Comparing Ehrhart Polynomials via T-Deformations

An interesting invariant of a lattice polytope is its Ehrhart polynomial, which counts the number of lattice points in its dilations. For normal polytopes, this polynomial is equal to the Hilbert polynomial of the associated projective toric variety. In this talk, I will present a combinatorial procedure for showing that two polytopes have equal Ehrhart polynomials. This procedure is in fact the combinatorial component involved in constructing certain homogeneous deformations of projective varieties with codimension-one torus action. One application is a combinatorial proof of the fact that geometric models of trivalent binary phylogenetic trees with equal number of leaves have equal Ehrhart-Hilbert polynomials.