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

黎曼曲面和卡-丘流形的模空间在从几何、拓扑、代数几何到数论的许多数学学科中起着重要的作用。它们也是弦理论中的重要对象。主要研究者建议通过将微分几何方法与其他新开发的技术相结合来进行深入研究，以解决黎曼曲面和Calabi-Yau流形的模空间的几何和拓扑的几个基本问题。微分几何方法与代数几何和组合方法相结合，在我们以前的工作中已经成功地证明了Marino-Vafa猜想、Faber相交数猜想和Labastilda-Marino-Ooguri-Vafa猜想等重要猜想。基于这些和其他几何结果，PI将进一步理解和解决几个重要问题，包括寻找黎曼曲面模空间的显式重言式环结构，证明一般弦对偶猜想，解决投影流形的一般Torelli问题，并澄清其与镜像对称的关系。弦理论是最有希望统一自然界中四种基本力的理论，而卡-丘流形在弦理论中是非常重要的。它们是满足弦理论的六个隐藏的空间维度的空间要求的形状，它们必须包含在比我们目前可观察的长度更小的空间中。黎曼曲面在弦理论中被称为世界面，是共形场论中最基本的对象。弦理论中弦对偶的最新发展激发了许多令人兴奋的新数学结果。弦理论和量子场论中的许多基本计算常常归结为黎曼曲面和卡-丘流形的模空间上的某些积分。通过比较不同弦理论的数学描述，人们经常会发现相当深刻和意想不到的数学命题，其中许多与黎曼曲面和卡-丘流形的模空间有关。这些理论的数学证明往往有助于验证今天通过传统实验无法实现的物理理论。我们的项目将对数学和理论物理的几个主要领域产生非常强烈的影响。这个程序不仅有助于验证弦理论中某些重要的物理理论，而且还可以产生数学中美丽而基本的结果。在实施该项目的过程中，我们还将培训一些年轻的学生和博士后，通过合作和讲座进行这些主题的研究。

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.