Counting on Toric Degenerations

How do you compute the dimension of a finite-dimensional vector space? For many vector spaces appearing in representation theory, this is a surprisingly difficult question. The answer often involves interesting combinatorics and frequently amounts to counting the number of integer-valued points inside of a convex body. Toric degeneration is a construction from algebraic geometry which provides an explanation for this phenomenon. The existence of a "polyhedral rule" for counting a quantity and combinatorial bijections between different ways of counting are related to deep questions about the geometry of certain algebraic varieties. I'll give an overview of the topic of toric degeneration, and describe its consequences for several questions in representation theory.