  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).
