## Talks KFU

#### Algebra and Number Theory Seminar

Title:
Arithmetic in finite-complement ideals
Speaker:
Nicholas R. Baeth (Franklin and Marshall College, USA)
Date:
Thursday, 6.6.2019, 15:15‒16:15
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Ext as a k[T] -module
Speaker:
Markus Schmidmeier (Florida Atlantic University)
Date:
Monday, 3.6.2019, 14:00‒15:00
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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)

#### Algebra and Number Theory Seminar

Title:
Symmetries on finite abelian groups and rational Dyck paths
Speaker:
Hanbin Zhang (Chinese Academy of Sciences, Beijing, China)
Date:
Thursday, 23.5.2019, 15:15‒16:45
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the commuting graphs of rings and semirings
Speaker:
David Dol\v{z}an (University of Ljubljana)
Date:
Thursday, 16.5.2019, 15:15‒16:15
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On Erds-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups.
Speaker:
JunSeok Oh (University of Graz)
Date:
Thursday, 9.5.2019, 15:15‒16:45
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
A lattice isomorphism theorem for cluster groups of type A
Speaker:
Isobel Webster (University of Leeds)
Date:
Thursday, 25.4.2019, 15:15‒16:45
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Speaker:
Nick Williams, Fabian Lenzen
Date:
Thursday, 4.4.2019, 15:15‒17:00
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:
• 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 $K_0(\mathscr{O})$.

#### Algebra and Number Theory Seminar

Title:
Higher tilting theory of type A
Speaker:
Jordan McMahon (University of Graz)
Date:
Thursday, 28.3.2019, 15:15‒16:45
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Maximal Cohen-Macaulay approximations
Speaker:
Dusko Bogdanic (University of Graz)
Date:
Thursday, 14.3.2019, 15:15‒16:45
Room:
SR. 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Real Algebra and Geometry: The Commutative and the Non-Commutative World
Speaker:
Tim Netzer (University of Innsbruck)
Date:
Tuesday, 12.3.2019, 14:15‒15:45
Room:
SR AE06, Steyrergasse 30, ground floor
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
What is a cluster algebra?
Speaker:
Charles Beil (University of Graz)
Date:
Thursday, 7.3.2019, 15:15‒16:45
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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".)

#### Algebra and Number Theory Seminar

Title:
Rigorosum
Combinatorial aspects of tilings
Speaker:
Lukas Markus Andritsch (University of Graz)
Date:
Tuesday, 5.2.2019, 10:30‒12:00
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
F.g.u., acyclic, and positive monoids: First steps into their arithmetic
Speaker:
Salvatore Tringali (University of Graz)
Date:
Thursday, 31.1.2019, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups.
Speaker:
JunSeok Oh (University of Graz)
Date:
Thursday, 24.1.2019, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Introduction to Cluster algebras and results about their factorization.
Speaker:
Ana Garcia Elsener (University of Graz)
Date:
Thursday, 17.1.2019, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On monoids of ideals of orders in quadratic number fields
Speaker:
Johannes Brantner (KFU Graz)
Date:
Thursday, 10.1.2019, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Knoten in der Mathematik - Ein Spiel mit Schnüren, Bildern und Formeln
Speaker:
Meike Akveld (ETH Zürich)
Date:
Wednesday, 12.12.2018, 15:00‒16:00
Room:
HS 02.11, Universitätsplatz 2
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Rank of elements of general rings
Speaker:
Nik Stopar (University of Ljubljana)
Date:
Thursday, 6.12.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Cluster structure on Grassmannians and root combinatorics
Speaker:
Karin Baur (University of Graz)
Date:
Thursday, 29.11.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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

#### Algebra and Number Theory Seminar

Title:
Self-injective Jacobian algebras
Speaker:
Andrea Pasquali (University of Uppsala)
Date:
Thursday, 22.11.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Generalized cluster algebras, orbifolds and Caldero-Chapoton functions
Speaker:
Diego Velasco (National Autonomous University of Mexico)
Date:
Thursday, 15.11.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the $x$-coordinates of Pell equations which are $k$-generalized Fibonacci numbers
Speaker:
Mahadi Ddamulira (Graz University of Technology)
Date:
Thursday, 8.11.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Hopf dreams and diagonal harmonics
Speaker:
Cesar Ceballos (University of Vienna)
Date:
Thursday, 25.10.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the arithmetic of C-monoids and domains
Speaker:
Qinghai Zhong (University of Graz)
Date:
Thursday, 18.10.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The occurrence of numerical semigroups in commutative algebra and algebraic geometry
Speaker:
Julio J. Moyano-Fern\'andez (Universitat Jaume I (Castell\'on de la Plana, Spain))
Date:
Thursday, 11.10.2018, 15:15‒16:45
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A survey of homological algebra
Speaker:
Irena Swanson (Reed College, USA; Fulbright - NAWI Graz Visiting Professor)
Date:
Thursday, 4.10.2018, 15:15‒16:45
Room:
HS 11.01 (ground floor, Heinrichstrasse 36)
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
From the Magic Square of reflections and rotations to the McKay correspondence
Speaker:
Eleonore Faber (University of Leeds)
Date:
Thursday, 26.7.2018, 11:00‒12:00
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Arc representations
Speaker:
Date:
Wednesday, 25.7.2018, 11:00‒12:00
Room:
SR 11.33, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Perfect matchings and modules over dimer algebras
Speaker:
Ilke Canakci (Newcastle University)
Date:
Tuesday, 3.7.2018, 15:15‒16:15
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Local Class Field Theory and the Formal Groups
Speaker:
Sofja Afanaseva (St. Petersburg State University, Russia)
Date:
Thursday, 28.6.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

