Positivity in Tropical Geometry and Combinatorics

Classical algebraic geometry studies solution spaces of systems of algebraic equations, which arise naturally in many areas of sciences and engineering. Although the solutions over complex numbers are better understood, the solutions over real numbers or positive numbers are often more meaningful in the contexts where the equations arise. The PI will use the modern technique of tropical geometry to study real and positive solutions of systems of polynomial equations in many unknowns. Tropical mathematics arises over the (max,plus)-algebra where addition is replaced by taking the maximum and multiplication is replaced by the usual addition. The tropical equations are often easier to solve, and some discrete features of the solution set over real or complex numbers can be computed from the solution set over the tropical numbers. This project aims at developing the tropical geometry specifically for solving equations over real numbers or positive numbers. Applications include development of new computational tools with applications in optimization. The PI will continue her work on mentoring postdocs, graduate students, and undergraduate students; organization of conferences; outreach to K-12 students; and promotion of inclusiveness and equity in the mathematical sciences. The PI will study important classes of real algebraic varieties and real semialgebraic sets using tropical geometry. These families include determinantal varieties, nonnegative and sums-of-squares polynomials, principal minors of positive semidefinite matrices, stable and Lorentzian polynomials, discriminants and resultants, and semialgebraic sets arising from positivity in polytope theory including Ehrhart theory and the theory of Minkowski weights. In particular, the PI will investigate computational problems, topology properties, and lifting problems for inequalities from tropical to classical algebraic geometry. The proposed work will promote interactions among various fields of mathematics and advance knowledge in foundations of tropical geometry, real algebraic geometry, and geometric combinatorics. The proposed problems have connections to optimization (low rank matrix completion, nonnegative and sum-of-square polynomials), computational algebra (principal minor assignment problem, discriminants and resultants), and convex geometry and polytope theory (Christoffel–Minkowski problem, weighted Ehrhart theory). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.