Polytopes and Real Tropical Geometry

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Polynomial (algebraic) equations and inequalities are ubiquitous in all areas of mathematics, sciences, and engineering, where it is interesting to ask: Are there any solutions? How many? What is the shape of the space of all solutions? What is the optimal solution? The answers depend on the chosen number system --- whether we consider whole, rational, real, or complex numbers; positive or negative numbers; and so on. Tropical geometry arises by considering algebraic equations and inequalities over the tropical (max-times) algebra where addition of two real numbers is replaced by taking their maximum. The tropical equations are easier to solve, and surprisingly, some geometric 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 for real algebraic geometry (which studies the geometry of algebraic equations and inequalities over real numbers) and polytope theory (which studies discrete properties of linear equations and inequalities). Applications include development of new computational tools for algebraic computations and optimization, which can be used to solve a variety of problems arising from statistics, economics, and engineering. As part of this award the PI will also mentor students, and will continue her outreach efforts as well as her work in promoting inclusiveness and equity in the mathematical sciences. During the last decade tropical geometry has grown into a powerful tool in combinatorics, particularly in matroid theory, and in algebraic geometry, particularly in log geometry, analytic geometry, and computational geometry. The project will solve combinatorial problems in three research directions at the intersection of polyhedral, real, and tropical geometry, with an emphasis on combinatorial theory. 1. On deformations of polytopes. Some classical problems and constructions will be studied using tropical geometry. The focus is on zonotopes and generalized permutohedra. 2. On real tropicalizations of semialgebraic sets. Foundations of real algebraic geometry will be developed. Important examples arise from combinatorics and optimization, such as oriented matroids, hyperbolic varieties, and the cone of non-negative polynomials. 3. On the relationship between algebraic matroids and Chow polytopes. The proposed research will give a better understanding of these classical constructions. The proposed work on polytopes has applications in statistics, in particular causal inference, and the work on real tropical geometry has applications in polynomial optimization. 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.