Interactions between homotopy theory and representation theory

• Gustavo Jasso (University of Cologne)

Title: The Derived Auslander-Iyama Correspondence
Abstract: In this lecture series we will study the general question of when the quasi-isomorphism type of a differential graded algebra A is uniquely determined by its graded cohomology algebra H*(A) plus a 'minimal' amount of additional data. We will focus on the case where A enjoys a strong regularity property as a generator of its perfect derived category, the so-called dZ-cluster tilting property. We will also study the case where, in addition, the differential graded algebra A is bimodule right Calabi-Yau.

In slightly more detail, the aim of the lecture series is to explain how techniques of homotopy theory can be leveraged to prove certain classification results of interest in representation theory---the Derived Auslander--Iyama Correspondence of the title and its bimodule Calabi--Yau variant---that seem to be out of reach by other methods. If time permits, we will also explain, following Keller, an application of these results to the (first) affirmative solution of the Donovan--Wemyss Conjecture in the context of Wemyss' Homological Minimal Model Programme in birrational geometry.

This lecture series is based on an ongoing project with Fernando Muro (Sevilla).

• Merlin Christ (Université Paris-Cité)

Title: A gentle introduction to sheaves of stable ∞-categories
Abstract: The derived category of a gentle algebra describes the partially wrapped Fukaya category of a marked surface with boundary. Further, this category can be described as the stable ∞-category of global sections of a constructible sheaf of stable ∞-categories. This constructible sheaf is defined on any choice of graph homotopy to the surface (the graph is sometimes called the skeleton of the surface). We will see how the abstract description as global sections, meaning the limit of a diagram of stable ∞-categories, can be used for concrete computations. To turn the universal property into what is essentially combinatorics, we will recall and employ the description of the limit via so-called coCartesian sections of the Grothendieck construction.

The sheaf theory perspective will naturally give rise to well known statements from the representation theory of gentle algebras. For instance, we will construct global sections (=objects in the derived category) from curves in the surface via the gluing of local sections corresponding to curve segments. Furthermore, some of the proofs will directly generalize from the base field to any base ring spectrum.
The setting of gentle algebras gives a particular accessible introduction to a circle of ideas. Other classes of examples were similar methods apply include many triangulated categories arising from surfaces and Fukaya-type categories. Finally, we remark that the cosheaf properties of topological Fukaya categories/derived categories of graded gentle algebras are originally due to Dyckerhoff-Kapranov ('15,'18) and Haiden-Katzarkov-Kontsevich ('17).

• Sira Helena Gratz (Aarhus University)

Title: Cluster categories and thick subcategories
Abstract: TBA

To be announced.

The registration will open later.

• Henrik Granau Holm <holm@math.ku.dk>
• Robert Szafarczyk <rs@math.ku.dk>