Topology of Algebraic Varieties and Singularities

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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.