Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

Tangent Quadrics in Real 3-Space

We examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of polynomial equations, also in the space of complete quadrics, and we solve them using certified numerical methods. Our aim is to show that Schubert's problems are fully real.

The degree of the central curve in semidefinite, linear, and quadratic programming

The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation to the complexity of these interior point algorithms, and for linear programs it was computed by De Loera, Sturmfels, and Vinzant in 2012. We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models. New results from the intersection theory of the space of complete qu…

minuscule量子K理论的正性

Positivity of minuscule quantum K-theory

本文证明了minuscule旗 manifold 或二次超曲面的量子K理论环的Schubert结构常数具有交替于余维数的符号，并探讨了cominuscule旗 manifold的量子cohomology或量子K理论环中Schubert类乘积的q的幂次区间。

Wronski 映射的通用 Plücker 坐标与实 Schubert 计算中的正性

Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus

本文通过对称群代数的显式算子公式，解决了给定 Wronskian 的逆 Wronski 问题，并证明实非负零点下 Plücker 坐标的非负性，验证了实 Schubert 计算中的多个正性猜想。

Applications of intersection theory: from maximum likelihood to chromatic polynomials

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, Ni…