GEOMETRY AND TOPOLOGY OF THE MODULI SPACES OF RIEMANN SURFACES AND CALABI-YAU MANIFOLDS

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Moduli spaces of Riemann surfaces and Calabi-Yau manifolds have played fundamental roles in many subjects of mathematics from geometry, topology, algebraic geometry, to number theory. They are also important objects in string theory. The principal investigator proposes to have an intensive study by combining differential geometric methods with other newly developed techniques to solve several fundamental problems about the geometry and topology of moduli spaces of Riemann surfaces and Calabi-Yau manifolds. Differential geometric methods combined with algebraic geometry and combinatorial methods have been very successful in proving various important conjectures such as the Marino-Vafa conjecture, the Faber intersection number conjecture and the Labastilda-Marino-Ooguri-Vafa conjecture in our previous work. Based on these and other geometric results, the PI will further understand and solve several important problems including finding the explicit tautological ring structure of the moduli spaces of Riemann surfaces, proving the general string duality conjecture and solving the general Torelli problem for projective manifolds and clarifying its relation to mirror symmetry. Calabi-Yau manifolds are very important in string theory, the most promising theory to unify the four fundamental forces in the Nature. They are the shapes that satisfy the requirement of space for the six hidden spatial dimensions of string theory, which must be contained in a space smaller than our currently observable lengths. Riemann surfaces are called world-sheet in string theory which are the most basic objects in conformal field theory. The recent development of string duality in string theory has motivated many exciting new mathematical results. Many fundamental computations in string theory and quantum field theory are often reduced to certain integrals on moduli spaces of Riemann surfaces and Calabi-Yau manifolds. By comparing the mathematical descriptions of different string theories, one often reveals quite deep and unexpected mathematical conjectures, many of which are related to moduli spaces of Riemann surfaces and Calabi-Yau manifolds. The mathematical proofs of these conjectures often help verify the physical theories which cannot be achieved today through traditional experiments. Our project will lead to very strong impacts on several major fields of mathematics and theoretical physics. This program will not only help verify certain important physical theories in string theory, but also produce beautiful and fundamental results in 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.