Polytopes and Real Tropical Geometry

多项式（代数）方程和不等式在数学、科学和工程的所有领域都是普遍存在的，在这些领域中，有趣的问题是：是否存在解？有几个？所有解的空间的形状是什么？什么是最佳解决方案？ 答案取决于所选择的数字系统-无论我们考虑整数、有理数、真实的数还是复数;正数还是负数;等等。热带几何是通过考虑热带（最大倍）代数上的代数方程和不等式而产生的，其中两个真实的数的相加被取它们的最大值所取代。 热带方程更容易求解，令人惊讶的是，真实的或复数上的解集的一些几何特征可以从热带数上的解集计算出来。 该项目旨在为真实的代数几何（研究真实的数上的代数方程和不等式的几何）和多面体理论（研究线性方程和不等式的离散性质）发展热带几何。 应用程序包括代数计算和优化的新计算工具的开发，可用于解决统计，经济和工程中出现的各种问题。作为该奖项的一部分，PI还将指导学生，并将继续她的外联工作以及她在促进数学科学的包容性和公平方面的工作。在过去的十年热带几何已成长为一个强大的工具，在组合学，特别是在拟阵理论，并在代数几何，特别是在日志几何，解析几何和计算几何。 该项目将在多面体，真实的和热带几何的交叉点解决三个研究方向的组合问题，重点是组合理论。关于多面体的变形。一些经典的问题和建设将研究使用热带几何。重点是地带性和广义排列面。关于半代数集的真实的热带化。真实的代数几何的基础将得到发展。 重要的例子来自组合学和优化，如定向拟阵，双曲品种，和锥的非负多项式。3.关于代数拟阵与Chow多面体的关系。拟议的研究将使这些经典的constructions.The拟议的工作多面体在统计学中的应用，特别是因果推理，和工作的真实的热带几何在多项式optimizations.This奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.