Tropical Geometry

热带几何是代数几何的一个新兴领域，其中一个品种是研究通过其组合阴影，被称为热带品种。最基本的是，热带几何是热带半环上的几何，其中乘法被加法取代，加法被最小值取代。热带多项式因此是分段线性函数：3x^2+2y^2变成最小（2x+3,2y+2）。一个（复仿射）代数变量是一组多项式方程的复数公解的集合。热带几何将一个品种变成多面体扇形，称为热带品种，是一个组合对象。该项目的总体目标是确定一个品种的哪些不变量可以从其热带品种计算出来。特别是，该项目的第一个目标是确定何时可以通过热带方法确定一个品种的净锥和有效锥。这些锥体是一个重要的不变量，来自于两族几何。该项目的第二个目标是了解何时可以从该品种中确定该品种的周环或上同环。最终目标是将这种理解应用于稳定地图的热带空间，通过将这些空间实现为原始空间的热带化。

Tropical geometry is an emerging area of algebraic geometry in which a variety is studied via its combinatorial shadow, known as the tropical variety. At its most basic, tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. Tropical polynomials are thus piecewise linear functions: 3x^2+2y^2 becomes min(2x+3,2y+2).A (complex affine) algebraic variety is the set of common solutions in the complex numbers to a set of polynomial equations. Tropical geometry turns a variety into a polyhedral fan, called the tropical variety, which is a combinatorial object.The overarching aim of this project is to determine which invariants of a variety can be computed from its tropical variety. In particular, the first goal of the project is to determine when the nef and effective cones of a variety can be determined via tropical methods. These cones are an important invariant of the variety coming from birational geometry. The second goal of the project is to understand when the Chow or cohomology rings of the variety can be determined from the variety. The final goal is to apply this understanding to the tropical space of stable maps, by realizing these spaces as tropicalizations of the original spaces.