See next page.

#### Algebra and Number Theory Seminar

Title:
A note on friezes of type $\Lambda_p$.
Speaker:
Lukas Andritsch (University of Graz)
Date:
Thursday, 21.6.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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 $p$.

#### Algebra and Number Theory Seminar

Title:
Dimer models on cylinders over Dynkin diagrams and cluster algebras
Speaker:
Maitreyee Kulkarni (Louisiana State University)
Date:
Friday, 8.6.2018, 14:00‒15:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

tba.

#### Algebra and Number Theory Seminar

Title:
Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type
Speaker:
Alireza Nasr-Isfahani (University of Isfahan, Iran)
Date:
Thursday, 7.6.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Structural properties of subadditive sequences with applications to factorization theory and additive combinatorics
Speaker:
Salvatore Tringali (University of Graz)
Date:
Thursday, 24.5.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
AR-quivers of CM modules from Grassmannians
Speaker:
Ana Garcia Elsener (University of Graz)
Date:
Thursday, 3.5.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Sets of Arithmetical Invariants in Transfer Krull Monoids
Speaker:
Qinghai Zhong (University of Graz)
Date:
Thursday, 26.4.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Power Monoids: Connecting Factorization Theory to Additive Combinatorics
Speaker:
Austin Antoniou (Ohio State University)
Date:
Thursday, 19.4.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On Null Polynomials of Higher Order
Speaker:
Amr Al-Maktry (Graz University of Technolgy)
Date:
Thursday, 12.4.2018, 15:15‒16:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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 $\big\{x_j^2: j=1,2,..,k \big\}$.

#### Algebra and Number Theory Seminar

Title:
Space-filling curves of fractal sets
Speaker:
Shuqin Zhang (University of Leoben)
Date:
Thursday, 15.3.2018, 15:15‒16:00
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The Representation Theorem of Persistent Homology Revisited and Generalized.
Speaker:
Michael Kerber (Graz University of Technology)
Date:
Thursday, 8.3.2018, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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)

#### Algebra and Number Theory Seminar

Title:
Product-one sequences over finite group.
Speaker:
JunSeok Oh (Universität Graz)
Date:
Thursday, 25.1.2018, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
$f$-vectors of simplicial and simple polytopes
Speaker:
Roswitha Rissner (Graz University of Technology)
Date:
Thursday, 18.1.2018, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the Spectrum of Rings of Functions
Speaker:
Sophie Frisch (Graz University of Technology)
Date:
Thursday, 11.1.2018, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
DK Day 2017
Speaker:
DK Day 2017 (Graz University of Technology)
Date:
Thursday, 14.12.2017, 10:30‒16:40
Room:
HS BE01, Steyrergasse 30/EG, TU Graz

#### Algebra and Number Theory Seminar

Title:
On the structure of unions of sets of lengths
Speaker:
Salvatore Tringali (Universität Graz)
Date:
Thursday, 7.12.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $L_1 + L_2 \subseteq L$.)

#### Algebra and Number Theory Seminar

Title:
The geometry of nonnoetherian coordinate rings
Speaker:
Charles Beil (Universität Graz)
Date:
Thursday, 30.11.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Integer-valued polynomials on rings of integers in number fields with prescribed sets of lengths
Speaker:
Sarah Nakato (Graz University of Technology)
Date:
Thursday, 23.11.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Long sets of lengths with maximal elasticity
Speaker:
Qinghai Zhong (Universität Graz)
Date:
Thursday, 16.11.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Pair correlations, additive combinatorics, and some linear algebra
Speaker:
Christoph Aistleitner (Graz University of Technology)
Date:
Thursday, 9.11.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Festkolloquium für Prof. Dr. Robert Tichy
Speaker:
Festkolloquium für Prof. Dr. Robert Tichy (Graz University of Technology)
Date:
Thursday, 19.10.2017, 9:00‒18:00
Room:
HS BE01, Steyrergasse 30/EG, TU Graz

