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无处不在，问这些问题很有趣：它们有解吗？有多少?所有解的空间形状是什么？最优解是什么？答案取决于所选择的数字系统——我们考虑的是整数、有理数、实数还是复数；正数或负数；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，其中两个实数的加法被取其最大值所取代。热带方程更容易求解，令人惊讶的是，实数或复数解集的一些几何特征可以从热带数解集计算出来。本项目旨在发展实代数几何（研究实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）的热带几何。应用包括开发用于代数计算和优化的新计算工具，可用于解决统计学，经济学和工程学中出现的各种问题。作为该奖项的一部分，PI还将指导学生，并将继续她的外展努力以及她在促进数学科学的包容性和公平性方面的工作。在过去的十年中，热带几何已经发展成为组合学，特别是在矩阵理论中，以及代数几何，特别是在对数几何、解析几何和计算几何中一个强大的工具。项目将解决多面体几何、实几何和热带几何交叉的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。一些经典的问题和结构将研究使用热带几何。重点是分带体和广义复面体。半代数集的实热带化。将发展真正代数几何的基础。重要的例子来自于组合学和最优化，如有向拟阵、双曲变型和非负多项式的锥。论代数拟阵与Chow多面体的关系。所提出的研究将有助于更好地理解这些经典结构。所提出的关于多面体的工作可以应用于统计，特别是因果推理，而关于实际热带几何的工作可以应用于多项式优化。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.