ell\in \mathbb Z\), we set \[||(x_1,\ldots,x_\ell)||_1=\min \{(ux_1)_n+\ldots+(ux_\ell)_n:\; u\in \mathbb Z,\; \gcd(n,u)=1\}.\] The goal is to find tight upper bounds for the projective ``norm'' \(||(x_1,\ldots,x_\ell)||_1\) assuming \(\sum_{i\in I}x_i\not\equiv 0\mod n\) for all nonempty \(I\subseteq [1,\ell]\), i.e., assuming the sequence \((x_1,\ldots,x_\ell)\) contains no non-empty zero-sum subsequence. The presence of such zero-sum subsequences causes the ``norm'' to become large for degenerate reasons, so the goal is to show this is the only way it can be so large. The first case of interest is \(\ell=3\), in which case it conjectured that \(||(x_1,x_2,x_3)||_1<n\) for a zero-sum free seqence of terms \(x_1,x_2,x_2\in \mathbb Z/n\mathbb Z\).

I will discuss the problem and its various equivalent formulations in detail and review the main cases for which the conjecture is known. The rest of the lecture will be spent describing a new perspective that yields simpler proofs of the known cases and how the prime power case be used to achieve a characterization result for certain subspaces in central simple algebras.

The polynomial method is a successful and promising approach for solving combinatorial problems. We will discuss this method via a theorem of Alon and Furedi and offer a new (and short) proof of a number-theoretic theorem of Ewald Warning from 1935, which concerns the number of zeros of a polynomial system over a finite field. We also offer a broad generalization of Warning's theorem. Further, we will discuss applications of this generalization to various zero-sum problems in additive combinatorics.

Conway and Coxeter introduced frieze pattern of positive numbers, these are arrays of positive integers arranged in a finite number of bi-infinite rows satisfying some relation. They showed that frieze pattern of positive numbers are closely related to triangulations of polygons and therefore to cluster algebras of type \(A\). In this talk, we introduce periodic infinite friezes where some of them have a geometric interpretation as triangulations of the disc with finitely many marked points on the boundary and one puncture in the interior. Moreover, for this special family of periodic infinite friezes we give an interpretation of the positive integers via matchings for periodic triangulations of the stripe. Finally we discuss a way how an infinite frieze can be modified to build ``bigger'' or ``smaller'' periodic infinite friezes and give the corresponding geometric interpretation of these two operations.

Let \(K\) be an algebraic number field with ring of integers \(\mathcal O_K\). An order \(\mathcal O\) in \(K\) is a subring of \(\mathcal O_K\) containing a \(\mathbb Q\)-basis for the field \(K\). The conductor \(\mathfrak f\) of \(\mathcal O\) is the largest ideal of \(\mathcal O_K\) which is contained in \(\mathcal O\). We investigate orders in pure cubic fields \(K\), i.e. \(K=\mathbb Q(\sqrt[3]{m})\) where \(m \in \mathbb Q\) is not a cube of a rational number. For any such a field \(K\), we will determine the number of all orders with given conductor \(\mathfrak f\), and describe how all these orders can be obtained.

Joint work with N.R. Baeth, A. Geroldinger, and D.J. Grynkiewicz.

Let \(R\) be a ring and let \(\mathcal C\) be a class of \(R\)-modules. Suppose there exists a set of representatives \(\mathcal V(\mathcal C)\) of isomorphism classes of \(\mathcal C\). Then the direct sum operation induces the structure of a commutative semigroup on \(\mathcal V(\mathcal C)\) by means of \([M] + [N] = [M \oplus N]\). The semigroup \(\mathcal V(\mathcal C)\) carries all information about direct sum decompositions in \(\mathcal C\), and hence the study of direct sum decompositions in \(\mathcal C\) can be reduced to the study of the factorization theory of the semigroup \(\mathcal V(\mathcal C)\).

If \(R\) is a one-dimensional reduced commutative Noetherian local ring, and \(\mathcal C\) denotes the class of all finitely generated \(R\)-modules, then \(\mathcal C\) is a Krull monoid with finitely generated class group \(G\), and factorization theoretical questions about \(\mathcal C\) can be studied as zero-sum problems in \(\mathcal B(G_P)\) with \(G_P \subset G\) the set of classes containing prime divisors.

On the other hand, Steinitz's Theorem implies that if \(R\) is a commutative Dedekind domain with non-trivial class group, and \(\mathcal C_{\text{proj}}\) is the class of finitely generated projective modules, then \(\mathcal V(\mathcal C_{\text{proj}})\) is not a Krull monoid, but a finitely primary monoid of rank \(1\) and exponent \(1\). Based on module-theoretic work of Lam, Levy, Robson, similar descriptions of \(\mathcal V(\mathcal C_{\text{proj}})\) are obtained for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings, and we study their arithmetic.

Joint work with N.R. Baeth, A. Geroldinger, and D.J. Grynkiewicz.

We study the Davenport constant \(\mathsf D (G_0)\) of subsets \(G_0\) of a finitely generated free abelian group \(G\). If \(G_0\) generates \(G\), then \(\mathsf D (G_0)\) is finite if and only if \(G_0\) is finite.

A main result is as follows. If \(G_r^+\) denotes the set of nonzero vertices of the \(r\)-dimensional hypercube in \(\mathbb Z^r\) and \(G_0 = G_r^+ \cup -G_r^+\), then \[\mathsf F_{r-2} \le \mathsf D(G_0) \le (r+2)^{\frac{r+2}{2}},\] where \(\mathsf F_{n}\) denotes the \(n\)-th Fibonacci number.

The lower bound is obtained by means of an inductive construction. The upper bound improves on a general upper bound of Diaconis, Graham, and Sturmfels. It is obtained through an auxiliary invariant, the elementary Davenport constant \(\mathsf D^{\textsf{elm}}(G_0)\), which is studied with methods from linear algebra over \(\mathbb Z\).

This research is motivated by questions about direct-sum decompositions of modules, and we will address these applications in a forthcoming talk.

In a noetherian ring, a non zero-divisor can be written as a product of finitely many atoms, but usually not uniquely so. In factorization theory, we study such non-unique factorizations, often by means of arithmetical invariants. In this thesis the machinery from Krull monoids is extended to a noncommutative setting, with the main application being to classical maximal orders in central simple algebras over global fields. This splits into two cases. In the main case, when every stably free left ideal is free, we obtain a transfer homomorphism to a monoid of zero-sum sequences over a ray class group. Factorizations in such monoids have been investigated in combinatorial number theory. By means of the transfer homomorphism, results on system of sets of lengths and catenary degrees carry over to classical maximal orders. In a further case it is impossible to construct such a transfer homomorphism, and some central arithmetical invariants are substantially different from the main case.

An integral domain is called an SP-domain if each of its ideals is a finite product of radical ideals. The main topic of this talk is to study certain properties (like being a Bézout domain) in the context of SP-domains. We prove that an integral domain \(R\) is a Bézout SP-domain if and only if \(\dim(R)\leq 1\), and the radical of every principal ideal of \(R\) is principal. Moreover, we study SP-domains with nonzero Jacobson radical, and investigate whether such domains are already Bézout domains. This problem is still unsolved, but we discuss recent progress towards a solution. For example, we show that every overring of an SP-domain with nonzero Jacobson radical is a ring of quotients. There are basically two types of constructions known that lead to SP-domains that are no Dedekind domains. We outline that they cannot be used to gain examples of non-Bézout SP-domains with nonzero Jacobson radical.

We introduce the idea of transfer of gradings between blocks of group algebras via stable equivalences. To do this we show how to grade Ext-spaces. We demonstrate this construction on several interesting examples.

