Calabi-Yau manifolds and mirror symmetry

摘要DMS - 0204326.M。格罗斯计划研究卡拉比-尧曼尼褶皱的镜像对称几何。这将从Strominger-Yau-Zaslow猜想和M.格罗斯和B。西伯特这些想法将追求的目标是找到一个一般的镜像对称结构，与任何极化的最大幂幺退化的Calabi-Yau流形的极化Calabi-Yau流形的镜像家庭。这些思想对理解Ricci平坦度量在最大幂幺简并附近的行为和使用对数Gromov-Witteninvariants的概念计数有理曲线的意义也将被追求。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小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不同的卡-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。格罗斯肖普斯对镜像对称现象给出了数学上的见解和解释。

ABSTRACT DMS - 0204326.M. Gross plans to study the geometry of mirror symmetry for Calabi-Yaumanifolds. This will be done from the perspective of the Strominger-Yau-Zaslowconjecture and algebro-geometric analogues of this conjecture currentlybeing developed by M. Gross and B. Siebert. These ideas will be pursuedwith a goal of finding a general mirror symmetry construction, associatingto any polarized maximally unipotent degeneration of Calabi-Yau manifolds a mirror familyof polarized Calabi-Yau manifolds. Implications of these ideas to understandingthe behaviour of Ricci-flat metrics near maximally unipotent degenerationsand the counting of rational curves using the notion of log Gromov-Witteninvariants will also be pursued.The work proposed by M. Gross lies at the intersection of string theoryand 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 mustbe 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 early1990s, string theorists proposed a remarkable association betweencompletely different Calabi-Yau manifolds: certain calculations extremelydifficult to perform on one Calabi-Yau manifold could be completedby performing completely different, and much easier, calculations ona 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. Grosshopes to give mathematical insight and explanation for the phenomenonof mirror symmetry.