#### Algebra and Number Theory Seminar

Title:
Maximal sets in $\mathbb{F}_q^n$ avoiding progressions
Speaker:
Christian Elsholtz (Graz University of Technology)
Date:
Thursday, 12.10.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

One of the most influential problems in contemporary mathematics is a question of Erds and Turán: Let $r_k(n)$ denote the maximal number of integers in $[1,2,3, \ldots , n]$ without any arithmetic progression of length $k$. Erds 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 $r_k(\mathbb{F}_q^n)$.

#### Algebra and Number Theory Seminar

Title:
Variations on Baur–Marsh’s determinant
Speaker:
Philipp Lampe (Durham University)
Date:
Monday, 7.8.2017, 14:00‒15:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
(Co)torsion via stabilization of additive functors. categories
Speaker:
Alex Martsinkovsky (Northeastern University)
Date:
Wednesday, 19.7.2017, 10:30‒11:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Speaker:
Zhi-Wei Sun, David J. Grynkiewicz
Date:
Thursday, 29.6.2017, 15:15‒17:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:
• 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

#### Algebra and Number Theory Seminar

Title:
An introduction to classical invariant theory and its application to zero-sum theory
Speaker:
Hanbin Zhang (University of Graz)
Date:
Thursday, 22.6.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Extensions and mapping cones for gentle algebras
Speaker:
David Pauksztello (Univ. Verona)
Date:
Friday, 16.6.2017, 14:00‒14:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Reduction for negative Calabi-Yau triangulated categories
Speaker:
Raquel Simoes (Univ. Lissabon)
Date:
Friday, 16.6.2017, 13:00‒13:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Gentle 2-Calabi-Yau tilted algebras
Speaker:
Ana Garcia Elsener (University of Graz)
Date:
Thursday, 8.6.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Dilogarithm identities in cluster algebras from Hamiltonian/Lagrangian point of view
Speaker:
Michael Gekhtman (University of Notre Dame)
Date:
Thursday, 1.6.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.)

#### Algebra and Number Theory Seminar

Title:
Periodic properties of directed families with applications to factorization theory
Speaker:
Salvatore Tringali (Universität Graz)
Date:
Thursday, 18.5.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Extensions between Cohen–Macaulay modules of Grassmannian cluster categories
Speaker:
Dusko Bogdanic (Universität Graz)
Date:
Thursday, 4.5.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Arithmetical invariants of local quaternion orders
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 6.4.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Braid groups and quiver mutation
Speaker:
Robert Marsh (University of Leeds)
Date:
Thursday, 30.3.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Transfinite mutations
Speaker:
Karin Baur (Universität Graz)
Date:
Thursday, 23.3.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Perfect matchings and the Grassmannian cluster algebra
Speaker:
Matthew Pressland (Max-Planck-Institut für Mathematik, Bonn)
Date:
Thursday, 9.3.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Linearizing the Word Problem in (some) Free Fields
Speaker:
Date:
Thursday, 26.1.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the algebraic and arithmetic structure of the monoid of product-one sequences over a finite group
Speaker:
Junseok Oh (Universität Graz)
Date:
Thursday, 19.1.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $\mathcal B (G)$.

#### Algebra and Number Theory Seminar

Title:
Power Monoids: A Bridge between Factorization Theory and Arithmetic Combinatorics
Speaker:
Salvatore Tringali (Universität Graz)
Date:
Thursday, 12.1.2017, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the monotone catenary degree
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thursday, 15.12.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Higher Frieze Patterns
Speaker:
Jordan McMahon (Universität Graz)
Date:
Thursday, 1.12.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Boundary algebra of a GL$_m$-dimer
Speaker:
Lukas Andritsch (Universität Graz)
Date:
Thursday, 17.11.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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. $e_b \Lambda_{Q_{T_1}} e_b \cong e_b \Lambda_{Q_{T_2}} e_b$.

#### Algebra and Number Theory Seminar

Title:
Research on Skew Polynomial Rings
Speaker:
Intan Muchtadi-Alamsyah (Institut Teknologi Bandung, Indonesia)
Date:
Thursday, 10.11.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Subword Complexes in Algebra, Combinatorics, and Geometry
Speaker:
Cesar Ceballos (Universität Wien)
Date:
Thursday, 3.11.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Sets of minimal distances and characterizations of class groups of Krull monoids
Speaker:
Qinghai Zhong (Universität Graz)
Date:
Thursday, 27.10.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Minimal relations and catenary degrees in Krull monoids
Speaker:
Yushuang Fan (Universität Graz)
Date:
Thursday, 20.10.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $H$.

