Applications of intersection theory: from maximum likelihood to chromatic polynomials
arXiv:2111.02057
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, Wiśniewski), 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).
arXiv comment37 pages; comments welcome!
References in corpus (10)
- Computing characteristic classes of projective schemes
- Estimating linear covariance models with numerical nonlinear algebra
- Segre Class Computation and Practical Applications
- On the birational geometry of spaces of complete forms II: skew-forms
- The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices
- Segre classes and invariants of singular varieties
- Maximum Likelihood Estimation for Nets of Conics
- Reciprocal maximum likelihood degrees of diagonal linear concentration models
- Inverting catalecticants of ternary quartics
- Proof of a conjecture of Sturmfels, Timme and Zwiernik