Talks KFU
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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:
- Konrad Schrempf (TU Graz)
- Date:
- Thursday, 26.1.2017, 15:15‒16:45
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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
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
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
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
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
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
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
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
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
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
Scientific Program:
10:00-10:30 Coffee
10:30-11:15 Weidong Gao (Nankai University, PR China)
Zero-sum subsequences of distinct lengths11:30-12:15 Wolfgang A. Schmid (University of Paris 8 & 13, France)
Transfer Krull monoids and weakly Krull monoids: a comparison of their sets of lengths14:00-14:45 Pedro A. García-Sánchez (University of Granada, Spain)
Numerical Semigroups15:00-15:45 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
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
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
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
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
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
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
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
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
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
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
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
Scientific Program:
15:00 Opening
15:00-15:15 Robert F.Tichy (Graz University of Technology)
Laudatio15:15-16:00 Nigel Byott (University of Exeter)
Scaffolds and Local Galois Module Structure16:00-16:30 Coffee Break
16:30-17:15 Ernst-Ulrich Gekeler (Saarland University)
Stochastic Structures in Number Theory17:20-18:05 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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:
- About special classes of SP-domains
- Speaker:
- Andreas Reinhart (Universität Graz)
- Date:
- Thursday, 15.5.2014, 15:15‒16:45
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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:
- Stable equivalences and gradings
- Speaker:
- Dusko Bogdanic (Universität Graz)
- Date:
- Thursday, 8.5.2014, 15:15‒16:45
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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
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
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
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
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
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
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
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
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:
- Additive structure of multiplicative functions
- Speaker:
- Christian Elsholtz (TU Graz)
- Date:
- Thursday, 19.12.2013, 16:00‒16:45
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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
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
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
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
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
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
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
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
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
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
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
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
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
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:
- Ademir Hujdurović (University of Primorska)
- Date:
- Thursday, 25.4.2013, 17:15‒18:00
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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
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
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
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
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
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
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
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
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:
- Lucas sequences and quadratic orders
- Speaker:
- Franz Halter-Koch (Universität Graz)
- Date:
- Thursday, 31.1.2013, 15:15‒16:45
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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
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
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
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
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
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
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
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
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
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
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:
- Sukumar Das Adhikari (Harish-Chandra Research Institute, Allahabad (India))
- Date:
- Thursday, 5.7.2012, 17:00‒18:00
- Room:
- SR 11.32, Heinrichstraße 36, Uni Graz
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:
- Finite Additive Theory
- 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
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
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
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
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
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
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
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
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
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
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
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
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
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
Talks by:
Pinzhi Yuan (South China N. Univ., Guangzhou)
Title: Ko Chao's method and its applications to Diophantine equationsFranz Halter-Koch (KFU Graz)
Title: The Fibonacci sequence and class numbersWolfgang 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
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
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
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
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
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
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
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
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
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:
- P.A. García-Sánchez (University Granada)
- Date:
- Thursday, 13.1.2011, 17:15‒18:00
- Room:
- SR 11.34, Heinrichstraße 36, Uni Graz
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
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
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
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
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
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
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
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
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.