Day 1 (April 14) @ 13:30 – 15:00 Subroom 2
Title: The adjoint of a polytope
This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, physics, and geometric modeling. First, we recall the definitions of the adjoint of a polytope given Wachspress in 1975 and by Warren in 1996 in the context of geometric modeling. They defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics (as the numerator of a generating function over all moments of the uniform probability distribution on a polytope), in intersection theory (as the central piece in Segre classes of monomial schemes), and in physics (when studying scattering amplitudes). Thirdly, we show that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either $K3$- or elliptic surfaces.
This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.
Title: to be announced
Elliptic curves and L-functions
Both the Birch-Swinnerton-Dyer conjecture and the Shafarevich-Tate conjecture provide ways of studying rational points on elliptic curves. Curiously, some basic properties of L-functions translate into out-of-reach statements about rational points, and vice versa. I will discuss several unexplained consequences that rational points, Selmer groups and L-functions imply about each other.