This will be an expository talk aimed at a broad audience, including beginning graduate students.
A polytope is a convex bounded polyhedron. There are many ways to measure a polytope: dimension, number of vertices, volume, number of lattice points inside the polytope, etc. A wide variety of problems in pure and applied mathematics involve measuring a polytope. For example, as we will see, even computing the dimension of a polytope can have geometric consequences for subvarieties of the flag variety. However, many of these measuring problems are very complex: there is no procedure that can efficiently measure any polytope. In this talk we will discuss these problems concentrating on families of polytopes which are of special interest due to their symmetries.
