Tropical Combinatorics and Applications

该项目旨在研究热带组合学的基本发展及其应用。组合学是一门艺术和科学，它将复杂的数学结构提炼成简单的属性，并由此发展出对原始结构的更深层次的理解。热带数学以圣保罗的巴西数学家伊姆雷·西蒙的名字命名，位于摩羯座的北回归线。热带数学是一个年轻的数学领域，它源于一个数字系统，它带有一对运算：加法和最大值，而不是更传统的加法和乘法对。热带数学在镜像对称性、计算代数、最优化、统计学、系统发育和经济学等领域有着广泛的应用。代数方程和不等式的解是代数几何的主要研究对象；同样，热带代数方程和不等式的解也是热带几何的主要参与者。在代数几何物体和它们的热带对应物体之间存在着精确的关系，因此热带数学也为经典问题提供了新的见解。它还为理解不同数学实体之间深刻的、有时令人惊讶的联系提供了一种自然语言和新的视角。热带几何的研究对象主要是热带凸集和热带代数集，它们可以类似于通常的凸集和代数集来构造，但使用的是极大代数。研究人员将研究与一个代数集有关的各种组合结构之间的关系，即它的热带品种、它的代数拟阵和它的Chow多面体。她将发展Ehrhart理论，用于列举热带多面体中的格点。在另一个方向，研究人员将使用热带观点研究组合学和凸几何中的经典对象。她将继续开发一种基于热带几何的多边形和三角剖分方法，并将研究正半定矩阵的锥形和仿射建筑之间的关系。在计算方面，将利用数值代数几何开发计算热带品种的新算法，由于其高度并行化的性质和数值计算的进步，数值代数几何已成为代数几何中最强大的工具之一。这些新方法对于科学和工程数学模型中普遍存在的多项式系统的问题将是有用的。具体应用包括在产品组合拍卖中寻找竞争性均衡价格，以及发现概率中随机变量之间的因果关系。

This project aims at fundamental development of tropical combinatorics as well as its applications. Combinatorics is the art and science of distilling a complex mathematical structure into simple attributes and developing from this a deeper understanding of the original structure. Tropical mathematics, named after Brazilian mathematician Imre Simon of Sao Paulo on the Tropic of Capricorn, is a young area of mathematics arising from a number system with a pair operations: addition and maximum, instead of the more traditional pair of addition and multiplication. Tropical mathematics has applications in a wide range of areas including mirror symmetry, computational algebra, optimization, statistics, phylogenetics, and economics. The solutions of algebraic equations and inequalities are the main objects of interest in algebraic geometry; likewise the solutions of tropical algebraic equations and inequalities are the main players in tropical geometry. There is a precise relationship between algebro-geometric objects and their tropical counterparts, so tropical mathematics provides new insights into classical problems as well. It also provides a natural language and a new perspective for understanding deep and sometimes surprising connections between different mathematical entities. This project will make contributions to the foundation of tropical geometry, focusing on the combinatorial structures.The main geometric objects of study in this field are tropically convex sets and tropical algebraic sets, which can be constructed analogously to the usual convex sets and algebraic sets, but using the max-plus algebra. The investigator will study relationships between various combinatorial structures associated to an algebraic set, namely, its tropical variety, its algebraic matroid, and its Chow polytope. She will develop Ehrhart theory for enumeration of lattice points in tropical polytopes. In the other direction, the investigator will use tropical point of view to study classical objects in combinatorics and convex geometry. She will continue developing an approach to polytopes and triangulations based on tropical geometry and will investigate the relationship between the cone of positive semidefinite matrices and affine buildings. On the computational front, new algorithms for computing tropical varieties will be developed using numerical algebraic geometry, which has become one of the most powerful tools in algebraic geometry due to its highly parallelizable nature and advancements in numerical computations. These new methods will be useful for problems concerning polynomial systems which are ubiquitous in mathematical models in science and engineering. Specific applications include finding competitive equilibrium prices in product-mix auctions and discovering causal relationships between random variables in probability.