Tropical Geometry: Foundations and Combinatorial Applications

热带几何是一个数学领域，它将代数几何中的问题转化为组合数学中的问题。 代数几何是研究代数簇，代数簇是由多项式方程描述的几何对象。 代数的种类包括熟悉的图形，如圆、线和抛物线，但也包括高维空间中更复杂的几何对象。 代数变种的特殊之处在于它们有很多结构。 组合数学是计数的理论，但包括离散对象的研究。 一类离散对象是多面体复合体，它是由点、线段、多边形及其高维类似物粘在一起组成的形状。 这些多面体配合物有丰富的理论，可以相当参与。 热带几何的主要技术是热带化，它以一个代数簇作为输入，输出一个多面体复形，多面体复形是该代数簇的组合影子。 这些多面体复形，称为热带簇，捕获了关于原始代数簇的惊人数量的数据。 特别是，热带品种反映了交叉理论，霍奇理论和枚举几何的原始代数品种。 热带几何在枚举几何、模空间研究、弦论中的镜像对称、数值代数计算理论、低维拓扑、数论和代数动力学等方面都有应用。

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.