Topology of Algebraic Varieties and Singularities

摘要奖：DMS-0405729主要研究员：Anatoly S.Libgober该提案研究了代数簇的拓扑学和奇点理论中的几个主题。作者计划考虑依附于可能的奇异代数簇的顶点算子代数。顶点算子代数是由Malikov，Scheck htman和Vaintrob在奇异簇上附加的，由Frenkel和Szczesny给出的。L.Borisv和作者在椭圆代数上的工作表明，对于具有Gorenstein对数终端奇点的簇，人们可以期待这样的代数。这些顶点算子代数将提供新的代数族的不变量，这些不变量的作用应该被研究。这些新的代数应该进一步阐明群作用和奇点之间的McKay对应关系。作者计划研究奇异变量的新的陈型不变量，这些不变量是由鲍里索夫与他合作提出的。其次，作者致力于寻找形式场理论与其他镜像对称性方法之间的联系，并计划在此基础上研究附加到流形上的顶点算子代数的椭圆亏格与其他不变量之间的关系以及同调镜像对称性。第三，研究了椭圆群和奇点之间的联系，以及对Herling关于孤立奇点的谱性质的猜想的可能应用。最后，讨论了补集的拓扑、作者介绍的Alexander类型不变量以及与前面部分介绍的不变量之间的联系。这项工作将澄清最近在理论物理和特别是弦理论中出现的新的数学结构。它符合正在进行的数学问题范围和解决方法的重组过程。这项研究的目的是寻找理论物理中出现的新结构在更广泛的以前没有解决的问题中的应用，并扩大数学和物理之间的联系。它将把新的方法带入数学领域，如奇点理论，它已经被证明是理论数学与科学和工程之间的重要纽带。该项目的工作将有助于将UIC关于研究生和本科生的指导提升到完全代表当代数学前沿的水平。

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.