Principal Investigator: Kefeng Liu
Moduli spaces have played fundamental roles in many subjects of mathematics from geometry, topology, algebraic geometry, to number theory. String duality from string theory has motivated many deep mathematical results and theories. The principal investigator proposes to have an intensive study on applying localization method combined with other newly developed geometric techniques to solve fundamental problems arisen from string duality about the geometry and topology of moduli spaces of Riemann surfaces and stable maps. There are several specific problems that we hope to solve under the support of the proposed research grant, these include the general recursion formulas for intersection numbers of general tautological classes 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 geometric results 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 grand unification of all fundamental forces in the nature, should include all previous theories like the Yang-Mills and the Chern-Simons theory. String duality which identifies different theories in string theory has produced many surprisingly beautiful mathematical formulas about the geometry and topology of moduli spaces of Riemann surfaces. The mathematical proofs of many of these conjectural formulas depend crucially on localization techniques. This program will not only help verify important physical theories, but also produce beautiful and fundamental mathematics. In carrying out the project we will also train several young students and post-doctors to conduct research in these subjects through collaboration and lectures.
