Topology of Algebraic Varieties

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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.