#### Algebra and Number Theory Day

Date:
Friday, 14.10.2016, 10:00‒15:45
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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

#### Algebra and Number Theory Seminar

Title:
Asymptotic triangulations and cluster algebras
(PhD defense)
Speaker:
Hannah Vogel (Universität Graz)
Date:
Thursday, 8.9.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Kac-Moody groups, generalized minors, and quiver representations
Speaker:
Salvatore Stella (Università degli studi di Roma "La Sapienza", Italy)
Date:
Monday, 11.7.2016, 15:00‒15:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On Lie algebras associated with representation-finite algebras
Speaker:
Alireza Nasr (IPM, Isfahan, Iran)
Date:
Monday, 11.7.2016, 14:00‒14:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Diagram categories old and new
Speaker:
Paul Martin (University of Leeds)
Date:
Thursday, 30.6.2016, 15:45‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Partial orders in Rickart rings
Speaker:
Janko Marovt (University of Maribor)
Date:
Thursday, 30.6.2016, 15:00‒15:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Monogenity of number fields
Speaker:
István Gaál (University of Debrecen)
Date:
Thursday, 23.6.2016, 15:00‒16:00
Room:
SR 11.34, Heinrichstraße 36, Uni Graz

#### Algebra and Number Theory Seminar

Title:
Multiserial algebras
Speaker:
Sibylle Schroll (University of Leicester)
Date:
Thursday, 16.6.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Matrices over commutative rings
Speaker:
Roswitha Rissner (TU Graz)
Date:
Thursday, 2.6.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Non-commutative resolutions of quotient singularities for reductive groups
Speaker:
Špela Špenko (University of Edinburgh)
Date:
Thursday, 19.5.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
General Topology, intersections of modules and flatness
Speaker:
Carmelo Finocchiaro (TU Graz)
Date:
Thursday, 12.5.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Products of $k$ atoms in Krull monoids
Speaker:
Yushuang Fan (Universität Graz)
Date:
Thursday, 28.4.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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 $k\geq k^*$.

#### Algebra and Number Theory Seminar

Title:
The set of distances in seminormal weakly Krull monoids
Speaker:
Qinghai Zhong (Universität Graz)
Date:
Thursday, 21.4.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Colloquium on the occasion of the 60th birthday of Günter Lettl

Date:
Thursday, 14.4.2016, 15:00‒18:05
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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

#### Algebra and Number Theory Seminar

Title:
Non-commutative Arithmetic
Speaker:
Wolfgang Rump (Universität Stuttgart)
Date:
Thursday, 17.3.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A survey on factorizations of elements in noncommutative rings
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 10.3.2016, 15:00‒16:30
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
The Ziegler spectrum of a derived-discrete algebra
Speaker:
Rosie Laking (University of Manchester, UK)
Date:
Thursday, 28.1.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Infinite friezes of positive integers
(PhD defense)
Speaker:
Manuela Tschabold (Universität Graz)
Date:
Thursday, 21.1.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The Hasse norm principle for abelian extensions
Speaker:
Christopher Frei (TU Graz)
Date:
Thursday, 14.1.2016, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
"Negative" cluster-tilted algebras and tiling algebras
Speaker:
Raquel Coelho Simões (University of Lisbon)
Date:
Thursday, 10.12.2015, 16:30‒17:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Growth behaviour of frieze patterns
Speaker:
Michaela Tschabold (Universität Graz)
Date:
Thursday, 10.12.2015, 15:15‒16:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Joseph ideals and minimal lisse W-algebras
Speaker:
Anne Moreau (Université de Poitiers, France)
Date:
Thursday, 26.11.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Infinite friezes
Speaker:
Mark Parsons (Universität Graz)
Date:
Thursday, 19.11.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A characterization of class groups via sets of lengths II
Speaker:
Alfred Geroldinger (Universität Graz)
Date:
Thursday, 12.11.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A characterization of class groups via sets of lengths
Speaker:
Qinghai Zhong (Universität Graz)
Date:
Thursday, 29.10.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Generalised cluster algebras and quantum affine algebras at roots of unity
Speaker:
Anne-Sophie Gleitz (Université de Caen, France)
Date:
Thursday, 22.10.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $\mathfrak{sl}_4$.

#### Algebra and Number Theory Seminar

Title:
Divisor-class groups of monadic submonoids of certain rings of integer-valued polynomials
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thu, 08.10.2015, 15:15‒16:45
Thu, 15.10.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The $\rho_k$ Invariants for Finite Abelian Groups
Speaker:
David J. Grynkiewicz (University of Memphis, USA)
Date:
Thursday, 25.6.2015, 16:45‒17:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $\rho_{2k+1}(G)$.