Let \(\mathfrak g\) be a Lie algebra over a field \(K\). We consider the invariant \(\mu(\mathfrak g)\), given by the minimal dimension of a faithful linear representation of \(\mathfrak g\). By Ado's and Iwasawa's theorem this invariant is finite. We give general upper and lower bounds for \(\mu(\mathfrak g)\) in terms of the dimension of \(\mathfrak g\) and other invariants, and present constructions of faithful Lie algebra representations of small degree using quotients of the universal enveloping algebra \(U(\mathfrak g)\) of \(\mathfrak g\). For reductive Lie algebras we can prove an explicit formula for \(\mu(\mathfrak g)\). This gives new insights into Dynkin's classification of maximal reductive Lie subalgebras in semisimple Lie algebras. Finally we give applications of good upper bounds for \(\mu(\mathfrak g)\) for left-invariant affine structures on Lie groups and affine crystallographic groups.

Assem, Dupont and Schiffler introduced the category of rooted cluster algebras, which has as objects pairs \((\mathcal A, \Sigma)\), where \(\mathcal A\) is a cluster algebra (of possibly infinite rank) and \(\Sigma\) a distinguished initial seed of \(\mathcal A\). We show that, though the category of rooted cluster algebras does not in general admit colimits, every rooted cluster algebra can be written as a colimit of rooted cluster algebras of finite rank. Colimits of rooted cluster algebras of Dynkin type \(A\) have a geometric interpretation as triangulations of the disc with infinitely many marked points. They correspond to infinite discrete cluster categories of type \(A\) or the continuous cluster category of type \(A\) as introduced by Igusa and Todorov.

We investigate the structure of generators of principal ideals in quadratic orders and relate them to algorithms of Lagrange, Mollin and Matthews for the representation of integers by indefinite binary quadratic forms.

I'll discuss results of joint work with R. Marsh regarding the Berenstein-Zelevinsky twist automorphism of the type A grassmannian; notably a combinatorial formula to calculate BZ-twisted plücker coordinates using perfect matchings on a class of bipartite graphs dual to Postnikov diagrams.

Calabi-Yau triangulated categories appear in many branches of mathematics and physics, for example as cluster categories in representation theory. Much work has been done on understanding triangulated categories of positive CY dimension, particularly those which are 2-CY or 3-CY.

Thus far, little is understood about triangulated categories of negative CY dimension. Examples of such categories arise out of the work of Riedtmann on the classification of selfinjective algebras and were one of the original motivations in the development of cluster-tilting theory. In this setting Hom-configurations are the natural objects of study, and their behaviour in a certain orbit category \(\mathcal C\) of the derived category with negative CY dimension is highly reminiscent of that of cluster-tilting objects.

In this talk, we consider a generalization of Hom-configurations in triangulated categories generated by spherical objects with negative CY dimension. We will give a combinatorial classification of these configurations and explain links with noncrossing partitions. Along the way, we obtain a geometric model for the higher versions of the orbit category \(\mathcal C\).

Let \(P\subseteq G\) be a standard parabolic subgroup of an algebraic group \(G\). We would like to study and understand the \(P\)-orbits in the corresponding nilradical \(\mathfrak n\).

In general, the structure of \(P\)-orbits in \(\mathfrak n\) is very complex and difficult to understand. Since we are more familiar with the concept of modules, we would like to translate this problem of orbit structure to the theory of modules and study the corresponding modules instead.

In this talk we will present two algebras \(\mathcal A_n\) and \(\mathcal D_n\) and show that \(P\)-orbits for \(P\subseteq GL_N\) and \(P\subseteq SO_N\) are in a bijective correspondence with certain classes of \(\Delta\)-filtered \(\mathcal A_n\)- and \(\mathcal D_n\)-modules.

The edge expansion (or isoperimetric number) of a finite graph is a well-known and studied object with diverse applications, the analog of Cheeger constant in Riemannian Geometry. It is defined to be the minimum of the ratio of the edge boundary by the (vertex) cardinality of a subgraph, where the minimum is taken over all finite subgraphs of cardinality at most half the cardinality of the whole graph. It was thoroughly investigated, in particular with regard to expander graphs, by algebraic means like the spectrum of the Laplace operator or by Kazhdan constant (for Cayley graphs), by probabilistic means like random walks, and more.

In the case of infinite graphs which are Cayley graphs of finitely generated infinite groups, the asymptotic invariant which was mainly studied was the isoperimetric profile of amenable groups. The edge expansion, i.e. the infimum, over all finite subgraphs, of the ratio of the edge boundary by the cardinality of the subgraph, was studied for specific groups.

In the talk we will describe our work of obtaining formulas and bounds for the edge expansion constant of infinite Cayley graphs with respect to basic constructions of the underlying groups.

Let \(K\) be an algebraic number field with ring of integers \(\mathcal{O}_K\). An order \(\mathcal{O}\) in \(K\) is a subring of \(\mathcal{O}_K\) containing a \(\mathbb{Q}\)-basis for the field \(K\). The conductor \(\frak{f}\) of \(\mathcal{O}\) is the largest ideal of \(\mathcal{O}_K\) which is contained in \(\mathcal{O}\). We investigate orders in purely cubic fields \(K\), i.e. \(K=\mathbb{Q}(\sqrt[3]{m})\) where \(m\in\mathbb{Q}\) is not a cube of a rational number. For any such a field \(K\), we will determine the number of all orders with given conductor \(\frak{f}\), where the norm of \(\frak{f}\) is a power of a rational prime, and describe how all these orders can be obtained.

Let \(f: N \rightarrow \{-1,1\}\) be a multiplicative function. We study questions on additive structures of the following type:

Assuming certain necessary hypotheses, and for any sign pattern \(\varepsilon_i\) there are (large) sets \(A\) and \(B\) such that for all \(a\in A, b\in B\) \[f(a)=\varepsilon_1, f(b)=\varepsilon_2, f(a+b)=\varepsilon_3.\] Also, (assuming necessary hypotheses), there are infinitely many \(n,d\) such that \[(f(n), f(n+d), f(n+2d), f(n+3d) )= (\varepsilon_1, \ldots , \varepsilon_4).\]

In additive combinatorics, one of the most basic and central problem is the following: estimate the minimal size of a sumset \(A+B\) of two sets with respect to the size of \(A\) and \(B\), two subsets of a given semigroup. To go a step further, one usually asks to improve the bound obtained when some families of pairs \((A,B)\) defined as precisely as possible are excluded. Pairs giving a small size for \(A+B\) lead to pairs of analogous sets, typically \(A\) and \(B\) are close to arithmetic progressions with the same difference. The sums of dilates problem studies the case where \(B\) is the set obtained from \(A\) after applying a dilation, \(B=t.A\), a typical middle case since \(B\) resembles \(A\) in some sense, but is also quite different from the point of view of arithmetic progressions.

In a recent work (2012) of Shigeki Akiyama and Attila Pethő the distribution of real polynomials with bounded roots is investigated. For given non-negative integers \(d\) and \(s\) the set \(v(d,s)\) of coefficient vectors of contractive polynomials of degree \(d\) having \(2s\) non-real zeros is considered and it is shown that these sets have rational Lebesgue measure. Furthermore it is conjectured that the quotient \(v(d,s)/v(d,0)\) is an integer for all non-negative integers \(d\) and \(s\). In this talk we will give a short introduction to the topic and then prove this conjecture for \(s = 1\) by a method which might be adaptable to the general case.

Representation theory is a part of mathematics that enables us to study abstract mathematical objects (such as groups, rings, Lie algebras etc.) by representing their elements as linear transformations of vector spaces. A representation makes an abstract object more concrete because matrices are more familiar objects. The aim of the talk is to present some basic ideas and open problems in representation theory of finite groups. The emphasis will be on the representations of symmetric groups and their connections with quantum groups.

The subword complexity of a word \(w\) over a finite alphabet \(\mathcal A\) is a function that assigns for each positive integer \(n\), the number of distinct subwords of length \(n\) in \(w\). The subword complexity of a word is a good measure of the randomness of the word and gives insight to what the word itself looks like. In this talk, we discuss the properties of subword complexity sequences, and consider different variables that influence their shape. We also compute the number of distinct subword complexity sequences for certain lengths of words over different alphabets, and state some conjectures about the growth of these numbers.

