Dimer algebras on surfaces

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 drawn on a surface such that its edge are crossing-free. The complement of the dimer model is a union of disks. To a dimer model, we can associated its dimer algebra by taking all the possible paths as a basis. If we only consider paths between vertices on the boundary, we obtain its so-called boundary algebra. The latter have been used to provide a combinatorial approach to certain 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.

The project focused on uncovering the structure of special mathematical objects called cluster algebras and dimer algebras. Cluster algebras originated about 20 years ago from looking at certain sequences of integers where miracles appear to occur, and generalize a remarkable relation of which the Pythagorean theorem is a special case. For example, consider the sequence of fractions s_1, s_2, s_3, ... defined recursively by the relation s_{n-1} s_{n+1} = s_n + 1, that is, s_{n+1} = (s_n + 1)/s_{n-1}. If we start with the values s_1 = s_2 = 1, then the sequence is 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, ... Amazingly, the sequence is periodic, and all the numbers in the sequence are integers (that is, whole numbers) -- no fractions appear! This is very surprising, since the sequence was defined using fractions. This sequence is known as the Pentagon recurrence, and cluster algebras were used to prove that this sequence, as well as many other similarly defined sequences, only consist of integers, and to characterize when such a sequence is periodic. Cluster algebras also generalize what is called "Ptolemy's theorem". Consider a quadrilateral (a 4-sided polygon) whose corners all lie on a circle. Label the four corners a, b, c, d, clockwise around the circle, and let ab be the length of the line segment from a to b (so either a side of the quadrilateral or a diagonal), and similarly for the other corners. Then Ptolemy's theorem says that ac x bd = ab x cd + bc x ad. In the special case where the quadrilateral is a rectangle, we get the Pythagorean theorem! Cluster algebras generalize this, by looking at quadrilaterals and their diagonals on surfaces such as a disc, an annulus, a donut, or a donut with many holes. In our project, we discovered a range of structural properties of geometric spaces called Grassmannians in the context of cluster algebras and their associated categories. A "dimer algebra" is a mathematical object constructed from an arrangement of arrows on a surface, such that the arrows form oriented polygons that cover the surface. These objects originated in string theory around 2005, and served as toy models to study the geometry of the extra six curled up dimensions of spacetime. They play an important role in the study of cluster algebras. In the project, we discovered new properties of dimer algebras, and generalized them to surfaces with any number of holes, using special geometric spaces that look like M. C. Escher's art work "Angels and Devils". In particular, we found intricate structures in their representation theory that are related, in unexpected ways, to certain properties of the surface.