#### Algebra and Number Theory Seminar

Title:
On labyrinth fractals
Speaker:
Ligia Loretta Cristea (Universität Graz)
Date:
Thursday, 25.6.2015, 15:15‒16:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Orders in cubic number fields
Speaker:
Günter Lettl (Universität Graz)
Date:
Thursday, 18.6.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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

#### Algebra and Number Theory Seminar

Title:
Mutations of Hom-configurations in negative Calabi-Yau triangulated categories
Speaker:
Raquel Coelho Simões (University of Lisbon)
Date:
Thursday, 21.5.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Polynomial functions on non-commutative rings
Speaker:
Sophie Frisch (TU Graz)
Date:
Thursday, 7.5.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Every abelian group is the class group of a (noncommutative) simple Dedekind domain
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 30.4.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Existence of gradings on associative algebras
Speaker:
Dusko Bogdanic (Universität Graz)
Date:
Thursday, 19.3.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Prime Spectra of Certain Two Dimensional Integral Domains – History and New Development
Speaker:
Aihua Li (Montclair State University)
Date:
Tuesday, 10.3.2015, 17:15‒18:00
Room:
SR 11.13, Heinrichstraße 36, Uni Graz

#### Algebra and Number Theory Seminar

Title:
Power integral bases in algebraic number fields
Speaker:
István Gaál (University of Debrecen)
Date:
Tuesday, 10.3.2015, 16:15‒17:00
Room:
SR 11.13, Heinrichstraße 36, Uni Graz

#### Algebra and Number Theory Seminar

Title:
A transfer homomorphism for (some) bounded Dedekind prime rings
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 29.1.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Periodic infinite friezes from punctured discs part II
Speaker:
Manuela Tschabold (Universität Graz)
Date:
Thursday, 22.1.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Bilinear pairings on elliptic curves
Speaker:
Sebastian Ramacher (TU Graz)
Date:
Thursday, 15.1.2015, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Asymptotic triangulations & Coxeter transformations of $C_{p,q}$
Speaker:
Hannah Vogel (Universität Graz)
Date:
Thursday, 11.12.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Explicit construction of companion bases
Speaker:
Mark Parsons (Universität Graz)
Date:
Thursday, 4.12.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Tilting Modules and Root Polytopes
Speaker:
Lutz Hille (Universität Münster)
Date:
Thursday, 20.11.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Equivariant completion and Dynkin quivers
Speaker:
Nils Carqueville (Universität Wien)
Date:
Tuesday, 11.11.2014, 14:00‒15:30
Room:
SR 15.33, Resowi Bauteil B, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Some recent results on additive decomposition of sets
Speaker:
Christian Elsholtz (TU Graz)
Date:
Thursday, 6.11.2014, 15:00‒16:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On conductor ideals
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thursday, 23.10.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Class semigroups
Speaker:
Thomas Wunderer (Universität Graz)
Date:
Thursday, 16.10.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Length Multiplicity Set Complete Krull Monoids
Speaker:
Paul Baginski (Fairfield University, USA)
Date:
Thursday, 9.10.2014, 15:15‒16:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The Index of Zero-Sum Free Sequences and Kummer Subspaces
Speaker:
David J. Grynkiewicz (University of Memphis, USA)
Date:
Thursday, 3.7.2014, 16:30‒17:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Warning's Second Theorem with Restricted Variables
Speaker:
John R. Schmitt (Middlebury College, USA)
Date:
Thursday, 3.7.2014, 15:15‒16:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Periodic infinite friezes from punctured discs
Speaker:
Manuela Tschabold (Universität Graz)
Date:
Thursday, 26.6.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Power integral bases in infinite families of quartic fields
Speaker:
Timea Szabó (University of Debrecen)
Date:
Tuesday, 17.6.2014, 15:00‒15:15
Room:
SR 11.34, Heinrichstraße 36, Uni Graz

#### Algebra and Number Theory Seminar

Title:
Calculating small solutions of relative Thue equations
Speaker:
István Gaál (University of Debrecen)
Date:
Tuesday, 17.6.2014, 14:00‒14:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz

