Kathlén Kohn: The adjoint polynomial of a polytope

Time: Wed 2019-11-06 10.15 - 11.00

Location: 3418

Abstract:
I will present open problems about polytopes that are relevant in
several areas such as intersection theory, statistics, and geometric
modeling.
The adjoint polynomial of a polygon was first defined by Wachspress in
1975 to generalize barycentric coordinates from triangles to arbitrary
polygons. This was generalized by Warren in 1996: he defined the adjoint
polynomial and barycentric coordinates for arbitrary polytopes. The
adjoint also has an intrinsic meaning from the point of view of
algebraic geometry: it is the unique hypersurface of minimal degree
which vanishes on the non-faces of a simple polytope. Moroever, the
adjoint appears both in statistics (as the numerator of a generating
function over all moments of the uniform probability distribution on a
polytope) and intersection theory (as the central factor in Segre
classes of monomial schemes). In the latter context, the adjoint
polynomial has positive integer cooefficients and it is an open problem
posed by Paolo Aluffi to understand the combinatorial meaning of these
numbers.
This talk is based on joint works with Kristian Ranestad, Boris Shapiro
and Bernd Sturmfels.