Tropical geometry and Grobner theory for spherical homogeneous spaces

Grobner theory and tropical geometry discuss ideals in the (Laurent) polynomial algebra (or subvarieties in the affine space). In these theories one assigns combinatorial/convex geometric objects, namely polyhedral cones, to the ideals which encode information about geometry and algebra. We discuss how to extend these ideas to much larger class of spherical homogeneous spaces. These include many familiar examples such as subvarieties in a reductive group itself. We will see that, in particular examples, these become connected with Singular Values Decomposition and Smith Normal Form of matrices. This is a work in progress with Chris Manon.