In this talk we discuss the ring of integer-valued polynomials and its ideals with a special focus on Skolem properties. For a domain \(D\) with field of fractions \(K\), the ring of integer-valued polynomials consists of all polynomials with coefficients in \(K\) which map \(D\) into \(D\), that is, \(\operatorname{Int}(D) = \{f \in K[X] \mid f(D) \subseteq D\}\).

Skolem properties characterize to what extent finitely generated ideals of \(\operatorname{Int}(D)\) are described by their ideals of values at elements of \(D\), that is, the ideals \(\mathcal{A}(a)=\{f(a)\mid f\in \mathcal{A}\}\) for an ideal \(\mathcal{A}\) of \(\operatorname{Int}(D)\) and elements \(a \in D\).

If all finitely generated ideals of \(\operatorname{Int}(D)\) which contain non-zero constants (so-called unitary ideals) are determined by their ideals of values at all elements \(a \in D\), the domain \(D\) is called an almost strong Skolem ring.

A one-dimensional, Noetherian, local domain \(D\) with finite residue field was previously known to be an almost strong Skolem ring if it is analytically irreducible (the \(\mathfrak{m}\)-adic completion of \(D\) is a domain where \(\mathfrak{m}\) is the maximal ideal of \(D\)). It was unknown whether this condition is necessary. In a joint work with Cahen we show that it is at least necessary for \(D\) to be unibranched (the integral closure of \(D\) is local).

In this talk we discuss special types of monoids which can be described via their monadic submonoids (i.e. their divisor-closed submonoids generated by one element). In particular, we introduce monoids whose monadic submonoids are Krull monoids (so called monadically Krull monoids). We present a characterization result for monadically Krull monoids and show that monadically Krull monoids can fail to be Krull by constructing various types of counterexamples. On the other hand we investigate rings of integer-valued polynomials and study their connections with monadically Krull monoids. Especially, we prove that the ring of integer-valued polynomials over a factorial domain is monadically Krull.

A commutative ring is called a Marot ring if every regular ideal is generated by its regular elements. First we discuss the theory of divisorial ideals in Marot rings, and then we study Marot C-rings (i.e., Marot rings whose monoid of regular elements is a C-monoid). We present the following theorem: Let \(A\) be a Marot Mori ring, \(R = \widehat{A}\) its complete integral closure and suppose that the conductor \(\mathfrak{f} = (A:R)\) contains a regular element. If the \(v\)-class group \(\mathcal{C}(R^\bullet)\) and \(R / \mathfrak{f}\) are finite, then \(A\) is a Marot C-ring. This was known for Mori domains.

The main character of this talk is the decorated mapping class group \(\operatorname{MCGp}(S, M)\) of a marked surface \((S, M)\). We show that (in most cases) it is isomorphic to a group formed by automorphisms of the cluster algebra associated to \((S, M)\), which can also be interpreted as a group of auto-equivalences of the corresponding cluster category \(\mathcal C(S, M)\). Moreover we describe the suspension functor of \(\mathcal C(S, M)\) geometrically, as an element in the decorated mapping class group of \((S, M)\).

We study dimer models arising from alternating strand diagrams (Postnikov, 2006). To such a diagram \(D\) we associate a quiver \(Q(D)\) with a potential \(W_D\) (giving rise to relations of the algebra of \(Q(D)\)). The quiver \(Q(D)\) embedded into a disk can be viewed as a dimer model with boundary. We show that the algebra associated to the Grassmannian by Jensen-King-Su can be realized as an idempotent subalgebra of the algebra of \(Q(D)\) (under the relations coming from \(W_D\)).

This is joint work with A. King (Bath) and R. Marsh (Leeds).

Let \(\Phi\) be a root system of finite type and let \(\mathcal{A}\) be the corresponding cluster algebra of finite type. Companion bases are associated with the exchange matrices of \(\mathcal{A}\). In particular, a companion basis is a \(\mathbb{Z}\)-basis of roots for the integral root lattice \(\mathbb{Z} \Phi\), whose associated matrix of inner products is a positive quasi-Cartan companion of an exchange matrix.

The initial motivation for the study of companion bases came from the work of Barot, Geiss and Zelevinsky on the recognition of cluster algebras of finite type. After first recalling the definition of a cluster algebra and the classification result of the cluster algebras of finite type (both due to Fomin and Zelevinsky), we focus on the key results of this work and how they led to the definition of companion bases.

In preparation for our examination of companion bases, we briefly introduce root systems and outline some of their properties. Our main results on companion bases include:

• A description of the relationship between different companion bases for the same exchange matrix.

• A companion basis mutation procedure that, given a companion basis for an exchange matrix, produces a companion basis for any mutation of that matrix.

• An explicit procedure for constructing a companion basis for any exchange matrix of a cluster algebra of finite type.

In addition, we establish that in Dynkin type \(A\), expressing the positive roots in terms of a companion basis yields the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra. This generalises part of Gabriel's Theorem.

Recently, Loper and Werner investigated a problem related to a question raised by Brizolis, namely, finding Prüfer domains properly contained between $\mathbb Z[X]$ and ${\rm Int}(\mathbb Z)$, the ring of integer-valued polynomials over $\mathbb Z$. Given a positive integer $n$, we denote by $\mathcal{A}_n$ the set of all the algebraic integers over $\mathbb Z$ of degree bounded by $n$. We consider the ring ${\rm Int}_{\mathbb Q}(\mathcal{A}_n)$ of polynomials with rational coefficients which are integer-valued over $\mathcal{A}_n$, that is, $f\in\mathbb Q[X]$ such that $f(\mathcal{A}_n)\subset \mathcal{A}_n$. Loper and Werner proved that this ring is a Prüfer domain. Obviously, ${\rm Int}_{\mathbb Q}(\mathcal{A}_n)\subset {\rm Int}_{\mathbb Q}(\mathcal{A}_{n-1})$ and ${\rm Int}_{\mathbb Q}(\mathcal{A}_1)={\rm Int}(\mathbb Z)$.

We show here that if $f\in\mathbb Q[X]$ is integer-valued over the set $A_n$ of the algebraic integers of degree equal to $n$, then $f(X)$ is integer-valued over $\mathcal{A}_n$, that is $${\rm Int}_{\mathbb Q}(\mathcal{A}_n)={\rm Int}_{\mathbb Q}(A_n,\mathcal{A}_n)\doteqdot\{f\in\mathbb Q[X]\,|\,f(A_n)\subset \mathcal{A}_n\}.$$ We use the fact that the integral closure of the ring ${\rm Int}(M_n(\mathbb Z))$ of integer-valued polynomials over the algebra of $n\times n$ integer matrices is equal to ${\rm Int}_{\mathbb Q}(A_n,\mathcal{A}_n)$.

Similarly, given a number field $K$ of degree $n$ over $\mathbb Q$, we show that the set $O_{K,n}$ of algebraic integers of $K$ of degree $n$ is polynomially dense in the ring of integers $O_K$, that is, given $f\in K[X]$ such that $f(O_{K,n})\subset O_K$, it follows that $f(O_K)\subset O_K$. We prove this result by means of a criterion of Gilmer which completely characterizes such polynomially dense subsets.

[1]

R. Gilmer. Sets that determine integer-valued polynomials, J. Number Theory 33 (1989), no.1, 95-100.

[2]

K. Alan Loper, Nicholas J. Werner. Generalized Rings of Integer-valued Polynomials, J. of Number Theory 132 (2012), 2481-2490.

[3]

G. Peruginelli. Integral-valued polynomials over the set of algebraic integers of bounded degree, arXiv:1301.2045.

Joint work with Nicholas Baeth (University of Central Missouri).

Some classical invariants of the theory of non-unique factorizations, in particular those defined in terms of lengths, immediately generalize to the non-commutative setting. For others, like the definition of a factorization itself, the notion of a distance between factorizations and invariants derived from these, in particular the catenary and tame degree, it is less obvious how to generalize them to a non-commutative setting in a meaningful way.

