Abstract Proposal: DMS 9803623 Principal Investigator: Anatoly Libgober This project concerns mirror symmetry. It aims to clarify the mathematical meaning and the scope of mirror correspondence and to find applications of mirror symmetry. While from mathematical viewpoint, the original discovery of mirror symmetry predicted a relation between the enumerative geometry of manifolds from the class of (Calabi-Yau) manifolds (i.e. the numerology of lines, quadrics, twisted cubics etc) and differential equations or more precisely the variation of the Hodge structure on another (Calabi-Yau) manifold, more recent developments, mainly in physics literature, suggest numerous additional properties of mirror correspondence. Mirror symmetry in the physical sense is a rather imposing condition on manifolds, defined as a certain isomorphism of conformal field theories associated with the manifolds and one would like to understand its exact mathematical meaning. The investigator is planning to search for additional topological and geometric properties of manifolds which will allow us to identify mirror partners, such as behavior of their characteristic classes, elliptic genera, or other properties which may facilitate identifying mirror partners in special situations, e.g. for manifolds with automorphisms. Another aspect of the proposed study is the investigation of algebraic structures associated with objects involved in the mirror symmetry such as differential equations for the periods of the families of Calabi-Yau manifolds. As part of the study of these differential equations the investigator plans to relate issues arising in mirror symmetry to previous work on the fundamental groups of the complements, the monodromy groups and the cohomology of local systems on certain quasiprojective varieties. Overall, the goal of the project is to attempt to bridge the gap between physical and mathematical understanding found in the early 1990's of the phenomenon of mirror sym metry. This will lead to better understanding of issues which have been the focus of mathematicians since the middle of 19th century, such as enumeration of geometric objects, using ideas from string theory, and we hope also to bring additional mathematical ideas to the understanding of mirror symmetry in physics.
