Applications of intersection theory: from maximum likelihood to chromatic polynomials

TL;DR

This paper reviews recent applications of algebraic intersection theory across various mathematical fields, highlighting a unified approach using varieties of complete forms and Segre classes.

Contribution

It provides a comprehensive overview of how intersection theory is applied to diverse problems like chromatic polynomials and algebraic statistics, unifying these results.

Findings

Unified approach to intersection theory applications

Connections between algebraic geometry and combinatorics

Insights into degrees of semidefinite programming

Abstract

Recently, we have witnessed tremendous applications of algebraic intersection theory to branches of mathematics, that previously seemed very distant. In this article we review some of them. Our aim is to provide a unified approach to the results e.g. in the theory of chromatic polynomials (work of Adiprasito, Huh, Katz), maximum likelihood degree in algebraic statistics (Drton, Manivel, Monin, Sturmfels, Uhler, Wi\'sniewski), Euler characteristics of determinental varieties (Dimca, Papadima), characteristic numbers (Aluffi, Schubert, Vakil) and the degree of semidefinite programming (Bothmer, Nie, Ranestad, Sturmfels). Our main tools come from intersection theory on special varieties called the varieties of complete forms (De Concini, Procesi, Thaddeus) and the study of Segre classes (Laksov, Lascoux, Pragacz, Thorup).