#### Algebra and Number Theory Seminar

Title:
Orders in pure cubic number fields
(PhD defense)
Speaker:
Chanwit Prabayak (Universität Graz)
Date:
Monday, 16.6.2014, 14:00‒15:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 12.6.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The Davenport constant of subsets of finitely generated free abelian groups
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 5.6.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Factorization Theory in Maximal Orders
(PhD defense)
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 22.5.2014, 15:15‒16:45
Room:
HS 10.01, Heinrichstraße 28, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thursday, 15.5.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Speaker:
Dusko Bogdanic (Universität Graz)
Date:
Thursday, 8.5.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Faithful Lie algebra representations of minimal degree
Speaker:
Dietrich Burde (Universität Wien)
Date:
Thursday, 10.4.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Cluster algebras of infinite rank as colimits
Speaker:
Sira Gratz (Leibniz Universität Hannover)
Date:
Thursday, 3.4.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Principal ideals in quadratic orders and Diophantine equations
Speaker:
Franz Halter-Koch (Universität Graz)
Date:
Thursday, 13.3.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Combinatorics of the Grassmannian's BZ-twist
Speaker:
Jeanne Scott (Boston, USA)
Date:
Thursday, 30.1.2014, 16:15‒17:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Hom-configurations in triangulated categories generated by spherical objects
Speaker:
Raquel Coelho Simões (Universität Lissabon)
Date:
Thursday, 30.1.2014, 15:15‒16:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $\mathcal C$.

#### Algebra and Number Theory Seminar

Title:
Parabolic orbits and $\Delta$-filtered modules
Speaker:
Jernej Pribosek (Universität Graz)
Date:
Thursday, 23.1.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the edge expansion of infinite Cayley graphs
Speaker:
Amnon Rosenmann (AIT Austrian Institute of Technology)
Date:
Thursday, 16.1.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Orders in purely cubic number fields
Speaker:
Chanwit Prabpayak (Universität Graz)
Date:
Thursday, 9.1.2014, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Speaker:
Christian Elsholtz (TU Graz)
Date:
Thursday, 19.12.2013, 16:00‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $(f(n), f(n+d), f(n+2d), f(n+3d) )= (\varepsilon_1, \ldots , \varepsilon_4).$

#### Algebra and Number Theory Seminar

Title:
Sums of dilates
Speaker:
Alain Plagne (École Polytechnique (Paris))
Date:
Thursday, 19.12.2013, 15:00‒15:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The distribution of real polynomials with bounded roots
Speaker:
Mario Weitzer (Montanuniversität Leoben)
Date:
Thursday, 28.11.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Representation Theory of Symmetric Groups
Speaker:
Dusko Bogdanic (Universität Graz)
Date:
Thu, 07.11.2013, 15:15‒16:45
Thu, 14.11.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the shape of subword complexity sequences of finite words
Speaker:
Hannah Vogel (Universität Graz)
Date:
Thursday, 31.10.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Skolem closure of ideals
Speaker:
Roswitha Rissner (TU Graz)
Date:
Thursday, 24.10.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Rings of integer-valued polynomials and monadic properties
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thursday, 17.10.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On Krull and Mori rings with zero-divisors
Speaker:
Sebastian Ramacher (Universität Graz)
Date:
Thursday, 10.10.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The decorated mapping class group of a marked surface
Speaker:
Thomas Brüstle (Université de Sherbrooke (Québec))
Date:
Friday, 12.7.2013, 14:00‒14:45
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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 $(S, M)$.

#### Algebra and Number Theory Seminar

Title:
Postnikov diagrams and categories associated to Grassmannians
Speaker:
Karin Baur (Universität Graz)
Date:
Thursday, 27.6.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Cluster algebras of finite type and companion bases
Speaker:
Mark Parsons (Universität Graz)
Date:
Thu, 13.06.2013, 15:15‒16:45
Thu, 20.06.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Integer-valued polynomials over sets of algebraic integers of bounded degree
Speaker:
Giulio Peruginelli (TU Graz)
Date:
Thursday, 6.6.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A study of factorizations in non-commutative semigroups
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 23.5.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Bounds and some exact values for plus-minus weighted Davenport constants and some related problems
Speaker:
Wolfgang A. Schmid (Université Paris 8)
Date:
Thursday, 2.5.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Pentavalent symmetric bicirculants
Speaker:
Date:
Thursday, 25.4.2013, 17:15‒18:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The Davenport Constant for Non-Abelian Groups
Speaker:
David J. Grynkiewicz (Universität Graz)
Date:
Thursday, 25.4.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Double centralizers of unipotent elements in simple algebraic groups
Speaker:
Iulian Simion (EPFL Lausanne)
Date:
Thursday, 4.4.2013, 16:00‒17:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Factorization Theory applied to Direct-sum Decompositions of Modules
Speaker:
Nicholas Baeth (University of Central Missouri)
Date:
Thursday, 14.3.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Chebyshev polynomials and tensor diagrams
Speaker:
Lisa Lamberti (ETH Zürich)
Date:
Thursday, 7.3.2013, 17:15‒18:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
From triangulated categories to module categories via localization
Speaker:
Robert Marsh (University of Leeds)
Date:
Thursday, 7.3.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Mutation of torsion pairs in triangulated categories
Speaker:
Sira Gratz (Universität Hannover)
Date:
Wednesday, 13.2.2013, 15:45‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
An introduction to co-$t$-structures and co-stability conditions
Speaker:
David Pauksztello (Universität Hannover)
Date:
Wednesday, 13.2.2013, 14:15‒15:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Maximal rigid objects as noncrossing bipartite graphs
Speaker:
Raquel Simoes (Leibniz Universität Hannover)
Date:
Thursday, 31.1.2013, 17:15‒18:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Speaker:
Franz Halter-Koch (Universität Graz)
Date:
Thursday, 31.1.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Seminormal orders and the half-factorial property
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thu, 17.01.2013, 15:15‒16:45
Thu, 24.01.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Polyhedral models for generalized associahedra via Coxeter elements
Speaker:
Salvatore Stella (North Eastern University (Boston))
Date:
Thursday, 10.1.2013, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Orbit Chern Classes
Speaker:
Mara D. Neusel (Texas Tech University (Lubbock, USA))
Date:
Thursday, 13.12.2012, 15:15‒16:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Parabolic orbits via $\Delta$-filtered modules
Speaker:
Jernej Pribosek (Universität Graz)
Date:
Thursday, 29.11.2012, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
What is a Frieze Pattern?
Speaker:
Manuela Tschabold (Universität Graz)
Date:
Thursday, 22.11.2012, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $A_n$.

