The workshop comprises three mini-courses and some research talks.
Mini-courses (Each one consisting of 3.5 to 4 hours):
Thomas Brüstle: Stability conditions and wall-chamber structures
Aaron Chan: Gentle algebras and surface combinatorics
Nathan Reading: Scattering diagrams and cluster algebras
Thomas Brüstle: Stability conditions and wall-chamber structures
The notion of (semi-)stability has been introduced in representation theory of quivers by Schofield and King, and it was formalised in the context of abelian categories by Rudakov. The concept has re-appeared in mathematical physics as scattering diagrams, and the same wall and chamber structure is also studied in the work of Bridgeland. It seems very natural to join two recent developments, the wall-chamber structure of scattering diagrams with the combinatorial structure of the fan associated with tau-tilting modules as described by Demonet, Iyama and Jasso.
We explain in this series of lectures how the tau-tilting fan can be embedded into King's stability manifold: Each support tau-tilting pair (M,P) yields a chamber C(M,P), and one can give a complete description of the walls bordering this chamber. Moreover, we associate to each chamber C a torsion class T(C). Considering Rudakov’s stability functions on abelian length categories, we relate them to the language of torsion classes. We also plan to present some recent contributions of Asai and Iyama to the theory of torsion classes.
Aaron Chan: Gentle algebras and surface combinatorics
Gentle algebras were introduced by Assem and Skowronski in the 80s as a generalisation of the path algebra of Dynkin and affine type A. Recent advances in cluster theory and homological mirror symmetry has unvealed that these algebras, along with their homological algebra, are encoded by geometry of topological surface. More precisely, each topological surface equipped with a dual pair of dissection corresponds to a gentle algebra, and curves on this surface (roughly) correspond to complexes of representations.
We will start by introducing gentle algebras and a few other closely related families of algebras, and then explain corresponding topological data, as well as surveying the several topological and geometric models that are used in different settings. We will then explain the correspondence between representations and curves, and how they can be used to classify torsion classes using certain refinement of the so-called laminations of surfaces.
Nathan Reading: Scattering diagrams and cluster algebras
This lecture series will focus on cluster scattering diagrams, cluster algebras, and the relationship between them. Cluster algebras were invented by Fomin and Zelevinsky to study total positivity, but were soon found to have connections to a wide variety of mathematical areas. Cluster scattering diagrams were constructed by Gross, Hacking, Keel, and Kontsevich to bring the powerful tool of scattering diagrams and theta functions to bear on cluster algebras (in particular proving some key structural conjectures).
I will review the definition of cluster algebras and explain scattering diagrams and the associated fans. Narrowing the focus to cluster scattering diagrams, I will establish their relationship to mutation fans (which encode the piecewise linear geometry of matrix mutation) and explore ways in which cluster scattering diagrams give insight into cluster algebras and vice versa. I will also make the case for the importance of models (combinatorial, representation-theoretic, etc.) for cluster scattering diagrams and describe some work on combinatorial models.