Abstracts and References

Abstracts of Research Talks

Categorical action of the braid group of the cylinder (Agnès Gadbled - Université Paris-Saclay, Orsay)

Abstract: Using the various definitions of the classical braid group (diagrammatic presentation, mapping class group, fundamental group of configuration space), Khovanov and Seidel constructed in their seminal article of 2000 an action of the braid group on a category of algebraic nature that categorifies the Burau representation. They proved the faithfulness of this action through the study of curves in a punctured disk (while Burau representation is not faithful for braids with five strands or more). In an article with Anne-Laure Thiel and Emmanuel Wagner, we extended this result to the braid group of the cylinder. The work of Khovanov and Seidel also had a faithful symplectic categorical action that we generalise to our cylindrical case in a joint work in progress. In this talk, I will explain some ingredients and strategy involved to obtain these categorical actions and faithfulness results.

Fukaya categories of symmetric products of disks (Gustavo Jasso- University of Bonn)

Abstract: Motivated by classical results and conjectures in representation theory of algebras, around 2004, Iyama introduced what nowadays is called higher Auslander–Reiten theory. On the other hand, motivated by the bordered Heegaard Floer homology of Lipshitz, Ozsváth and Thurston, around 2010, Auroux introduced what nowadays are called partially wrapped Fukaya categories associated to symmetric products of Riemann surfaces. In this talk, I will explain how these two subjects relate to each other in the simplest possible instance of each: the higher Auslander algebras of type A on the one hand, and symmetric products of disks on the other hand. This is a report of joint work with Dyckerhoff and Lekili.

A geometric model of the derived category of skew-gentle algebras (Yadira Valdivieso - University of Leicester)

Geometric models of certain kinds of categories have been helpful to link several branches of mathematics. For example, the geometric model of the derived categories of the well-known gentle algebras provides a nice like between representation theory of algebras and symplectic topology. In recent work, we provided a geometric model of the objects of the bounded derived category of a skew-gentle algebra (a natural generalisation of gentle algebras) in the form of graded curves in an orbifold dissection with orbifold points of order two with boundary and punctures.

In this talk, we show that this model also provides a nice interpretation of morphisms between complexes. More precisely, we show that intersections of graded curves in orbifolds dissections induce maps between their respective associated complex. We also show that not every map between two complexes arises of this form. This is a work in progress.

References