Mirror Symmetry and Modular Functions

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Lian 9619884 This research is concerned with work on mirror symmetry and modular functions. The principal investigator will work on extending known results for toric hypersurfaces to the case of toric complete intersections. He will try to construct all large radius limit points. He will also work on extending work on modularity of the mirror map for K3 surfaces to arbitrary 1-parameter families and to multiparameter families. Finally, he will study S-duality for K3-fibered threefolds. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.