Combinatorics and Commutative Algebra

I study lattice polytopes. I'm particularly interested in the Ehrhart polynomial of a lattice polytope, which is the enumerative function on the number of lattice points contained in a lattice polytope. For a lattice polygon P, we can compute the area of P by counting the lattice points contained in P, and this is known as Pick's formula. The theory of Ehrhart polynomials of lattice polytopes can be understood as a higher-dimensional analogue of Pick's formula.

Moreover, I'm also working on lattice polytopes appearing in the context of commutative ring theory and algebraic geometry. For example, for a toric ring, which is a kind of graded commutative algebra arising from a lattice polytope, or a toric Fano variety corresponding to a Fano polytope (a kind of lattice polytopes), I try to obtain some properties of the original algebraic objects via combinatorial discussions on the corresponding lattice polytopes. Especially, I'm currently interested in "mutation of Fano polytopes". This is the new combinatorial concept derived from mirror symmetry for Fano manifolds.