Darius Dramburg: Are there polymer models for toric algebras?

Abstract.

A dimer model is a bipartite graph embedded on a 2-torus. This setup originated in statistical mechanics, but has far-reaching connections and applications to representation theory, algebraic geometry, and commutative algebra. In this talk, I want to give a brief overview of what Dimer models are, and then focus on the relationship with affine Gorenstein toric algebra of dimension 3. It turns out that every such toric algebra R can be obtained from a Dimer model, and the corresponding Dimer algebra A is a so-called noncommutative crepant resolution of its center R, which is the toric algebra in question. The extra noncommutativity allows the Dimer algebra to encode subtle geometric and algebraic information of the toric algebra, and I will give some examples of this. The main focus will then be on the question of how to generalise this definition to higher dimensions, so that we can study Gorenstein toric algebras of arbitrary dimension. While we have no general answer to this question, recent joint work with Oleksandra Gasanova points towards a generalisation in case R is a quotient singularity. I want to explain the precise challenges we face, and suggest some future work to extend this to arbitrary Gorenstein toric singularities.