Tropical Combinatorics and Applications

该项目旨在热带组合学及其应用的基础发展。 组合数学是一门将复杂的数学结构提炼成简单属性，并由此对原始结构有更深入理解的艺术和科学。 热带数学（英语：Tropical mathematics），以位于南回归线上的巴西数学家、圣保罗的伊姆雷·西蒙（英语：Imre Simon）的名字命名，是一个年轻的数学领域，起源于一个具有成对运算的数字系统：加法和最大值，而不是更传统的加法和乘法。 热带数学在许多领域都有应用，包括镜像对称、计算代数、最优化、统计学、遗传学和经济学。代数方程和不等式的解是代数几何中感兴趣的主要对象;同样，热带代数方程和不等式的解是热带几何中的主要角色。 代数几何对象和它们的热带对应物之间存在精确的关系，因此热带数学也为经典问题提供了新的见解。 它还提供了一种自然语言和一种新的视角，用于理解不同数学实体之间深刻的，有时令人惊讶的联系。 本计画将对热带几何的基础作出贡献，著重于组合结构，此领域的主要几何研究对象是热带凸集与热带代数集，它们可以类似于通常的凸集与代数集，但使用极大代数来建构。研究者将研究与代数集相关的各种组合结构之间的关系，即其热带变种，代数拟阵和Chow多面体。 她将发展Ehrhart理论在热带多面体的格点枚举。在另一个方向，研究者将使用热带的观点来研究组合数学和凸几何中的经典对象。她将继续开发一种方法，以多面体和三角形的基础上，热带几何，并将调查锥的半正定矩阵和仿射建筑物之间的关系。在计算方面，将使用数值代数几何来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。这些新方法对于解决科学与工程中数学模型中普遍存在的多项式系统问题具有重要意义。 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.