Applications of intersection theory: from maximum likelihood to chromatic polynomials

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).