I will talk about some work in progress in this direction, and discuss the proposed invariants for some examples (upper triangular matrices over an integral domain, maximal orders in central simple algebras over a number field and some abstract semigroups).

Let \(G\) be a finite abelian group. One has says that a sequence \(g_1\dots g_n\) has a plus-minus weighted zero-subsum if there exists a non-empty subset \(I \subset \{1, \dots , n \}\) and \(\epsilon_i \in \{1,-1\}\) such that \(\sum_{i \in I} \epsilon_i g_i = 0\). The plus-minus weighted Davenport constant of \(G\), denoted \(\mathsf{D}_{\pm}(G)\), denotes the smallest integer such that each sequence of that length has a plus-minus weighted zero-subsum. Recently Adhikari–Grynkiewicz–Sun gave bounds that allow to estimate the value of this constant up to an error bounded by the rank of \(G\), and for certain types of groups the bounds even yield the exact value.

We present some additional results on the constant \(\mathsf{D}_{\pm}(G)\). Among others we present a construction yielding another type of lower bound, which improves the existing ones for certain types of groups, while being worse for others.

Time permitting related problems will be discussed as well.

A graph \(X\) is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices \((u,v)\) and \((u',v')\), there is an automorphism \(f\) of \(X\), such that \(f(u)=u'\) and \(f(v)=v'\). A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. The most famous example of a symmetric biciruculant is the Petersen graph, which is cubic. All cubic and tetravalent symmetric bicirculants are know. In this talk I will present a classification of pentavalent symmetric bicirculants.

This is a joint work with Iva Antončič and Klavdija Kutnar.

Let \(G\) be a finite group written multiplicatively. By a sequence over \(G\), we mean a finite sequence of terms from \(G\) which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of \(G\). The (large) Davenport constant \(\mathsf D (G)\) is the maximal length of a minimal product-one sequence. This is a product-one sequence whose terms cannot be factored (i.e., partitioned) into two nonempty product-one subsequences. When \(G\) is abelian, \(\mathsf D(G)\) has been well-studied and plays an important role in other areas of mathematics (for instance, when studying non-unique factorization questions over Krull monoids), though its estimation remains one of the most challenging problems in Combinatorial Number Theory. In this talk, we will focus on a (relatively) new direction with regards to \(\mathsf D(G)\). Namely, we will talk about what is known for non-abelian groups \(G\). We will focus on certain non-abelian groups for which \(\mathsf D(G)\) can be determined exactly, including those with a cyclic index \(2\) subgroup and the non-abelian group of order \(pq\) (with \(p\) and \(q\) prime), on some general upper bounds, and on some potential connections with aspects of Invariant Theory.

Let \(u\) be a unipotent element in a simple algebraic group \(G\) defined over an algebraically closed field \(k\) of characteristic \(p\). If \(p\) is good for \(G\) the connected component of the double centralizer \(Z(u):=C_G C_G(u) = Z(C_G(u))\) is a canonically defined connected abelian unipotent overgroup for \(u\). We give a characteristic independent description of \(Z(u)^{\circ}\) which can be used to calculate this group explicitly if both the rank of \(G\) and \(p\) are small. The obtained method allowed for the determination of \(Z(u)^{\circ}\) in the case of exceptional \(G\) in bad characteristic.

Over the past fifteen years, the study of direct-sum decompositions of modules over certain classes of rings has been viewed through the lens of factorization theory in commutative monoids. That is, when \(\mathcal C\) is a class of modules over a ring \(S\) closed under isomorphism, direct sums and direct summands, all information about direct-sum decompositions of modules in \(\mathcal C\) is contained in the structure of the monoid \(H\) whose elements are isomorphism classes \([M]\) of modules in \(\mathcal C\) with operation given by \([M]+[N]= [M\oplus N]\). In particular, when \((R,\mathfrak m)\) is a commutative Noetherian local ring with unique maximal ideal \(\mathfrak m\) and with \(\mathfrak m\)-adic completion \(\widehat R\), the map from the monoid \(M(R)\) of isomorphism classes of finitely generated \(R\)-modules to the monoid \(M(\widehat R)\) of finitely generated \(\widehat R\)-modules given by \([M] \mapsto [M\otimes_R \widehat R]\) is a divisor homomorphism. In this talk I will restrict to the submonoid \(T(R)\) of isomorphism classes of finitely generated torsion-free \(R\)-modules and will summarize what is know about \(T(R)\) when \(R\) is a one-dimensional analytically unramified local ring and when \(R\) is a two-dimensional normal domain. We will discuss when \([M] \mapsto [M\otimes_R \widehat R]\) is a divisor theory, computations of the divisor class group of \(T(R)\), the set of classes containing prime divisors, and what is known about the arithmetic of this monoid.

In recent work S. Fomin and P. Pylyavskyy showed that the ring of \(\mathrm{SL}(V)\) invariants of configurations of vectors and linear forms have cluster algebra structures whenever \(V\) is a three dimensional complex vector space. In this talk we discuss how a class of tensor diagrams can be expressed as (normalized) Chebyshev polynomials.

Joint work with Aslak Bakke Buan (NTNU, Trondheim).

We consider a triangulated category \(\mathcal C\) under mild additional assumptions, together with a rigid object \(T\) in \(\mathcal C\) (i.e. an object with no self-extensions). We show that the category \(\mathrm{End}(T)\)-mod of finite dimensional modules over the endomorphism algebra of \(T\) can be realised by localizing \(\mathcal C\) at a collection of morphisms determined by \(T\) - i.e. by formally inverting these morphisms.

If, in addition, \(\mathcal C\) has Serre duality, then \(\mathrm{End}(T)\)-mod can be obtained by localization of a preabelian quotient of \(\mathcal C\) at the class of maps which are both monomorphisms and epimorphisms. In this case, the class of maps admits a calculus of left and right fractions.

The results will be illustrated with an example from cluster theory, using Auslander-Reiten quivers, which give a graphical depiction of the categories involved.

Torsion pairs in triangulated categories were introduced by Iyama and Yoshino in [IY08]. They were classified in the cluster category of $A_n$ by Holm, Jørgensen and Rubey in [HJR11] and in the cluster category of $A_{\infty}$ by Ng in [Ng] via Ptolemy diagrams. In [ZZ11], Zhou and Zhu defined the mutation of torsion pairs in triangulated categories, thus generalizing the notion of mutation of clusters in cluster categories. Furthermore, they provided a geometric model for the mutation of Ptolemy diagrams in the cluster categories of $A_n$ and $A_{\infty}$ which agrees with the mutation of torsion pairs on the categorical level. Very recent work by Holm, Jørgensen and Rubey classifies torsion pairs in the cluster category of type $D_n$ via Ptolemy diagrams of type D. If time permits, the talk will discuss a geometric realization of mutations of torsion pairs in cluster categories of type $D_n$.

[BMRRT06]

A. B. Buan, R. J. Marsh, M. Reineke, Reiten and G. Todorov. Tilting theory and cluster combinatorics, Adv. Math., 204: 572--618, 2006.

[HJR11]

T. Holm, P. Jørgensen, M. Rubey. Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type $A_n$. J. Algebraic Combin. 34 (2011), 507--523.

[IY08]

O.Iyama and Y. Yoshino. Mutations in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172 (2008), no.1,117--168.

[Ng]

P.Ng. A characterization of torsion theories in the cluster category of Dynkin type $A_{\infty}$. Preprint. arXiv:1005.4364

[ZZ11]

Y. Zhou, B. Zhu. Mutation of torsion pairs in triangulated categories and its geometric realization. Preprint arXiv:1105.3521v1 [math.RT], 2011.

In this talk we introduce the ideas of co-\(t\)-structures and co-stability conditions and compare and contrast with \(t\)-structures and stability conditions. We show that the space of co-stability conditions on a triangulated category forms a complex manifold, and give some examples. Part of this talk is joint work with Peter Jorgensen (Newcastle-upon-Tyne).

