Geometry and Topology

几何和拓扑

• 批准号: 9803623
• 负责人: Anatoly Libgober
• 金额: $ 7.01万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803623&HistoricalAwards=false
• 关键词: Geometry; Topology

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.

摘要 提案：DMS 9803623主要研究者：Anatoly Libgober 这个项目涉及镜像对称。旨在阐明镜像对应的数学意义和范围，并寻找镜像对称的应用。而从数学的观点来看，镜像对称的最初发现预示了流形的枚举几何与（Calabi-Yau）流形（即线，二次曲面，扭曲三次曲面等的数字学）和微分方程或更精确地说，霍奇结构在另一个（Calabi-Yau）流形，最近的发展，主要是在物理学文献中，提出了许多额外的镜像对应性质。 物理意义上的镜像对称是流形上一个相当苛刻的条件，定义为与流形相关的共形场论的某种同构，人们希望理解它的确切数学含义。 研究人员计划寻找流形的其他拓扑和几何性质，这将使我们能够识别镜像伙伴，例如其特征类的行为，椭圆属或其他性质，这些性质可能有助于识别特殊情况下的镜像伙伴，例如具有自同构的流形。 拟议研究的另一个方面是调查与镜像对称中涉及的对象相关的代数结构，例如卡-丘流形家族周期的微分方程。 作为研究的一部分，这些微分方程的调查计划有关问题所产生的镜像对称以前的工作基本群体的补充，monodromy群体和上同调的局部系统的某些quasiprojective品种。 总体而言，该项目的目标是试图弥合差距之间的物理和数学的理解发现在90年代初的现象，镜面对称。这将导致更好地理解自19世纪中叶以来一直是数学家关注的问题，例如几何对象的枚举，使用弦论的思想，我们也希望为物理学中镜像对称的理解带来更多的数学思想。