Recently, strong links have been uncovered between homological mirror symmetry and the representation theory of certain 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 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. Knot theory is also linked to these objects, notably via skein algebras and categorical braid group actions. This conference is based on these interactions which arise from the interplay between homological mirror symmetry, representation theory, knot theory, cluster algebras, and the geometry of polytopes.