Clusters, Homological Algebra, Representations and Mirror Symmetry – CHARMS

In his address at the ICM in 1994, Maxim Kontsevich stated his Homological Mirror Symmetry Conjecture. This conjecture relates two categories, one from symplectic geometry (the Fukaya category), the other from algebraic geometry (the category of coherent sheaves), via an equivalence of suitably defined derived categories. The conjecture remains wide open to this day.

Recently, strong links have been uncovered between homological mirror symmetry and the representation theory of certain classes of associative algebras, called "gentle algebras". This relationship opens new paths to attack long-standing problems in both worlds, allowing the application of well-understood representation theory to the study of Fukaya categories (such as in the search for good generators of the category), and that of geometric tools to study the homological properties of associative algebras (such as the search for numerical derived invariants).

In addition to their links with homological mirror symmetry, the class of gentle algebras enjoys a deep relationship with the world of combinatorics and polyhedral geometry. The homological and representation-theoretic properties of these algebras naturally lead to the study of combinatorial objects, such as words on graphs, and geometric ones, such as polyhedra and fans. These objects are closely related to those appearing in the very active field of cluster algebras.

This project aims at exploiting interactions between homological mirror symmetry, representation theory of quivers, combinatorial models and polyhedral geometry, brought to light in part by the theory of cluster algebras. The vast knowledge on cluster algebras acquired in the past twenty years opens new directions of research in each of these fields. In the achievement of its objectives, the project will bring together leading experts in the numerous and varied fields involved.

More specifically, the aims of the project are the following:
- Understand the Fukaya category of certain surfaces using the representation theory of associative algebras. Apply this understanding, among other things, to obtain derived invariants of algebras, to study the Fukaya category of a surface in the non-homologically smooth case and to obtain categorical representations of braid groups.
- Push the boundaries of knowledge on the representation theory of gentle algebras and associative algebras in general. In particular, develop tau-tilting theory for infinite-dimensional gentle algebras and more general classes of infinite-dimensional algebras and describe the derived categories of such algebras using cominatorial models.
- Describe and study the combinatorial objects related to gentle algebras and cluster algebras. In particular, study their type cones, scattering diagrams, cluster complexes and g-vector fans, and construct polyhedral realisations in finite or infinite types.