Affine Manifolds, Log Geometry, and Mirror Symmetry

格罗斯计划研究Calabi-Yau流形的镜像对称几何。这将从M.Gross和B.Siebert提出的Strominger-Yau-Zaslow猜想的代数几何版本的角度进行。与Calabi-Yau流形的某些大型复结构极限退化相联系，我们可以定义一个交复形，它是一个具有奇点的仿射流形。相反，给定一个奇点仿射流形，Gross和Siebert证明了根据奇点仿射流形上的“热带”数据显式地建立这样的退化是可能的。M·格罗斯计划研究这种结构的应用，特别是打算朝着表明这种结构为镜像对称性提供了几何解释的目标而努力。最终，相同的热带结构将决定镜像对称两侧有理曲线的数量和周期计算。M.格罗斯的工作位于弦理论和几何的交叉点。弦理论用一个穿越时空的小弦环取代了传统的点粒子的概念。要使弦理论与量子力学兼容，时空必须是十维的。由于时空看起来是四维的，人们预计其中的六个维度会是一个非常小的“卷曲”的几何物体。这些几何对象称为Calabi-Yau流形。在20世纪90年代初，弦理论家提出了一个完全不同的Calabi-Yau流形之间的显著联系：在一个Calabi-Yau流形上执行某些极其困难的计算，可以通过在不同的Calabi-Yau流形上执行完全不同且容易得多的计算来完成。这一发现被称为镜面对称性。从那时起，许多几何学家一直在试图理解这一奇迹观测背后的数学原理。M·格罗斯的工作希望对镜像对称现象给予数学上的洞察和解释。

M. Gross plans to study the geometry of mirror symmetry for Calabi-Yau manifolds. This will be done from the perspective of an algebro-geometric version of the Strominger-Yau-Zaslow conjecture introduced by M. Gross and B. Siebert. Associated to certain sorts of large complex structure limit degenerations of Calabi-Yau manifolds one can define an intersection complex, which is an affine manifold with singularities. Conversely, given an affine manifold with singularities, Gross and Siebert showed it is possible to build explicitly such a degeneration in terms of "tropical" data on the affine manifold with singularities. M. Gross plans to study applications of this construction, and in particular intends to work towards the goal of showing that this construction provides a geometric explanation for mirror symmetry. Ultimately, it will be shown that the same tropical structures determine both numbers of rational curves and period calculations on the two different sides of mirror symmetry.The work proposed by M. Gross lies at the intersection of string theory and geometry. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time.To make string theory compatible with quantum mechanics, space-time must be ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small "curled up" geometric object. These geometric objects are called Calabi-Yau manifolds. In the early 1990s, string theorists proposed a remarkable association between completely different Calabi-Yau manifolds: certain calculations extremely difficult to perform on one Calabi-Yau manifold could be completed by performing completely different, and much easier, calculations on a different Calabi-Yau manifold. This discovery was known as mirror symmetry.Since this time, many geometers have been trying to understand the mathematics behind this miraculous observation. The work of M. Gross hopes to give mathematical insight and explanation for the phenomenon of mirror symmetry.