Arising in the context of cluster categories, the association of geometric models to triangulated categories is an important tool to understand the representation theory of these categories. In particular, these models allow us, in many cases, to classify certain classes of representation-theoretic objects, such as maximal rigid objects, which play a key role in (cluster-)tilting theory. In this talk, we consider another orbit category of the derived category of the path algebra of a quiver of Dynkin type \(A\), with respect to the autoequivalence \(\tau \Sigma^2\), where \(\tau\) is the Auslander-Reiten translate and \(\Sigma\) is the shift functor. Using work of Riedtmann on selfinjective algebras, we construct a geometric model of this category and use it to classify maximal rigid objects in terms of certain noncrossing bipartite graphs. Moreover, we describe the corresponding endomorphism algebras in terms of quivers with relations.

Lucas sequences are second-order linear recurrence sequences with characteristic polynomial \(X^2 - PX + Q\) and discriminant \(\Delta = P^2 - 4Q\). We assume that \(\Delta\) is not a square, use the quadratic order of discriminant \(\Delta\) to study divisibility and periodicity properties of the sequence, and we show connections with class numbers relations.

In this series of talks we study seminormal orders in algebraic number fields and their generalizations. We investigate several factorization theoretical invariants and concepts, like the catenary degree, the monotone catenary degree, the set of distances and special unions of sets of lengths (in the context of seminormal orders). Moreover, we discuss the connections between seminormal orders and block monoids resp. T-block monoids by using transfer-homomorphisms. We present a characterization of the half-factorial property for special types of seminormal weakly Krull domains. Furthermore, we prove that every intermediate ring of a half-factorial seminormal order and its integral closure is half-factorial. Finally, we discuss the ascent of the half-factorial property in a more general context.

Motivated by the theory of cluster algebras, S. Fomin and A. Zelevinsky have associated to each finite type root system a simple convex polytope called generalized associahedron. It turns out that this purely combinatorial gadget encodes many informations on the associated cluster algebra making it an interesting object to study.

I will describe, after recalling the basic definitions, a family of geometric realizations of these polytopes, parametrized by orientations of the corresponding Dynkin diagram. I will also show that this construction agrees with the one given by C. Hohlweg, C. Lange and H. Thomas in the setup of Cambrian fans developed by N. Reading and D. Speyer.

Let \(F\) be a field, \(G\) a finite group, and \(\rho\colon G \hookrightarrow GL(n, F)\) a faithful representation thereof. Via \(\rho\) the group \(G\) acts on \(V=F^n\), hence on the dual space \(V^*\), and thus on the symmetric algebra on the dual, \(F[V]\). The subring of \(G\)-invariants is denoted by \(F[V]^G\). In this talk we study the question how to construct this ring. A classical method to construct invariants is by orbit Chern classes: For a linear form \(l\in V^*\), its orbit \(o[l]\) consists of finitely many linear forms. The \(i\)-th orbit Chern class of \(l\) is then the \(i\)-th elementary symmetric function in the orbit elements of \(l\). In the classical (characteristic zero) case these Chern classes are enough to generate \(F[V]^G\). We explain this result, and why its proof cannot be generalized to finite characteristic. Then we present what is known in the case that the characteristic of \(F\) is positive.

We define a quasi-hereditary algebra \(\mathcal A_n\) via quiver and relations and we consider a class of \(\Delta\)-filtered \(\mathcal A_n\)-modules. A result from Brüstle and Hille shows that the class of these modules is in a bijective correspondence to the nilpotent orbits for some parabolic group \(P\subset GL_n\).

Similarly, we consider a quasi-hereditary algebra \(\mathcal D_n\) (the skew group algebra of \(\mathcal A_n\)), obtained from the double cover of the above quiver. Our motivation is now to find similar bijection between nilpotent orbits for \(P \subset SO_n\) and a class of \(\Delta\)-filtered \(\mathcal D_n\)-modules.

In this talk we are going to explain the problem and present some approaches toward our main goal.

In mathematics frieze pattern arose from Coxeter's study of metric properties of polytopes, inspired by arts and architecture. In this talk, we will see a relation between frieze patterns with positive integers and triangulated polygons. Furthermore, we will discuss the connection with cluster algebras of type \(A_n\).

In this talk we study the maximal dimension \(d\) such that an iterated sumset of the form \(a_0+\{0,a_1\}+ \cdots +\{0,a_d\}\) is contained in the set \(S\), where \(S\) is a multiplicatively defined set, such as the set of primes, squares, or sums of two squares.

Given a prime \(p\) and a sequence \(A=(a_1,...,a_l)\) of nonzero elements in \(\mathbb F_p\), we consider the set \(S_A\) of all \(0\)-\(1\) valued solutions to the equation \(a_1 x_1 + ... + a_l x_l = 0\). In more geometrical terms, this solution set is the intersection of a certain hyperplane of \((\mathbb F_p)^l\) with the \(l\)-cube. In the mid-eighties, Olson proved a lower bound conjectured by Erdős on the size of such a set. In this talk, I will describe how some standard tools from additive combinatorics, when combined with the polynomial method, lead us to discover more precise structural properties of \(S_A\). This is joint work with Eric Balandraud.

In additive combinatorics, the following general problem is central: let \(G\) be a group and \(S\) be a subset of \(G\). Let (E) be a linear equation of the type \(a_1 x_1 +.. + a_s x_s =0\) where the \(a_i\)'s are fixed integers and the \(x_i\)'s are the unknowns. What is the maximal cardinality of a subset \(X\) of \(S\) such that (E) has no solution if the unknowns \(x_i\) are restricted to belong to \(X\)? In this talk, we will review the state of knowledge for (some aspects of) this problem and some closely related ones.

An order \(\mathcal{O}\) in an algebraic number field \(K\) is a subring of \(K\), which is also a finitely generated \(\mathbb{Z}\)-module that contains a basis of \(K\). Let \(\frak{a}\) be a nonzero ideal of \(\mathcal{O}_K\). Then the ring \(\mathbb{Z}+\frak{a}\) is an order in \(\mathcal{O}_K\). One can show that \(\mathbb{Z}+\frak{f}\) is the smallest order in \(\mathcal{O}_K\) containing conductor \(\frak{f}\). If \(\frak{p}\) is a prime ideal in \(\mathcal{O}_K\) with \(\mathcal{N}(\frak{p})=p^f\) then there exist \(\tau(f)-1\) different orders in \(\mathcal{O}_K\) with conductor \(\frak{p}\). Let \(p\in\mathbb{Z}\) be a prime that splits into a product of prime ideals of \(\mathcal{O}_K\) as \((p)=\frak{p}_1^{e_1}\cdots\frak{p}_g^{e_g}\). We study and investigate conditions for the ideal \(\frak{f}=\frak{p}_1^{k_1}, \ldots,\frak{p}_g^{k_g}\) (\(k_1, \ldots,k_g\) are positive integers) in order that \(\frak{f}\) is a conductor. In this way we obtain a complete characterization of all ideals of \(\mathcal{O}_K\), which are conductors of some orders in \(K\).

Let \(H\) be a Krull monoid with finite class group \(G\), and let \(u \in H\) be an atom (an irreducible element). Then the local tame degree \(\mathsf t (H, u)\) is the smallest integer \(N \in \mathbb N_0\) with the following property: for any multiple \(a\) of \(u\) (so, for any \(a \in uH\)) and any factorization \(a = v_1 \cdot \ldots \cdot v_n\) of \(a\) into atoms, there is a short subproduct which is a multiple of \(u\), say \(v_1 \cdot \ldots \cdot v_m\), and a refactorization of this subproduct which contains \(u\), say \(v_1 \cdot \ldots \cdot v_m = u u_2 \cdot \ldots \cdot u_{\ell}\), such that \(\max \{\ell, m\} \le N\).

