Affine Manifolds, Log Geometry, and Mirror Symmetry

M.格罗斯计划研究卡拉比-丘流形的镜像对称几何。这将从M.格罗斯和B。西伯特与Calabi-Yau流形的某些大复结构极限退化相联系，可以定义一个交复形，它是一个具有奇点的仿射流形。相反，给定一个具有奇异性的仿射流形，格罗斯和西伯特证明了可以在具有奇异性的仿射流形上明确地建立这样的退化。M.格罗斯计划研究这一结构的应用，特别是打算努力实现这一目标，即证明这一结构为镜像对称提供了几何解释。最后，我们将证明，相同的热带结构决定了镜像对称的两个不同侧面上的有理曲线的数目和周期的计算。格罗斯处于弦理论和几何学的交叉点。弦理论用一个在时空中运动的弦环取代了传统的点粒子概念，为了使弦理论与量子力学兼容，时空必须是10维的。既然时空看起来是四维的，那么我们可以预期其中的六个维度是一个非常小的“卷曲”的几何对象。这些几何对象被称为卡-丘流形。在20世纪90年代早期，弦理论家提出了完全不同的卡-丘流形之间的一种显著联系：某些在一个卡-丘流形上极难执行的计算，可以通过在另一个卡-丘流形上执行完全不同的、容易得多的计算来完成。这一发现被称为镜像对称。从那时起，许多几何学家一直试图理解这一奇迹般的观察背后的数学原理。M的工作。格罗斯希望能对镜像对称现象给出数学上的解释。

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.