Kathlén Kohn (UiO): The adjoint of a polytope
Abstract: This talk brings many areas together: discrete geometry, statistics, algebraic geometry, and geometric modeling.
First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes.
Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a simplicial polytope.
Finally, we prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. In particular, we see that some polytopes can be deformed to singular Calabi-Yau hypersurfaces such that the adjoints of these polytopes are special cases of the classical notion of the unique adjoint associated to a singular Calabi-Yau.
This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.