#### Algebra and Number Theory Seminar

Title:
Subsetsums/Hilbert cubes in multiplicatively defined sets
Speaker:
Christian Elsholtz (TU Graz)
Date:
Thursday, 8.11.2012, 16:45‒17:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the intersection of a hyperplane of $(\mathbb F_p)^l$ with the $l$-cube
Speaker:
Benjamin Girard (Université Pierre et Marie Curie (Paris 6))
Date:
Thursday, 8.11.2012, 15:15‒16:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Sets with forbidden linear constraints
Speaker:
Alain Plagne (École Polytechnique, Paris)
Date:
Thursday, 25.10.2012, 15:15‒16:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Characterization of conductors in algebraic number fields
Speaker:
Chanwit Prabpayak (Universität Graz)
Date:
Thursday, 18.10.2012, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $K$.

#### Algebra and Number Theory Seminar

Title:
Local and global tameness in Krull monoids
Speaker:
Alfred Geroldinger (Universität Graz)
Date:
Thursday, 11.10.2012, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The Davenport constant with weights and related questions
Speaker:
Date:
Thursday, 5.7.2012, 17:00‒18:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Speaker:
David Grynkiewicz (Universität Graz)
Date:
Thu, 14.06.2012, 15:00‒16:30
Thu, 21.06.2012, 15:00‒16:30
Thu, 28.06.2012, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
About $r$-SP-monoids and radical factorial monoids
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thursday, 10.5.2012, 16:45‒17:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Direct-sum decompositions of modules
Speaker:
Nicholas Baeth (University of Central Missouri)
Date:
Thu, 10.05.2012, 15:15‒16:15
Thu, 24.05.2012, 15:15‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The set of distances in totally definite quaternion algebras over number fields
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 3.5.2012, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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 $\Delta(R) = \mathbb N$.

#### Algebra and Number Theory Seminar

Title:
Gaussian Shift Radix Systems - Pethö's Loudspeaker
Speaker:
Mario Weitzer (Montanuniversität Leoben)
Date:
Thursday, 29.3.2012, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On non-unique factorization in the ring of integer-valued polynomials
Speaker:
Sophie Frisch (TU Graz)
Date:
Thursday, 22.3.2012, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On square solutions of the Diophantine equations $D_1 X^2 - D_2 Y^2 = c$, $\,c \in \{1,2,4\}$
Speaker:
Günter Lettl (Universität Graz)
Date:
Thursday, 15.3.2012, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
An introduction to cluster categories
Speaker:
Karin Baur (Universität Graz)
Date:
Thu, 12.01.2012, 15:00‒16:30
Thu, 19.01.2012, 15:00‒16:30
Thu, 26.01.2012, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Elasticity of rings of integer-valued polynomials
Speaker:
Giulio Peruginelli (TU Graz)
Date:
Thursday, 15.12.2011, 16:45‒17:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On products of two irreducible elements and characterizations of class group
Speaker:
Alfred Geroldinger (Universität Graz)
Date:
Thu, 24.11.2011, 15:00‒16:30
Thu, 01.12.2011, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On generalizations of factorial monoids
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thu, 10.11.2011, 15:00‒16:30
Thu, 17.11.2011, 15:00‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Factorization in central simple algebras
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thu, 06.10.2011, 15:00‒16:30
Thu, 13.10.2011, 15:00‒16:30
Thu, 20.10.2011, 15:00‒16:30
Thu, 03.11.2011, 15:00‒16:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A Generalization of Kneser's Theorem to nonabelian groups
Speaker:
Matthew DeVos (Simon Fraser University, Vancouver)
Date:
Thursday, 7.7.2011, 17:15‒18:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
On the index of minimal zero-sum sequences
Speaker:
Pingzhi Yuan (South China N. Univ., Guangzhou)
Date:
Friday, 1.7.2011, 14:00‒14:45
Room:
HS 11.02, Heinrichstraße 36, Uni Graz

