Principal Investigator: Anatoly S. Libgober
The proposal studies several topics in the topology of algebraic varieties and singularity theory. The author plans to consider the vertex operator algebras attached to possibly singular algebraic varieties. Vertex operator algebras were attached to non singular varieties by Malikov, Schechtman and Vaintrob and to orbifolds by Frenkel and Szczesny. L.Borisov and author's work on elliptic genera suggests that one may expect such algebras for varieties with Gorenstein log terminal singularities. These vertex operator algebras will provide new invariants of algebraic varieties which role should be investigated. These new algebras should provide further clarification of McKay correspondence between the group actions and singularities. The author plans to study new Chern class type invariants for singular varieties which are suggested by the joint work with L.Borisov. Secondly, the author works on finding connections between the conformal field theory and others approaches to mirror symmetry and, as part of this, he plans to study the relationship between elliptic genera and other invariants of vertex operatror algebras attached to manifolds and homological mirror symmetry. Thirdly the author proposes to study connection between the elliptic genera and isolated singularities and possible applications to Herling's conjecture on properties of spectra of isolated singularities. Final part deals with problems about the topology of the complements, the Alexander type invariants introduced by the author and the connections with the invariants introduced in previous parts of the proposal.
Work on this project will clarify new mathematical structures which appeared recently in theoretical physics and in particular in string theory. It fits into ongoing process of restructuring the scope of mathematical problems and methods for their solutions. The puropose of this research is to find applications of the new structures which emerged in theoretical physics to a wider range of previously unsolved problems and to expand connections between mathematics and physics. It will bring new methods into areas of mathematics such as singularity theory which already proved to be important link between theoretical mathematics and science and engineering. Work on this project will help to upgrade the instructions at UIC on graduate and undergraduate level to the level fully representing frontiers of contemporary mathematics.
