Algebra and geometry seminar 30-04-2021

[Francesco Galuppi]

Dear colleagues,

this is to advertise the weekly seminar of the Algebra and geometry 
group of University of Trieste. On Friday, April 30th at 10:30, Kathlén 
Kohn (KTH Stockholm) will talk about "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.

The seminar will take place online. The Zoom link is 
https://us02web.zoom.us/j/87239350702.

You are welcome to extend the invitation to anyone interested.