In the theory of toric varieties, the combinatorics of a convex integral polytope $\Delta$ is intimately linked with the geometry of an associated toric variety $X(\Delta)$; indeed, the polytope $\Delta$ fully encodes the geometry of $X(\Delta)$ in this case. In more general situations arising in geometry, one can often associate combinatorial data to a group action on a manifold, but usually the combinatorics doesn't completely encode the original geometric data. Recently, however, Kaveh-Khovanskii and Lazarsfeld-Mustata generalized the theory of toric varieties by associating a combinatorial object to an algebraic variety in a very general setting.

This talk will be an introduction to this relatively recent theory, with a particular focus on its relationships with symplectic geometry and representation theory.