This project studies interactions between geometrical objects such as curves on surfaces and categories of representations. The main focus of the project is the development of a general theory of surface categories capturing properties of their geometry. The project will provide a novel combinatorial geometric approach to the study of module categories and new approaches to important problems in plane and surface geometry from an algebraic perspective. At the heart of all mathematical modelling is representation theory; and at the heart of representation theory lies quiver algebras. These are algebras defined from oriented graphs, a key notion of the proposal. A dimer model is an oriented graph embedded in a surface such that its complement is a union of disks. This data naturally associates an algebra to a dimer model, the dimer algebra, with basis all the paths in the graph (infinitely many), with multiplication arising from concatenation of paths (we work over the complex numbers) and where the relations are given by a so-called potential. The boundary algebra of a dimer model is the idempotent subalgebra of the dimer algebra with respect to boundary vertices. Sources for dimer algebras and their boundaries are abundant, most relevant for the proposal are the dimers models arising from Postnikov's strand diagrams on disks. More generally, dimer algebras arise from arbitrary tilings of surfaces. With this proposal, we want to study categories which arise from dimers, i.e. from graphs embedded on surfaces, capturing the essential properties of the surface. We call such categories surface categories. Examples for these are the Grassmannian cluster categories and the (higher) cluster categories. The main aim of the proposal is to study dimer algebras on surfaces and the boundary algebras arising from them. It is supported by five objectives.

(1) Determine boundary algebras for surfaces with punctures, for surfaces with several boundary components, and for higher genus.

(2) Explore module categories of boundary algebras and their stable parts. Study homological properties of algebras of infinite global dimension.

(3) Determine boundary algebras for infinity-gons, for surfaces with asymptotic arcs.

(4) Associate dimer algebras to rhombic tilings, study algebras for Grassmann permutations. Explore the exchange graph of Yang-Baxter moves.

(5) Explore the interactions between noncommutative resolutions, nonnoetherian geometry, and the homological properties of dimer algebras on surfaces.

Article in IMN, 2016 link

Austrian press:

We propose to study a new class of algebras which we call surface algebras. To a given surface we associate a collection of algebras, the so-called surface algebras. For our project, we have Riemann surfaces in mind, over the complex numbers, with boundary components. In recent research, algebras arising from surfaces have been studied in various instances. A class of algebras (which we call boundary algebras) arising from surfaces, with interesting connections to cluster theory, were introduced in Jensen-King-Su and subsequently studied in detail by Baur-King-Marsh. Our surface algebras provide a generalisation of these algebras. We will approach our study of surface algebras from a number of different angles. One is to study algebras associated to a surface, incorporating the geometric properties of that surface. A second approach is to closely examine the links of surface algebras to boundary algebras, thereby providing valuable insights into these algebras. A third approach is to vary the set-up and restrict the generators of the algebras we study.

The project is set in pure mathematics in the areas of representation theory of associative algebras and Lie theory. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. A representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, which is well understood. The algebraic objects that can be represented in such a way include groups, associative algebras and Lie algebras. One of the most important classes of associative algebras are group algebras. Their structure depends on the structure of the group involved. If the group is finite, then the group algebra is of finite dimension. This means that, as a vector space, this group algebra has finite dimension. Such an algebra can be written as a product of algebras that cannot be reduced any further. These indecomposable summands are called blocks. The goal of our project is to contribute to the structure theory of the blocks of group algebras of symmetric groups with non-abelian defect groups.