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

黎曼曲面和卡拉比-丘流形模空间的几何和拓扑

• 批准号: 1007053
• 负责人: Kefeng Liu
• 金额: $ 32.4万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007053&HistoricalAwards=false
• 关键词: GEOMETRY; TOPOLOGY; MODULI; SPACES; RIEMANN

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.

黎曼曲面和Calabi-Yau流形的模空间在从几何、拓扑学、代数几何到数论的许多数学学科中都扮演着重要的角色。它们也是弦理论中的重要对象。主要研究人员建议通过将微分几何方法与其他新发展的技术相结合来深入研究Riemann曲面和Calabi-Yau流形的模空间的几何和拓扑的几个基本问题。在我们以前的工作中，微分几何方法结合代数几何和组合方法已经非常成功地证明了各种重要的猜想，如Marino-Vafa猜想、Faber交数猜想和Labastilda-Marino-Ooguri-Vafa猜想。在这些几何结果和其他几何结果的基础上，PI将进一步理解和解决几个重要问题，包括寻找黎曼曲面的模空间的显式重言环结构，证明一般的弦对偶猜想，解决射影流形的一般Torelli问题，并澄清其与镜像对称的关系。Calabi-Yau流形在弦理论中是非常重要的，弦理论是最有希望统一自然界中四种基本力的理论。它们是满足弦理论的六个隐藏空间维度的空间要求的形状，该空间必须包含在比我们当前可观察到的长度更小的空间中。黎曼曲面在弦论中被称为世界薄片，是共形场理论中最基本的物体。弦理论中弦对偶的最新发展激发了许多令人振奋的新的数学结果。弦论和量子场论中的许多基本计算往往归结为黎曼曲面和Calabi-Yau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猜想，其中许多与黎曼曲面和Calabi-Yau流形的模空间有关。这些猜想的数学证明往往有助于验证今天无法通过传统实验实现的物理理论。我们的项目将在数学和理论物理的几个主要领域产生非常强烈的影响。这个程序不仅将有助于验证弦理论中的某些重要物理理论，而且还将在数学中产生美丽的基本结果。在实施该项目时，我们还将培训几名年轻学生和博士后，通过合作和讲座在这些主题上进行研究。