Thus the local tame degree \(\mathsf t (H, u)\) measures the distance between an arbitrary factorization of \(a\) and a factorization of \(a\) which contains the atom \(u\). By definition, the atom \(u\) is a prime element if and only if \(\mathsf t (H, u) = 0\). The (global) tame degree \(\mathsf t (H)\) is the supremum of the local tame degrees over all atoms \(u \in H\). Again we get that the monoid \(H\) is factorial if and only if \(\mathsf t (H) = 0\). Moreover, the finiteness of the class group easily implies that the finiteness of the tame degree.

We discuss upper and lower bounds for \(\mathsf t (H)\), and the relationship between \(\mathsf t (H)\) and the tame degree \(\mathsf t \big( \mathcal B (G) \big)\), where \(\mathcal B (G)\) is the monoid of zero-sum sequences over the class group \(G\).

This is joint work with W. Gao and Wolfgang A. Schmid.

The Davenport constant of a finite abelian group \(G\) is defined as the smallest integer \(\ell\) such that every sequence over \(G\) having at least \(\ell\) elements has a nontrivial zero-sum subsequence. This is a classical constant in Combinatorial Number Theory, and its study was one of the starting points of zero-sum theory in the 1960s. Another starting point is the classical zero-sum result of Erdős, Ginzburg and Ziv, which says that every sequence of length \(2n-1\) of elements in an abelian group of order \(n\) has a zero-sum subsequence of length \(n\). A constant, whose definition is suggested by this theorem, was later shown to be related to the Davenport constant by a theorem of Gao.

In the last few years a new type of so-called weighted zero-sum problems has attracted a lot of interest, and the present talk will be devoted to these new developments. Among others, we will discuss weighted versions of the Davenport constant, a weighted version of the above mentioned theorem of Gao, and of some related zero-sum constants.

Given an abelian group \(G\) and finite, nonempty subsets \(A,\,B\subseteq G\), we define their sumset to be \(A+B=\{a+b:\; a\in A\quad b\in B\}\). Given a sequence \(S\) of terms from \(G\) and an integer \(n\geq 0\), we let \(\Sigma_n(S)\) denote all elements that are expressible as the sum of terms in an \(n\)-term subsequence of \(S\). This series of talks is intended as a survey of some of the fundamental results concerning sumsets and subsequence sums. We will start with classical results, such as Kneser's Theorem and the Cauchy-Davenport Theorem, and then move on to more modern era achievements, including generalizations of Pollard's Theorem, the DeVos-Goddyn-Mohar Theorem, and the Partition Theorem. In the first lecture, we will focus on sumsets, moving on to subsequence sums in the subsequent lectures.

Let \(H\) be a commutative (multiplicative) monoid that possesses a zero element, such that every non-zero element of \(H\) is cancellative. We call \(H\) radical factorial if each principal ideal of \(H\) is a finite product of radical principal ideals of \(H\). If \(r\) is a finitary ideal system on \(H\), then \(H\) is called an \(r\)-SP-monoid if every \(r\)-ideal of \(H\) is a finite \(r\)-product of radical \(r\)-ideals of \(H\). In this talk we discuss the properties of these types of monoids and we specify the relations between them. In particular we show that every radical factorial monoid that is an \(r\)-Prüfer monoid or that has \(r\)-dimension one is already an \(r\)-SP-monoid. We will provide a characterization result for \(r\)-SP-monoids in the context of almost \(r\)-Dedekind monoids. Moreover we introduce monoids with many primary \(r\)-ideals, and show that \(r\)-SP-monoids that have many primary \(r\)-ideals are almost \(r\)-Dedekind monoids. Besides, we prove that polynomial rings over radical factorial GCD-domains are radical factorial.

Let \(R\) be a commutative ring with identity. It often happens that \(M_1 \oplus \cdots \oplus M_s \cong N_1 \oplus \cdots \oplus N_t\) for indecomposable \(R\)-modules \(M_1, \ldots , M_s\) and \(N_1, \ldots, N_t\) with \(s \not=t\). This behavior can be captured by studying the commutative monoid \(\{[M] \mid M\text{ is an $R$-module}\}\) of isomorphism classes of \(R\)-modules with operation given by \([M]+[N]=[M \oplus N]\). In the first talk we will discuss why it is reasonable to restrict to considering only local Noetherian rings and finitely generated modules over these rings. In particular we will show (1) that with these restrictions, proofs of unique direct-sum decomposition follow closely the proof of unique factorization in the ring of integers and (2) that if \(R\) is in addition complete, then all finitely generated modules decompose uniquely over \(R\). In the second talk we will discuss how the monoid of isomorphism classes of \(R\)-modules sits inside the monoid of isomorphism classes of \(\hat R\)-modules, where \(\hat R\) denotes the completion of \(R\) with respect to its unique maximal ideal. We will illustrate that this natural inclusion is a divisor homomorphism and use the monoid structure to answer questions about direct-sum decompositions.

If \(R\) is a maximal order in a central simple algebra \(A\) over a number field \(K\), and every stably free left \(R\)-ideal is free (a statement that is true unless \(A\) is a totally definite quaternion algebra), then there exists a transfer homomorphism from the semigroup of cancellative elements of \(R\) to a monoid of zero-sum sequences over a ray class group of \(K\), by means of which the sets of lengths and in particular the set of distances \(\Delta(R)\) can be described.

We show by means of an explicit construction: If there exist non-free but stably-free left \(R\)-ideals then there exists no transfer homomorphism to a monoid of zero-sum sequences over an abelian group, and we determine some arithmetic invariants, in particular \(\Delta(R) = \mathbb N\).

Shift Radix Systems (SRS) turned out to be the missing link between two generalizations of positional notation systems - Beta-Expansions and Canonical Number Systems (CNS) - which have been studied extensively during the last decade, but still leave behind many open questions and unsolved problems. In distinction from positional notation systems, where all integers greater than 1 can serve as a basis, in the general cases only the elements of complicated sets satisfy certain natural finiteness conditions. One of these sets is known as Pethö's Loudspeaker. An introduction to SRS is given and first results in relation to the characterization of Pethö's Loudspeaker are being presented.

In a ring \(R\) with non-unique factorization, the set of lengths of an element \(r\) is the set of natural numbers \(n\) such that \(r\) is representible as a product of \(n\) irreducible elements in \(R\). We show in a constructive way that every finite set of integers greater \(1\) occurs as the set of lengths of an element when \(R=\textrm{Int}(\mathbb Z)\) is the ring of integer-valued polynomials.

We discuss how division polynomials of the cosine are connected with Chebyshev polynomials and how these might be used for a unified treatment of the Diophantine problem mentioned in the title.

Cluster categories can be viewed as a categorification of cluster algebras. Cluster algebras have been introduced by Fomin and Zelevinsky around 2002. The motivations were to provide a tool for understanding phenomena of universal canonical bases and of total positivity. Cluster algebras are iteratively defined commutative rings with a distinguished set of generators (the so-called cluster variables) grouped into overlapping subsets of a fixed finite cardinality (the rank of the cluster algebra). Examples of cluster algebras are coordinate rings of algebraic varieties, e.g. homogeneous coordinate rings of Grassmannians.

Cluster categories were invented around 2005 by Buan-Marsh-Reineke-Ringel-Todorov (hereditary case) and Caldero-Chapoton-Schiffler (type \(A_n\)). They are orbits in bounded derived categories and have a structure of triangulated categories. In the Dynkin cases, the cluster categories have finitely many isomorphism classes of indecomposable objects. These are in bijection with the cluster variables of a cluster algebra of the same type.

This observation provides the link between cluster algebras and cluster categories. The theory of cluster algebras and categories has a lot of exciting connections and applications to other areas, such as quiver representations, Calabi-Yau categories, Teichmüller theory, discrete integrable systems, Poisson geometry, ...

We will explain the concepts of cluster algebras and of cluster categories and then go on to describe some recent results.

