The goal of this course is to introduce the audience to the emerging field of perverse schobers in the language of Lurie's stable infinity-categories and describe applications to the representation theory of dg-algebras.
In general, perverse schobers are a conjectural categorification of perverse sheaves, proposed by Kapranov-Schechtman. In the case of perverse schobers on surfaces with boundary, there has been much progress in developing a theory of perverse schobers, formulated in terms of constructible sheaves of enhanced triangulated categories on graphs. In the first part of the course, we will introduce this framework for the description of perverse schobers. We will see that perverse schobers offer the possibility of studying their categories of global sections via powerful local-to-global methods.
In the second part of the course, we will discuss examples of categories of global sections of perverse schober appearing in representation theory. These include famously the derived categories of (graded) gentle algebras, which are also called topological Fukaya categories. The primary focus will however be on (relative) Ginzburg algebras of surfaces. In particular, we will see how local-to-global methods allow the description (of parts) of the derived categories of these Ginzburg algberas in terms of curves in the surfaces, leading to the so called geometric models.
Eight lectures, each 2x45 minutes.
•Lecture1: Constructible sheaves on graphs and perverse sheaves 2022-2-23
•Lecture2: Crash course on stable infinity categories 2022-3-2
•Lecture3: Spherical adjunctions 2022-3-9
•Lecture4: Perverse schobers 2022-3-16
•Lecture5: Global sections of perverse schobers 2022-3-23
•Lectures6-8: relative Ginzburg algebras of triangulated or n-angulated surfaces via perverse schobers 2022-3-30&2022-4-6/13