Topology of Algebraic Varieties

代数簇的拓扑

• 批准号: 0103651
• 负责人: Anatoly Libgober
• 金额: $ 9.72万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103651&HistoricalAwards=false
• 关键词: Topology; Algebraic; Varieties

DMS-010Anatoly S. LibgoberConcrete issues proposed in this project are the studyof 2-variable elliptic genus and chiral deRham complexfor non-singular and singular varieties,mathematical foundations of the second quantized ellipticgenus and orbifold elliptic genus,ODE and PDE appearing in mirror symmetry, the Chernclasses of mirrors and the topology of thecomplements as well as the study of the fundamental groupsof the complements.The goal of this project is to analyze frommathematical point of view several methods andtechniques which were used in theoretical physics,more precisely in string theory. Such mathematicalinvestigation should lead to new results inthe physical theories involved and produce abroader understanding of the reasons of effectivenessof the methods used by physicists.Particular problems which we are planning to solveare motivated by dualities in string theoryand more specifically by mirror correspondence.One of the problems is to find new propertiesof mirror correspondence and its characterization.This involved the study in topology, algebraic geometryand differential equations.

本课题主要研究非奇异和奇异变异体的2变量椭圆格和手性deRham复形，二次量化椭圆格和轨道椭圆格的数学基础，镜像对称中的ODE和PDE，镜像的Chernclasses和补的拓扑以及补的基本群的研究。本项目的目的是从数学的角度分析理论物理，更准确地说是弦理论中使用的几种方法和技术。这样的数学研究应该会导致有关物理理论的新结果，并对物理学家使用的方法的有效性的原因产生更广泛的理解。我们计划解决的特殊问题是由弦理论中的对偶性引起的，更具体地说是由镜像对应引起的。其中一个问题是发现镜像对应的新性质及其表征。这涉及到拓扑学、代数几何和微分方程的研究。