Tropical Geometry

热带几何是代数几何中的一个新兴领域，通过它的组合阴影来研究一个簇，也就是所谓的热带簇。最基本的热带几何是热带半环上的几何，其中乘法用加法代替，加法用最小代替。因此，热带多项式是分段线性函数：3x^2+2y^2变成min(2x+3，2y+2)。(复仿射)代数族是一组多项式方程的复数公共解的集合。热带几何将一个变种转化为一个多面体扇形，称为热带变种，这是一个组合对象。这个项目的主要目标是确定一个变种的哪些不变量可以从它的热带变种计算出来。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球果和有效球果。这些锥体是来自二次几何的变化的一个重要不变量。该项目的第二个目标是了解何时可以从品种中确定该品种的Chow或上同源环。最终目标是将这种理解应用于稳定映射的热带空间，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.