Geometry and Topology

摘要提案：DMS 9803623首席研究员：Anatoly Libgober本项目涉及镜像对称。它旨在阐明镜像对应的数学意义和范围，并找到镜像对称的应用。虽然从数学的角度来看，镜像对称的最初发现预测了（Calabi-Yau）流形（即线条、二次曲线、扭曲立方等的命理学）和微分方程之间的枚举几何关系，或者更准确地说，是另一个（Calabi-Yau）流形上Hodge结构的变化，但最近的发展，主要是在物理文献中，提出了镜像对应的许多附加性质。镜像对称在物理意义上是流形的一个非常重要的条件，它被定义为与流形相关的共形场论的某种同构，人们想要理解它的精确数学意义。研究者正计划寻找流形的其他拓扑和几何性质，这些性质将使我们能够识别镜像伙伴，例如它们的特征类，椭圆属的行为，或其他可能有助于在特殊情况下识别镜像伙伴的性质，例如具有自同构的流形。该研究的另一个方面是研究与镜像对称相关的对象的代数结构，如Calabi-Yau流形族周期的微分方程。作为这些微分方程研究的一部分，研究者计划将镜像对称中出现的问题与先前关于补的基本群、某些拟射影变体上的局部系统的单群和上同调的工作联系起来。总的来说，该项目的目标是试图弥合20世纪90年代初对镜像对称现象的物理和数学理解之间的差距。这将使我们更好地理解自19世纪中叶以来一直是数学家关注的问题，比如几何物体的枚举，使用弦理论的思想，我们也希望为物理学中的镜像对称的理解带来更多的数学思想。

Abstract Proposal: DMS 9803623 Principal Investigator: Anatoly Libgober This project concerns mirror symmetry. It aims to clarify the mathematical meaning and the scope of mirror correspondence and to find applications of mirror symmetry. While from mathematical viewpoint, the original discovery of mirror symmetry predicted a relation between the enumerative geometry of manifolds from the class of (Calabi-Yau) manifolds (i.e. the numerology of lines, quadrics, twisted cubics etc) and differential equations or more precisely the variation of the Hodge structure on another (Calabi-Yau) manifold, more recent developments, mainly in physics literature, suggest numerous additional properties of mirror correspondence. Mirror symmetry in the physical sense is a rather imposing condition on manifolds, defined as a certain isomorphism of conformal field theories associated with the manifolds and one would like to understand its exact mathematical meaning. The investigator is planning to search for additional topological and geometric properties of manifolds which will allow us to identify mirror partners, such as behavior of their characteristic classes, elliptic genera, or other properties which may facilitate identifying mirror partners in special situations, e.g. for manifolds with automorphisms. Another aspect of the proposed study is the investigation of algebraic structures associated with objects involved in the mirror symmetry such as differential equations for the periods of the families of Calabi-Yau manifolds. As part of the study of these differential equations the investigator plans to relate issues arising in mirror symmetry to previous work on the fundamental groups of the complements, the monodromy groups and the cohomology of local systems on certain quasiprojective varieties. Overall, the goal of the project is to attempt to bridge the gap between physical and mathematical understanding found in the early 1990's of the phenomenon of mirror sym metry. This will lead to better understanding of issues which have been the focus of mathematicians since the middle of 19th century, such as enumeration of geometric objects, using ideas from string theory, and we hope also to bring additional mathematical ideas to the understanding of mirror symmetry in physics.