Viktoriia Borovik - Graphical Scattering Equations

This lecture was part of the Thematic Programme on "Amplitudes and Algebraic Geometry" held at the ESI February 19 - March 27, 2026.

In this talk, I will introduce the graphical scattering equations — an instance of the likelihood equations in algebraic statistics and a generalisation of the CHY scattering equations in particle physics, obtained by restricting the kinematic space via a fixed simple graph. We study these equations through the lens of computational commutative algebra.

For this purpose, we define four notions of when a graph has “enough edges’’ for the corresponding scattering equations to behave well: geometric, algebraic, matroidal, and topological copiousness. A key result is that under minor assumptions all four notions are equivalent.

We classify copious graphs up to eight vertices, compute their ML degrees and the degrees of the logarithmic discriminant, and show that the ML degree is the topological Euler characteristic of a certain very-affine variety, which partially compactifies the moduli space M0,n. Our main conjecture concerns the stratification of this variety and a resulting closed formula for the ML degree.

This talk is based on joint work with Barbara Betti, Bernd Sturmfels, Bella Finkel and Bailee Zacovic.

Видео Viktoriia Borovik - Graphical Scattering Equations канала Erwin Schrödinger International Institute for Mathematics and Physics (ESI)