Potential research problems include:
- Investigate (weighted) Ehrhart theory, which is about counting the number of integer points in dilates of integral polytopes.  For a fixed polytope, this lattice point count is known to be a polynomial function in the dilation factor, and its generating function is a rational function.  Explore properties of these polynomials for various weights and families of polytopes.
- Investigate how the Ehrhart polynomials change as polytopes are deformed, either by parallel translations of facets or as Minkowski sums of other polytopes.
- Investigate deformations and realization spaces of polytopes.   For a polytope, what is the degree of freedom in deforming it while preserving the edge directions?  In other words, which collections of edge lengths are realizable on the same polytope?  Similarly, which face volumes are realizable?