LOCALIZATION, STRING DUALITY AND MODULI SPACES

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AbstractAward: DMS-0705284Principal Investigator: Kefeng LiuModuli spaces have played fundamental roles in many subjects ofmathematics from geometry, topology, algebraic geometry, to numbertheory. String duality from string theory has motivated many deepmathematical results and theories. The principal investigatorproposes to have an intensive study on applying localization methodcombined with other newly developed geometric techniques to solvefundamental problems arisen from string duality about the geometryand topology of moduli spaces of Riemann surfaces and stable maps.There are several specific problems that we hope to solve under thesupport of the proposed research grant, these include the generalrecursion formulas for intersection numbers of general tautologicalclasses which include the Faber conjecture as special case, the Virasoro conjecture for general projective manifolds,the understanding of topology, algebraic geometry of moduli spaces,and the computing of the holomorphic anomaly by using our geometricresults about moduli spaces.The interactions of mathematics and physics have motivated many fundamental advances in both disciplines.String Theory, as the most promising candidate for the grandunification of all fundamental forces in the nature, should includeall previous theories like the Yang-Mills and the Chern-Simonstheory. String duality which identifies different theories in stringtheory has produced many surprisingly beautiful mathematicalformulas about the geometry and topology of moduli spaces of Riemannsurfaces. The mathematical proofs of many of these conjecturalformulas depend crucially on localization techniques. This programwill not only help verify important physical theories, but alsoproduce beautiful and fundamental mathematics. In carrying out theproject we will also train several young students and post-doctors to conduct research inthese subjects through collaboration and lectures.