Summer Semester 2023

17.04 Haibo Jin

Title: A localisation theorem for singularity categories of proper dg algebras

Abstract: Given a recollement of three proper dg algebras, e.g. three finite-dimensional algebras, which extends one step downwards, it is shown that there is a short exact sequence of their singularity categories. This allows us to recover and generalise some known results on singularity categories of finite-dimensional algebras. This is a joint work with Dong Yang and Guodong Zhou.

24.04 Severin Barmeier

Title: From cluster categories to scattering amplitudes

Abstract: In this talk I will give explain how cluster categories can be viewed as a "categorification" of scattering amplitudes. Scattering amplitudes are physical observables in scattering experiments of elementary particles and the amplituhedron programme of Arkani-Hamed et al. reveals a deep connection to combinatorial structures such as positive Grassmannians and cluster algebras. From the point of view of cluster categories, the summands of the "canonical forms", whose integrals compute the scattering amplitudes, are in one-to-one correspondence with triangulations of a polygon, and thus to cluster tilting objects for cluster categories of type A quivers. I will sketch several directions in which this correspondence may be extended which are interesting both from a mathematical and from a physical viewpoint. This talk is based on joint work with Koushik Ray, Aritra Pal, Prafulla Oak and Hipolito Treffinger.

08.05 Lang Mou

Title: Quivers with relations associated to skew-symmetrizable matrices

Abstract: Skew-symmetric matrices are equivalent to 2-acyclic quivers, whose representations play an essential role in the categorification of skew-symmetric cluster algebras and the Hall algebra realization of symmetric Kac-Moody Lie algebras. In this talk, I will report on recent progress in these lines, extending from the (skew-)symmetric realm to the (skew-)symmetrizable one. This is based on joint projects with Daniel Labardini-Fragoso, Pierre-Guy Plamondon, and Xiuping Su.

15.05 Haibo Jin (14:00 - 15:00, Paris algebra seminar)

Title: A complete derived invariant and silting theory for graded gentle algebras

Abstract: We show that among the derived equivalent classes of homologically smooth and proper graded gentle algebras there is only one class whose perfect derived category does not admit silting objects.

As one application we give a sufficient and necessary condition for any homologically smooth and proper graded gentle algebra under which all pre-silting objects in its perfect derived category may be complete into silting objects.

As another application we confirm a conjecture by Lekili and Polishchuk that the geometric invariants which they construct for homologically smooth and proper graded gentle algebras are a complete derived invariant. Hence, we obtain a complete invariant for partially wrapped Fukaya categories of surfaces with stops.

This is a report on joint work with Sibylle Schroll and Zhengfang Wang.

22.05 Lang Mou

Title: Quivers with relations associated to skew-symmetrizable matrices II

Abstract: This is subsequent to my last talk on 08.05. Felikson, Shapiro and Tumarkin introduced orbifold models for a class of skew-symmetrizable cluster algebras, which form a major part of mutation-finite cluster algebras. I will explain a construction of quivers with relations associated to orbifold models and how their representations categorify cluster algebras.

05.06 Guodong Zhou

Title: An introduction to algebraic Morse theory

Abstract: We give an introduction to algebraic Morse theory and two-sided Anick resolutions following Skoeldberg et al. and illustrate this theory by concrete examples.

19.06 Maximilian Kaipel

Title: On a category associated to a partitioned fan

Abstract: The τ-cluster morphism category was first introduced by Igusa-Todorov for hereditary algebras and then extended to τ-tilting finite and all finite dimensional algebras by Buan-Marsh and Buan-Hanson, respectively. In the τ-tilting finite case, Hanson-Igusa have shown that the classifying space of this category is a cube complex whose fundamental group is the picture group associated to the algebra.

A recent construction of this category from the wall-and-chamber structure by Schroll-Tattar-Treffinger-Williams provides a simplified proof that the category is well-defined. By forgetting the associated τ-tilting theory we extend their construction to an arbitrary fan with an admissible partition of its cones and generalise the results of Hanson-Igusa to this setting.

26.06 Aran Tattar

Title: Weak stability conditions and the space of chains of torsion classes

Abstract: Joyce introduced the concept of weak stability conditions for an abelian category as a generalisation of Rudakov’s stability conditions. We show an explicit relationship between chains of torsion classes and weak stability conditions. In particular, up to a natural equivalence, they coincide. We also discuss topological properties of the space of chains of torsion classes and its quotient given by this equivalence relation.

03.07 Aran Tattar

Title: Weak stability conditions and the space of chains of torsion classes, Part 2

Abstract: Joyce introduced the concept of weak stability conditions for an abelian category as a generalisation of Rudakov’s stability conditions. We show an explicit relationship between chains of torsion classes and weak stability conditions. In particular, up to a natural equivalence, they coincide. We also discuss topological properties of the space of chains of torsion classes and its quotient given by this equivalence relation.

10.07 Ralph Kaufmann (Purdue University)

Title: Algeraic operations from surfaces with arcs or surface marked ribbon graphs

Abstract: we construct operations of surfaces with arcs on the tensor algebra of an algebra. These dualitze to operations on Hochschild chains and cochains and descend to the Tate-Hochschlid complex. In the talk, we will give examples for products, coproducts, pre Lie and double Poisson brackets. This is joint work with Manuel Rivera and Zhengfang Wang.

17.07 Zheng Xin (Qufu University)

Title: Stability and Covering Relations in Torsion Theory

Abstract: In this talk, I talk about the construction of torsion pairs by using stability functions on an exact category. Besides, the cover relations of torsion classes in the exact category are characterized by certain indecomposable object. This is my Master's thesis under the guidance of Prof. Dingguo Wang  and Dr. Tiwei Zhao.

24.07 Luigi Caputi (University of Turin)

Title: Reachability categories and commuting algebras of quivers

Abstract: In the talk, we will introduce the notion of reachability categories. These categories are obtained from path categories of quivers by taking quotients under the "reachability" relation. We will compare reachability categories to path categories, from both a topological and a categorical viewpoint. Then, we will focus on the category algebras of reachability categories, also known as commuting algebras. As application, we will prove that commuting algebras are Morita equivalent to incidence algebras of posetal reflections of reachability categories, a result previously obtained by E. L. Green and S. Schroll. This is joint work with H. Riihimäki.