Tropical Geometry: Foundations and Combinatorial Applications

热带几何是将代数几何问题转化为组合几何问题的数学领域。代数几何是研究代数变量的学科，代数变量是用多项式方程描述的几何对象。代数变体包括熟悉的图形，如圆、线和抛物线，但也包括高维空间中更复杂的几何对象。代数变量的特殊之处在于它们有很多结构。组合学是计数理论，但也包括对离散对象的研究。一类离散对象是多面体复合体，它是由点、线段、多边形和它们的高维类似物粘合在一起组成的形状。这些多面体配合物有丰富的理论，可以相当复杂。热带几何的主要技术是热带化，它以一个代数变量作为输入，输出一个多面体复合体，即该变量的组合阴影。这些被称为热带变种的多面体复合物捕获了大量关于原始代数变种的数据。特别是热带品种反映了原始代数品种的交集理论、霍奇理论和枚举几何。热带几何应用于枚举几何、模空间研究、弦理论中的镜像对称、数值代数计算理论、低维拓扑、数论和代数动力学。

Tropical geometry is an area of mathematics that translates problems in algebraic geometry to problems in combinatorics. Algebraic geometry is the study of algebraic varieties which are geometric objects that are described by polynomial equations. Algebraic varieties include familiar figures like circles, lines, and parabolas but also more complicated geometric objects in high dimensional spaces. Algebraic varieties are special in that they have a lot of structure. Combinatorics is the theory of counting but includes the study of discrete objects. One class of discrete objects is that of polyhedral complexes which are shapes made up of points, line segments, polygons, and their higher-dimensional analogues glued together. These polyhedral complexes have a rich theory and can be quite involved. The main technique of tropical geometry is tropicalization which takes as input an algebraic variety and outputs a polyhedral complex which is a combinatorial shadow of the variety. These polyhedral complexes, called tropical varieties capture a surprising amount of data about the original algebraic varieties. In particular, the tropical variety reflect the intersection theory, Hodge theory, and enumerative geometry of the original algebraic variety. Tropical geometry has applications to enumerative geometry, the study of moduli spaces, mirror symmetry in string theory, the theory of numerical algebraic computations, low dimensional topology, number theory, and algebraic dynamics.