Topology of Algebraic Varieties and Singularities

代数簇和奇点的拓扑

• 批准号: 0405729
• 负责人: Anatoly Libgober
• 金额: $ 16.46万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405729&HistoricalAwards=false
• 关键词: Topology; Algebraic; Varieties; Singularities

AbstractAward: DMS-0405729Principal Investigator: Anatoly S. LibgoberThe proposal studies several topics in the topology of algebraicvarieties and singularity theory. The author plans to considerthe vertex operator algebras attached to possibly singularalgebraic varieties. Vertex operator algebras were attached tonon singular varieties by Malikov, Schechtman and Vaintrob and toorbifolds by Frenkel and Szczesny. L.Borisov and author's work onelliptic genera suggests that one may expect such algebras forvarieties with Gorenstein log terminal singularities. Thesevertex operator algebras will provide new invariants of algebraicvarieties which role should be investigated. These new algebrasshould provide further clarification of McKay correspondencebetween the group actions and singularities. The author plans tostudy new Chern class type invariants for singular varietieswhich are suggested by the joint work with L.Borisov. Secondly,the author works on finding connections between the conformalfield theory and others approaches to mirror symmetry and, aspart of this, he plans to study the relationship between ellipticgenera and other invariants of vertex operatror algebras attachedto manifolds and homological mirror symmetry. Thirdly the authorproposes to study connection between the elliptic genera andisolated singularities and possible applications to Herling'sconjecture on properties of spectra of isolated singularities.Final part deals with problems about the topology of thecomplements, the Alexander type invariants introduced by theauthor and the connections with the invariants introduced inprevious parts of the proposal.Work on this project will clarify new mathematical structureswhich appeared recently in theoretical physics and in particularin string theory. It fits into ongoing process of restructuringthe scope of mathematical problems and methods for theirsolutions. The puropose of this research is to find applicationsof the new structures which emerged in theoretical physics to awider range of previously unsolved problems and to expandconnections between mathematics and physics. It will bring newmethods into areas of mathematics such as singularity theorywhich already proved to be important link between theoreticalmathematics and science and engineering. Work on this projectwill help to upgrade the instructions at UIC on graduate andundergraduate level to the level fully representing frontiers ofcontemporary mathematics.

摘要：项目负责人：Anatoly S. libgober，主要研究代数变异拓扑和奇点理论中的几个问题。作者打算考虑可能的奇异代数变种的顶点算子代数。Malikov， Schechtman和Vaintrob将顶点算子代数附加到奇异变体上，Frenkel和Szczesny将顶点算子代数附加到奇异变体上。L.Borisov和作者关于椭圆属的工作表明，对于具有Gorenstein对数终端奇点的变体，可以期望有这样的代数。这些算子代数将提供新的代数变量不变量，其作用有待进一步研究。这些新代数将进一步澄清群作用与奇点之间的McKay对应关系。本文拟研究与L.Borisov联合提出的奇异变量的新的chen类类型不变量。其次，寻找共形场理论与其他镜像对称方法之间的联系，并计划研究流形上的顶点算子代数的椭圆属与其他不变量与同调镜像对称的关系。第三，作者提出研究椭圆属与孤立奇点之间的联系，以及在孤立奇点谱性质的赫林猜想中的可能应用。最后讨论了补的拓扑问题、本文引入的Alexander型不变量以及与前几部分不变量的联系。这个项目的工作将澄清最近在理论物理特别是弦理论中出现的新的数学结构。它符合正在进行的重构数学问题范围及其解决方法的过程。本研究的目的是寻找理论物理中出现的新结构的应用，以更广泛地解决以前未解决的问题，并扩大数学与物理之间的联系。它将为数学领域带来新的方法，例如奇点理论，奇点理论已经被证明是理论数学与科学和工程之间的重要联系。该项目的工作将有助于将ucic研究生和本科生的教学水平提升到充分代表当代数学前沿的水平。