**Abstract:**

If \$P\$ is a lattice polytope (i.e., \$P\$ is the convex hull of finitely many integer points in \${\bf R}^d\$), Ehrhart's theorem asserts that the integer-point counting function \$L_P(m) = \#(mP \cap {\bf Z}^d)\$ is a polynomial in the integer variable \$m\$. Our goal is to study structural properties of Ehrhart polynomials—essentially asking variants of the (way too hard) question *which polynomials are Ehrhart polynomials?* Similar to the situations with other combinatorial polynomials, it is useful to express \$L_P(m)\$ in different bases. E.g., a theorem of Stanley (1980) says that \$L_P(m)\$, expressed in the polynomial basis \$\binom m d, \binom{m+1} d, \dots, \binom{m+d} d\$, has nonnegative coefficients; these coefficients form the *\$h^*\$-vector* of \$P\$. More recent work of Breuer (2012) suggests that one ought to also study \$L_P(m)\$ as expressed in the polynomial basis \$\binom{m-1} 0, \binom{m-1}1, \binom{m-1} 2, \dots\$; the coefficients in this basis form the *\$f^*\$-vector* of \$P\$. We will survey some old and new results (the latter joint work with Danai Deligeorgaki, Max Hlavaczek, and Jéronimo Valencia) about \$f^*\$- and \$h^*\$-vectors, including analogues and dissimilarities with \$f\$- and \$h\$-vectors of polytopes and polyhedral complexes.

**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**