Toric varieties are defined abstractly as algebraic varieties with the action of an open dense torus; in practice, they arise from the very concrete combinatorial data of a collection of cones. The utility of toric varieties lies in the fact that many familiar varieties are toric, and many basic operations and definitions can be recast in terms of the associated combinatorial data. Thus toric varieties give a useful class of examples to elucidate more abstract, complex theories.

In this talk, we'll examine basic definitions and examples of toric varieties, and cover the general correspondences between the geometric, algebraic, and combinatorial aspects of the theory. We'll then focus on toric resolution of singularities as an example of how the special setting of toric varieties can give useful examples for general theory. In particular, we'll describe the types of singularities that can occur on an affine toric variety, and show how to resolve them via the associated combinatorial data. We'll focus on surface singularities, but time permitting may sketch the general approach in higher dimensions. Speaker(s): Devlin Mallory (UM)