Collaborative Research: The Geometry of Duality in Mathematics and Physics

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AbstractAward: DMS-0073657Principal Investigator: Sheldon KatzThis research project assembles a team at two campuses to work onstring theory and related geometry. The project is led by twomathematicians and a physicist, and the award supportspostdoctoral research fellows, graduate students, and thecollaborations and communications of a broadly based researchnetwork. The research program supported by this awardconcentrates on dualities in string theory and M theory, a topicin which a broad range of geometric and physical concepts cometogether. The best known of these dualities is mirror symmetry,which has had dramatic consequences for algebraic and symplecticgeometry, and this project intends to explore new dualities andconstraints along with mirror symmetry.String theory is a promising candidate for a unifying theory ofthe universe at its most fundamental levels. The basic idea issimple - elementary particles should be modeled as mathematicalloops of string rather than as points - but working out thedetails of this theory has involved and inspired somesophisticated mathematical tools and ideas. Constraints andobservations from physics, sometimes posed as a claim that twodistinct geometries must generate the same physical theory, canhave large numbers of consequences for geometry since quantitiesof physical interest are often expressed as the average value ofan observable quantity over space, or as a way of counting thenumber of times two objects meet. The program in Algebra, NumberTheory, and Combinatorics is cofunding this award.