Graz Algebra and Geometry Seminar

Graz Algebra & Geometry Seminar

In summer term 25, we meet on Wednesdays 11:15 -- 12:15 in SR 11.34 (Heinrichstr. 36, 3rd floor).

05.03.2025

Arf rings versus ADE singularities

Abstract: In a pandemic era preprint, Hailong Dao proved two interesting properties for Arf rings: that under suitable mild conditions Arf rings have only finitely many indecomposable reflexive modules up to isomorphism and that every reflexive module is isomorphic to its own dual. Arf rings are defined in Krull dimension one and he said it would be fun to extend the latter result to higher dimensions. In this talk, I will show you the following: 1) if a local ring of depth at least two satisfies the second property (and is not regular), then it has to be a hypersurface ring and it has Krull dimension at most four and 2) if a local ring of depth at least two satisfies both of the properties above (and is not regular), then it has to be a simple singularity.

12.03.2025

String algebras over local rings

Abstract: String algebras are usually defined by path algebras over fields. Path algebras have also been considered over noetherian local ground rings, and Raggi-Cardenas and Salmeron generalised the definition of an admissible ideal in this context. By characterising such ideals, I will motivate the definition of a string algebra over a local ring. I will then show how to construct them over any regular local ring, highlighting an example from homotopy theory studied by Baues and Drozd. I will also explain why the representation theory of string algebras over local rings is well-behaved. Time permitted; I will discuss a (clannish) algebra over the 11-adic integers, related to modular representations of the Matheiu 11-group. This talk is based on a paper found at doi.org/10.1016/j.jalgebra.2024.12.019.

19.03.2025

TBA

26.03.2025

The surprising distribution of the global dimension of linear Nakayama algebras

Abstract: For a positive integer $n$, an $n$-Nakayama algebra $A$ is a finite-dimensional algebra over some field $\mathbb{F}$ that can be constructed as a quotient algebra $A = \mathbb{F}Q / I$, where $Q$ is a linear or cyclic quiver on $n$ vertices (i.e., $0 \to 1 \to \cdots \to n-1$ or $0 \to 1 \to \cdots \to n-1 \to 0$), $\mathbb{F}Q$ is the corresponding path algebra, and $I$ is a suitable two-sided ideal. A quantity that is particularly interesting for algebraists is the global dimension of $A$, which is defined as the maximal projective dimension of a simple module of $A$.

There is a well-known bijection that maps a "linear" $n$-Nakayama algebra to a Dyck path. Under this correspondence, the simple modules relevant for determining the global dimension of $A$ are mapped to even-parity integer points on and under the Dyck path. From this representation, their dimensions can be determined following a simple recursive scheme. Intriguingly, experiments suggest that the global dimension follows the same distribution as the height of Dyck paths -- for which we do not yet have a proper explanation.

In this talk we consider the global dimension from a purely combinatorial point of view. After walking through known results, we discuss the evidence for our conjecture linking path height and global dimension. In particular, we present a hand full of strategies that have not (yet) led to a successful proof.

02.04.2025

09.04.2025

Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields

Abstract: An innocent question of Erdős and Turán (1936) asked about the maximal size of sets in {1,2,..., N} without 3 integers in arithmetic progression.

The upper bound on the size of such sets, when N is large, has seen influential ideas, and the problem became one of the corner stones of combinatorial number theory/additive combinatorics. (The list of contributors includes Roth, Szemeredi, Furstenberg, Bourgain, Gowers, Sanders, Bloom and Sisask, and most recently Kelley and Meka.)

On lower bounds the situation remained unclear: Szekeres (1936) conjectured (based on small examples) that the set of integers avoiding the digit 2 in the ternary system might be best possible. Salem and Spencer (1942) and Behrend (1946) gave better constructions. These constructions required a new idea, but are quite simple otherwise. Behrend's construction was named "fairly unimprovable" and the status was described as "the lower bound has remained virtually stationary".

In this talk we describe a novel idea. This new method allows to surpass a natural threshold for progression-free sets in (Z_m)^n, and also eventually improves Behrend's construction.

This is joint work with Zach Hunter, Laura Proske, Lisa Sauermann

30.04.2025

n-exact categories and (non)commutative algebraic geometry.

Abstract: We give the simple modules over commutative regular local rings and Artin--Schelter regular algebras a categorical meaning. This gives rise to a variety of examples of n-exact structures (a generalization of exact structures) on their categories of finitely generated (graded) projective modules. We apply our categorical viewpoint to show that the Auslander-Buchbaum formula for higher Auslander-algebras (which has recently been shown Cruz--Marczinzik) can be deduced from work of Martinéz-Villa--Solberg and present our attempt of giving a categorical meaning to this formula.

07.05.2025 UNUSUAL TIME (11:45--12:45)

Jacobians of graphs and degenerating Riemann surfaces

Abstract: There is a compelling interplay between the theories of Riemann surfaces and graphs. On the one hand, graphs serve as analogs of Riemann surfaces in tropical geometry and, in the last two decades, were discovered to satisfy analogs of many classical theorems on Riemann surfaces (e.g., Riemann--Roch and Torelli theorems). On the other hand, properties of Riemann surfaces which are degenerating to a singular Riemann surface (in the sense of moduli spaces) can often be understood using graphs.

In this talk, we discuss the behavior of two important notions, the rank of divisors and polarized Jacobians, on degenerating Riemann surfaces. We show how these can be understood using corresponding notions for graphs.

Based on joint work with Omid Amini (École Polytechnique).

14.05.2025

TBA

21.05.2025 UNUSUAL TIME (11:45--12:45)

Factorization-Theoretic Aspects of Generalized Cluster Algebras

Abstract: The class group of a commutative domain encodes the failure of unique factorization and is a central invariant in factorization theory. In the setting of cluster algebras, this group exhibits particularly regular behavior: it is always a finitely generated free abelian group. However, when extending the framework to generalized cluster algebras(Chechov, Shapiro 2014) this regularity breaks down.

In this talk, I will explore how the more flexible structure of generalized cluster algebras affects their arithmetic properties. In particular, I will show that torsion can appear in their class groups, and that every finitely generated abelian group can be realized as the class group of a suitably constructed generalized cluster algebra. These results highlight a fundamental divergence between cluster and generalized cluster structures from the viewpoint of ideal theory and factorization, and open the door to further exploration of their arithmetic complexity.

28.05.2025

Categorical resolutions of A_2 singularities

Abstract: In this talk I will explain that there exists a particularly small (“crepant”) categorical resolution of the derived category of a projective variety with an isolated A_2/cuspidal singularity. More importantly, one can describe generators of its kernel in a very explicit way: In the case of an even dimensional variety with an isolated A_2 singularity, the kernel can be generated by two 2-spherical objects, which are related to spinor sheaves on a nodal quadric and induce autoequivalences on the categorical resolution.

04.06.2025 ONLINE

Presentations of (complex) braid groups via triangulations

Abstract: Coxeter classified the finite reflection groups and showed that they have beautiful presentations, now known as Coxeter presentations. More recently, cluster algebras and their surface models have been used by several authors to construct new families of presentations of reflection and braid groups. This talk will give an overview of a class of such families and introduce its complex analogue for complex braid groups. This is based on joint work with Bethany Marsh.

11.06.2025

TBA

18.06.2025

TBA

25.06.2025

TBA

Organizers: E. Faber, M. Kalck