Title: Quivers, brane tilings and toric Calabi-Yau/Sasaki-Einstein geometry

Abstract: In string theory, there is a remarkable conjecture that the study of quivers is in a sense "equivalent" to toric Sasaki-Einstein geometry or its cone toric Calabi-Yau geometry.
In this talk, I will report on possible mathematical formulation of this conjecture. First, I will present categorical formulation of this equivalence (joint work with Kazushi Ueda). In this discussion, bipartite graphs on two-dimensional torus, known as brane tilings, play crucial roles. I will also describe the relation of quivers and brane tilings to Sasaki-Einstein geometry, and explain the equivalence of the "a-function" of quiver with relations and the volume of Sasaki-Einstein manifold.