The term elasticity was introduced by Valenza in 1990, in the context of ring of integers \(R\) of a number field. The elasticity of an integral domain \(R\) is a function that in part measures the failure of \(R\) to be a unique factorization domain. Let \(h\) be the class number of a number field \(K\) with ring of integers \(R\). Carlitz proved that \(h \leq 2\) if and only if the elasticity of \(R\) is 1; this corresponds to say that given any non-unit \(x\) in \(R\) the number of irreducible factors of any factorization of \(x\) is constant (depending on \(x\)). Valenza discovered other relations between \(h\) and the elasticity of \(R\). After recalling the main definitions and known results, in this talk we will see how these notions apply in the non-noetherian case, in particular in the case of the ring of integer-valued polynomials. We will show a result of Chapman and McClain, that in particular implies that this ring has infinite elasticity.

Let \(H\) be a Krull monoid with finite class group \(G\) (e.g., the ring of integers of an algebraic number field). The Davenport constant \(\mathsf D(G)\) of \(G\) is the maximal length of a minimal zero-sum sequence over \(G\). A straightforward observation shows that a product of two irreducible elements (atoms) of \(H\) can be written as a product of \(\mathsf D(G)\) irreducible elements at most. We study this extremal case and consider the set \(\mathcal V_{\{2, \mathsf D(G)\}}(H)\) of all possible \(l \in \mathbb N\) such that there is an equation \[u_1u_2 = v_1 \cdot \ldots \cdot v_l = w_1 \cdot \ldots \cdot w_{\mathsf D(G)} \,,\] where all \(u_1, u_2, v_i, w_j\) are irreducible elements. This is the same as studying the set of all \(l \in \mathbb N\) for which there exist a minimal zero-sum sequence \(U\) over \(G\) of length \(|U| = \mathsf D(G)\) and minimal zero-sum sequences \(V_1, \ldots, V_l\) over \(G\) such that \[U (-U) = V_1 \cdot \ldots \cdot V_l \,.\] It turns out that in many cases this set \(\mathcal V_{\{2, \mathsf D(G)\}}(H)\) is characteristic for the group \(G\). The main result is based on the recent characterization of all minimal zero-sum sequences of length \(\mathsf D(G)\) over groups of rank two.

In this series of talks we introduce several weaker notions than that of a factorial monoid (or a Krull monoid) and discuss the connections between them. In particular we investigate well known classes of monoids like atomic monoids, completely integrally closed monoids, FF-monoids and idf-monoids. We define and study the notion of radical factorial monoids and elaborate its relationship with not so well known concepts like the idpf-property or the quasi idpf-property. We will show that every radical factorial monoid satisfies a (weak) monoid theoretical version of Krull's principal ideal theorem. Furthermore we prove that a monoid is factorial iff it is weakly factorial and radical factorial iff it is a radical factorial weakly Krull monoid whose \(v\)-class group is a torsion group. We replenish these results by a few counterexamples.

We study arithmetical properties of maximal orders in central simple algebras over number fields. We begin by recalling the concept of (one-sided) fractional divisorial ideals in non-commutative semigroups, and following a paper of Asano, study the factorization of integral elements in certain lattice-ordered Brandt-groupoids (the necessary concepts will be introduced). We apply this to the groupoid of normal ideals in a central simple algebra over a number field, and connect the factorization of regular elements into irreducible elements to the factorization of integral ideals. For the sets of lengths, we obtain a transfer principle to a block monoid over a ray class group unless the algebra is a totally definite quaternion algebra. Partial results for the totally definite quaternion algebras, where the set of distances may be infinite, will be discussed.

A classical problem in additive number theory is to understand the structure of a pair of finite sets \(A\), \(B\) from a multiplicative group with the property that \(|AB| < |A| + |B|\). This problem was solved for groups of prime order by Vosper, and for general abelian groups by Kempermann. Here we give a complete characterization for arbitrary groups. As a corollary, we obtain a generalization of Kneser's addition theorem to nonabelian groups.

Talks by:

• Pinzhi Yuan (South China N. Univ., Guangzhou)
  Title: Ko Chao's method and its applications to Diophantine equations

• Franz Halter-Koch (KFU Graz)
  Title: The Fibonacci sequence and class numbers

• Wolfgang A. Schmid (Ecole Polytechnique, Palaiseau Cedex)
  Title: Towards a more precise understanding of the structure of factorizations of algebraic integers via an associated (inverse) zero-sum problem

Organizers: A. Geroldinger and G. Lettl

Detailed program: as pdf. file here

We continue our series of lectures on Non-Commutative Krull rings. The focus of these two lectures will be on various types of class groups.

We continue our series of lectures on Non-Commutative Krull rings. Among others, we will deal with two-sided divisorial ideals and various types of class groups.

In the International Congress of Mathematics which took place in 1962 in Stockholm, Shafarevich proved that there are finitely many isomorphism classes of elliptic curves defined over a number field \(K\) having Good Reduction outside a prescribed finite set of finite places of \(K\).

In this talk I will introduce a recently new notion of good reduction introduced by Szpiro and Tucker for endomorphisms \(\varphi\) of the projective line defined over \(K\) at a finite place \(v\) of \(K\), called Critically Good Reduction. They proved that, given an integer \(n\geq2\), a number field \(K\) and a finite set \(S\) of finite places of \(K\), there are only finitely many equivalence classes of endomorphisms of the projective line of degree \(n\) defined over \(K\) which ramify at three or more points and having Critically Good Reduction at all places outside \(S\). They also showed how this theorem in particular implies Shafarevich's theorem.

Here we show how under the assumption of separability of the reduction map \(\varphi_v\), the Critically Good Reduction at \(v\) implies the Standard Good Reduction at \(v\), namely \(\varphi_v\) and \(\varphi\) have the same degree.

This is a joint work with J.-K. Canci and D. Tossici.

In a series of talks—given by several members of the research group—we are going to study maximal orders in central simple algebras over commutative Krull domains. We will partly follow the monograph by F. Van Oystaeyen and A. Verschoren on Relative Invariants of Rings. The first two talks gather some basic material on non-commutative noetherian rings, in particular on non-commutative Dedekind and Krull rings.

To understand the breathtaking formation of swarms by birds, F. Cucker and S. Smale developed in 2007 a mathematical model which has been of growing influence since then. In this model the interaction of birds is assumed to have a symmetric and global structure and a special kind of intensity. In the talk swarm formation is obtained under the following less restrictive assumptions:

• The structure of interaction need to be local only in that any two birds interact just with some third bird. Moreover, the structure of interaction may change in time admitting for different flight regimes.

• The intensity of interaction may be of any kind as long as it decreases not too fast in time.

• The proof for swarm formation under these assumptions uses tools from the theory of positive dynamical systems which will be explained in the talk.

We consider product-like binary operators on classes of finite graphs (with or without loops). Even if we want these operators to be associative and commutative and to distribute over the disjoint union operator, we still have the choice between the Cartesian, direct, and strong products. Obviously, this provides us with many new examples of commutative monoids, for which it would be interesting to determine the cancellation and factorisation properties. I will start by introducing the above-mentioned products, and also present the strong results that are already known this area. Most results, however, are restricted to connected graphs. If we consider the factorisation properties of possibly disconnected graphs, it turns out that the structure is isomorphic to the monoid of polynomials with nonnegative integer coefficients. I will show how the isomorphism works, and also present some new results which completely characterise the nonunique factorisation phenomena in this monoid for polynomials with at most 12 terms.

We introduce C-like monoids, give examples and study their arithmetic.

