Calabi-Yau manifolds and mirror symmetry

DMS - 0204326.m。格罗斯计划研究卡拉比-尤流形的镜像对称几何。这将从斯特罗明格-尤-扎斯洛猜想和目前由格罗斯和希伯特提出的这个猜想的代数-几何类似物的角度来完成。这些想法将以寻找一个一般的镜像对称结构为目标，将任何极化最大单能退化的Calabi-Yau流形与极化Calabi-Yau流形的镜像家族联系起来。这些思想对理解里奇平面度量在极大单幂退化附近的行为和利用log Gromov-Witteninvariants的概念计算有理曲线的意义也将被探讨。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，人们预计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：在一个Calabi-Yau流形上执行极其困难的某些计算可以通过在不同的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。格罗斯先生的工作旨在对镜像对称现象提供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.