Mirkovic-Vilonen polytopes from geometry and algebra

A famous foundational problem concerns finding combinatorial
expressions for tensor products of irreducible representations.  One
way to solve this problem is to use the combinatorics of
Mirkovic-Vilonen polytopes.  These polytopes were originally defined
as moment map images of MV cycles, but they also arise from the study
of submodules of preprojective algebra modules.  With Pierre Baumann
and Allen Knutson, we defined measures
supported on these polytopes.  These measures allow us to perform
computations which  distinguish among different bases for representations.