**Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.**

**Join Zoom Meeting: https://washington.zoom.us/j/ 91547335974Meeting ID: 915 4733 5974**

The Ehrhart quasipolynomial of a rational polytope \$P\$ encodes the number of integer lattice points in dilates of \$P\$, and the \$h^*\$ -polynomial of \$P\$ is the numerator of the accompanying generating function. We provide two decomposition formulas for the \$h^*\$-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke--McMullen formula to provide a novel proof of Stanley's Monotonicity Theorem for the \$h^*\$-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the \$h^*\$-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes. This is joint work with Matthias Beck (San Francisco State Univ. & FU Berlin) and Ben Braun (Univ. of Kentucky).