# 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.