We shall present a survey of two famous classical conjectures concerning the characterization of topological (\(n > 2\))-manifolds: the Busemann Conjecture from the 1950's asserts that every Busemann \(G\)-space, is a topological manifold, whereas the Bing-Borsuk Conjecture from the 1960's asserts that every homogeneous absolute neighborhood retract (ANR), is a topological manifold. The key object in both cases are so-called generalized manifolds, i.e. Euclidean neighborhood retracts (ENR) which are also Z-homology manifolds. We shall look at their history, from the early beginnings to the present day, concentrating on those geometric properties of these spaces which are particular for dimensions 3 and 4, in comparison with generalized (\(n > 4\))-manifolds. In the second part of the talk we shall discuss the present state of these conjectures, concentrating on the work by V.Berestovskii, D.Halverson, P.Thurston and the speaker. We shall also list some interesting open problems and several related (and also unsettled) conjectures.

It is well known that Krull monoids with finite class group satisfy nice factorization properties. In particular such monoids are locally tame and they fulfill the structure theorem for sets of lengths. Many important monoids (e.g. non-principal orders in algebraic number filds) fail to be Krull monoids, since they are not completely integrally closed. This problem leads to the investigation of \(C\)-monoids (which are special submonoids of factorial monoids), seminormal Mori monoids (which are generalizations of Krull monoids) and \(C_0\)-monoids (which are special \(C\)-monoids). On the first talk on March 10th we will give a summary on well-known concepts (e.g. \(C\)-monoids, \(C_0\)-monoids, transferhomomorphisms and class groups). Moreover we provide essential results for the second talk. Especially we discuss Cohen-Kaplansky domains and integral domains whose group of divisibility is finitely generated. On the second talk we present recent results. We prove a characterization theorem for integral domains that are \(C_0\)-monoids (especially in the seminormal case and in the noetherian case). Beyond this we investigate the seminormal closure of a monoid and discuss some problems related to local tameness.

Uniquely presented monoids are those having a unique minimal presentation (up to rearrangement of the components of the relators). Since presentations are systems of generators of the kernel congruence of the factorization homomorphism, these monoids have peculiar factorization properties. A particular class of uniquely presented monoids are those having a generic presentation. This condition forces the tame and catenary degree to be the same. We will also review how to construct examples of uniquely presented monoids.

If S is a sequence of terms from an abelian group, then it is a classical question from Combinatorial Number Theory to study the set of elements representable as a sum of terms of some subsequence of S. In line with recent interest in the area, we consider a weighted variation by looking at what elements can be represented by summing all terms of S each multiplied by a different element taken from an arithmetic progression of the same length as S. We present a general lower bound sufficient to determine how long S must be to guarantee every group element can be represented as such a weighted subsequence sum.

In this talk, we will discuss a new finiteness property called “tautness”, which lies between half-factoriality and bounded factorization. We will give some sufficient conditions for a monoid to be taut and show that many familiar monoids possess this property. We will indicate how tautness can be used to infer elasticity properties about the monoid. More restricted versions of this property (notably “strong tautness”) have forced certain classes of monoids to be half-factorial, and imposed strong conditions on other classes. In one particular example (block monoids over certain infinite groups), the analysis induced from the point of view of strong tautness has pointed the way to arithmetic criteria for half-factoriality. As this is a new property, many open questions will be posed.

The maximal order \(\mathcal O_K\) of an algebraic number field is a Dedekind domain, and its arithmetic is completely determined by its Picard group \({\rm Pic}(\mathcal O_K)\). In particular, \(\mathcal O_K\) is factorial if and only if its Picard group is trivial. In contrast, non-principal orders are not integrally closed, hence they are never factorial, and their arithmetic depends not only on their Picard group but also on the localizations at singular primes. A non-principal order \(\mathcal O\) with \(|{\rm Pic}(\mathcal O)| \ge 3\) inherits many arithmetical properties from the maximal order. In contrast, only little is known about the arithmetic of non-principal orders whose Picard group has at most two elements, even if all localizations are half-factorial. In this case, we formulate a new semi-group theoretic approach based on monoids of relations. Using this machinery and common transfer principles, we are able to give a quite explicit description of various arithmetical invariants in various situations such as the elasticity \(\rho(\mathcal O)\), the set of distances \(\min\Delta(\mathcal O)\), the catenary degree \(\mathsf c(\mathcal O)\), the monotone catenary degree \(\mathsf c_{\mathrm{mon}}(\mathcal O)\), and the tame degree \(\mathsf t(\mathcal O)\). Under a weak assumption, we prove, for example, that \(\rho(\mathcal O)=\frac12\mathsf c(\mathcal O)\in\lbrace 1,\frac32,2\rbrace\) and \(\triangle(\mathcal O)\subset\lbrace 1,2\rbrace\).

Let \(\mathsf s(C_n^r)\) denote the minimal number such that every sequence of \(\mathsf s(C_n^r)\) elements in \(C_n^r\) contains a zerosum of length \(n\). The well known Erdős-Ginzburg-Ziv theorem states that \(\mathsf s(C_n)=2n-1\). In this talk we discuss the currently known lower and upper bounds in this and related problems, and explain how to construct nontrivial examples (giving lower bounds).

This talk will explore (noncommutative) factorization as applied to various important classes of matrices over the integers. All machinery will be built as needed. No background in either matrix theory or factorization theory is assumed. Many open problems will be presented.

The distribution of algebraic points in affine space is best described in terms of the height \(H:\mathbb A^n(\overline{\mathbb Q})\longrightarrow [1,\infty)\). To any subset \(S\) of \(\mathbb A^n(\overline{\mathbb Q})\) of uniformly bounded degree one can associate a counting function \(N(S,X)=|\{\mathbf \alpha\in S;H(\mathbf \alpha)\leq X\}|\). Let \(K\) be a number field with ring of integers \(\mathcal O_K\). In this talk we present a fairly general result on the distribution of points in \(\mathcal O_K^n\) whose conjugates satisfy certain inequalities. As a corollary we deduce counting results for the integral points in \(n\) dimensions having prescribed degree over a fixed number field \(k\), and for the Pisot numbers in \(K\). We shall also explain some ideas of the proof.

Let \(K\) be an algebraic number field with non-trivial class group \(G\) and \(\mathcal O_K\) be its ring of integers. For \(k \in \mathbb N\) and some real \(x \ge 1\), let \(F_k (x)\) denote the number of non-zero principal ideals \(a\mathcal O_K\) with norm bounded by \(x\) such that \(a\) has at most \(k\) distinct factorizations into irreducible elements. It is well known that \(F_k (x)\) behaves, for \(x \to \infty\), asymptotically like \(x (\log x)^{-1/|G|} (\log\log x)^{\mathsf N_k (G)}\). We study \(\mathsf N_k (G)\) with new methods from Combinatorial Number Theory.

This is a series of lectures introducing and overviewing recently refined ideas from Additive Combinatorics/Number Theory, particularly involving sumsets—given two subsets \(A\) and \(B\) of an abelian group, their sumset is \(A+B=\{a+b:\, a\in A,\,b\in B\}\)—and subsequence sums of sequences—given a sequence \(S\) of terms from an abelian group, \(\Sigma_n(S)\) denotes the set of all sums of subsequences of \(S\) of length \(n\). Topics are open to suggestion and may include

• An introduction to the Isoperimetric Method, which is a method with graph theoretical roots for tackling problems regarding sumsets \(A+B\). The fundamental definitions, basic bounds, and new applications to direct additive questions involving Sidon sets will be discussed.

• An introduction to Freiman Homomorphisms. In many fields—e.g., group theory, graph theory, number theory, etc.—the maps which preserve the fundamental properties of the studied structures play a fundamental role in the theory. The same is true of Additive Theory, for which these maps are known as Freiman Homomorphisms, though it is only very recently that even the basics regarding these maps have begun to be understood. Worse, in many cases, the theory is only presented for symmetric sumsets \(A+A\). The aim here is to present the theory in its more general context, including fundamental definitions, the existence of (a normalized form of) the universal ambient group for \(\sum_{i=1}^{n}A_i\), more general notions of restricted Freiman isomorphisms, and explicit calculations of the universal ambient group for small torsion-free sumsets.

• An overview of recent improvements to the Partition Theorem, which is a theorem relating sumsets and subsequence sums related to the Devos-Goddyn-Mohar Theorem.