2020 Thematic Program (I)

Surfaces and representation theory


January 05-10, 2020

W303  School of Mathematics, Sichuan University

[minicourse III]0104-09Sibylle Schroll-01.png


Sibylle Schroll (U Leicester, UK)


We will begin with an overview of the advent of surfaces in representation theory in recent years, focussing on algebras and categories in the context of cluster algebras. We introduce Jacobian algebras of quivers arising from triangulations of marked bounded oriented surfaces, and their representation theory in terms of the surface. In particular, if the surface has no marked points in its interior, the Jacobian algebra associated to any triangulation is gentle and much information about the algebra is known in terms of the surface. In the second part of the course, we will focus on gentle algebras in general. This well-studied class of algebras has recently received renewed interest in particular in connection with Fukaya categories of surfaces with boundaries and stops in the work of Haiden, Katzarkov and Kontsevich. We will give a geometric model for the bounded derived category of a gentle algebra and in the case of finite global dimension, we will link it with the partially wrapped Fukaya categories in Haiden, Katzarkov and Kontsevich’s work. Finally, we give a complete invariant in terms of the surface to distinguish between derived equivalence classes of gentle algebras.   

1. Cluster algebras and their classification 

2. Jacobian algebras of quiver with potential from triangulations of surfaces and cluster categories 

3. Gentle algebras  

4. Derived model for gentle algebras and link with partially wrapped Fukaya categories of surfaces with stops 

5. Winding numbers, Arf invariants and derived invariants