Affine Manifolds, Log Geometry, and Mirror Symmetry

格罗斯计划研究卡拉比-丘流形的镜像对称几何。这将从M. Gross和B. Siebert引入的strominger - you - zaslow猜想的代数-几何版本的角度来完成。结合某些大型复杂结构Calabi-Yau流形的极限退化，可以定义一个交复，它是具有奇点的仿射流形。相反，给定一个具有奇点的仿射流形，Gross和Siebert证明了在具有奇点的仿射流形上用“热带”数据明确地建立这样的退化是可能的。格罗斯先生计划研究这种结构的应用，并特别打算朝着表明这种结构为镜像对称提供几何解释的目标努力。最后，将证明相同的热带结构决定了镜面对称两侧的有理曲线数和周期计算。格罗斯先生提出的工作是弦理论和几何学的交叉。弦理论用一个在时空中运动的小环弦取代了传统的点粒子概念。为了使弦理论与量子力学兼容，时空必须是十维的。由于时空看起来是四维的，人们期望其中的六个维度是一个非常小的“卷曲”的几何物体。这些几何物体被称为Calabi-Yau流形。在20世纪90年代早期，弦理论家提出了完全不同的Calabi-Yau流形之间的显著联系：某些在一个Calabi-Yau流形上极其困难的计算可以通过在另一个Calabi-Yau流形上执行完全不同的、更容易的计算来完成。这一发现被称为镜像对称。从那时起，许多几何学家一直在试图理解这一神奇观察背后的数学原理。格罗斯先生的工作希望为镜像对称现象提供数学见解和解释。

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.