The lecture notes can be found at https://www.math.uni-hamburg.de/home/christ/schobers.pdf
After the lectures, there is the option to meet in gather town for an informal discussion.
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.
•Lecture 1: Constructible sheaves on graphs and perverse sheaves 2022-2-23 https://cloud.tsinghua.edu.cn/f/4c321d2f00024a83b02a/
•Lecture 2: Crash course on stable infinity categories 2022-3-2 https://cloud.tsinghua.edu.cn/f/6a500a5d841242c7befb/
•Lecture 3: Spherical adjunctions 2022-3-9 https://cloud.tsinghua.edu.cn/f/cc30d7471c164a69a9b7/
•Lecture 4: Perverse schobers 2022-3-16 https://cloud.tsinghua.edu.cn/f/917b03d6a85748839eb1/
•Lecture 5: Global sections of perverse schobers 2022-3-23 https://cloud.tsinghua.edu.cn/f/66ed7d896fc042d2aeaa/
•Lecture 6: Relative Ginzburg algebras 2022-3-30 https://cloud.tsinghua.edu.cn/f/304ab7fc28084553b9a0/
•Lecture 7: Geometric models, part I 2022-4-6 https://cloud.tsinghua.edu.cn/f/c52413d7e9f34554be96/
•Lecture 8: Geometric models, part II 2022-4-13 https://cloud.tsinghua.edu.cn/f/94879a83185045cd8156/
Required: Familiarity with the notion of triangulated category. Basic familiarity with the language of simplicial sets and very basic familiarity with infinity-categories.
Helpful: Knowledge about dg-categories and their model structures.
J. Lurie.HigherToposTheory. Annals of Mathematics Studies, vol. 170, Princeton Uni- versity Press, Princeton, NJ. MR 2522659, 2009.
J. Lurie.Higher Algebra. preprint,availableon the author’swebsite, 2017.
M. Kapranov and V.Schechtman.Perverse Schobers.arXiv:1411.2772, 2014.
T. Dyckerhoff, M. Kapranov, V.Schechtman,and Y. Soibelman. Spherical adjunctions of stable∞-categories and the relative S-construction.arXiv:2106.02873, 2021.
M. Christ. Spherical monadic adjunctions of stable infinity categories.arXiv:2010.05294, 2020.
M. Kapranov andV.Schechtmann.Perverse sheaves and graphs on surfaces.arXiv:1601.01789, 2016.
M.Christ. Ginzburg algebras of triangulated surfaces and perverseschobers.arXiv:2101.01939, 2021.
S. Opper,P.Plamondon, and S. Schroll. A geometric model of the derived category of a gentle algebra.arXiv:1801:09659, 2018.
Y. Qiu. Decorated marked surfaces (part B): topological realizations.Math. Z. 288,39–53., 2018.
Y. Qiu and Y. Zhou. Decorated marked surfaces II: Intersection numbers and dimensions of Homs.Trans.Amer.Math.Soc.372, 635–660, 2019.
M. Christ. Geometric models for Ginzburg algebras ofn-angulated surfaces via local-to- global principles.arXiv:2107.10091, 2021.