#### Mini-Colloquium on Number Theory

Date:
Thursday, 16.6.2011, 15:00‒18:00
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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

#### Algebra and Number Theory Seminar

Title:
Non-commutative Krull rings III
Speaker:
Florian Kainrath (Universität Graz)
Date:
Thu, 26.05.2011, 15:15‒16:45 (SR 11.32)
Tue, 31.05.2011, 15:30‒17:00 (HS 11.01)
Room:
Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Non-commutative Krull rings II
Speaker:
Daniel Smertnig (Universität Graz)
Date:
Thursday, 19.5.2011, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Good reductions for endomorphisms of the projective line
Speaker:
Giulio Peruginelli (TU Graz)
Date:
Thursday, 12.5.2011, 17:15‒18:15
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Non-commutative Krull rings I
Speaker:
Alfred Geroldinger (Universität Graz)
Date:
Thu, 07.04.2011, 15:15‒16:45
Thu, 14.04.2011, 15:15‒16:45
Thu, 12.05.2011, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Swarms of birds and positive dynamical systems
Speaker:
Ulrich Krause (Universität Bremen)
Date:
Thursday, 31.3.2011, 17:00‒18:00
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Factorization in monoids of finite graphs
Speaker:
Christiaan van de Woestijne (Montanuniversität Leoben)
Date:
Thursday, 31.3.2011, 15:15‒16:30
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Arithmetic of Mori domains and monoids: the global case
Speaker:
Florian Kainrath (Universität Graz)
Date:
Thursday, 24.3.2011, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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

#### Algebra and Number Theory Seminar

Title:
The Busemann and the Bing-Borsuk Conjectures
Speaker:
Dusan Repovs (University of Ljubljana)
Date:
Thursday, 17.3.2011, 17:15‒18:00
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
C-monoids and seminormality
Speaker:
Andreas Reinhart (Universität Graz)
Date:
Thu, 10.03.2011, 15:15‒16:45
Thu, 17.03.2011, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Factorizations in uniquely presented (finitely generated) monoids
Speaker:
Date:
Thursday, 13.1.2011, 17:15‒18:00
Room:
SR 11.34, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Using Arithmetic Progressions to Weight Subsequence Sums
Speaker:
David Grynkiewicz (Universität Graz)
Date:
Thursday, 13.1.2011, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Tautness: Stretched Between Half-Factoriality and Bounded Factorization
Speaker:
Paul Baginski (Université Lyon)
Date:
Thursday, 9.12.2010, 16:45‒18:00
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Non-unique factorizations — A semi-group theoretic approach with an application to non-principal orders in algebraic number fields
Speaker:
Andreas Philipp (Universität Graz)
Date:
Thursday, 9.12.2010, 15:15‒16:30
Room:
HS 11.02, Heinrichstraße 36, Uni Graz
Abstract:

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 $\triangle(\mathcal O)\subset\lbrace 1,2\rbrace$.

#### Algebra and Number Theory Seminar

Title:
Upper and lower bounds in zero sum problems
Speaker:
Christian Elsholtz (TU Graz)
Date:
Thursday, 25.11.2010, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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).

#### Algebra and Number Theory Seminar

Title:
Factorization Theory of Matrices
Speaker:
Vadim Ponomarenko (San Diego State University)
Date:
Thursday, 18.11.2010, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
The distribution of integral points in affine space
Speaker:
Martin Widmer (TU Graz)
Date:
Thursday, 11.11.2010, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
A quantitative aspect of non-unique factorizations: the Narkiewicz constants
Speaker:
Alfred Geroldinger (Universität Graz)
Date:
Thursday, 4.11.2010, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.

#### Algebra and Number Theory Seminar

Title:
Some Recent Developments in Additive Theory
Speaker:
David Grynkiewicz (Universität Graz)
Date:
Thu, 07.10.2010, 15:15‒16:45
Thu, 14.10.2010, 15:15‒16:45
Wed, 20.10.2010, 15:15‒16:45 (SR 11.34)
Thu, 28.10.2010, 15:15‒16:45
Room:
SR 11.32, Heinrichstraße 36, Uni Graz
Abstract:

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.