Positively and Non-Positively Curved Manifolds

正曲流形和非正曲流形

• 批准号: 0504534
• 负责人: Xiaochun Rong
• 金额: $ 11.6万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504534&HistoricalAwards=false
• 关键词: Positively; Non; Curved; Manifolds

AbstractAward: DMS-0504534Principal Investigator: Xiaochun RongPositive curvature and non-positive curvature have been frequentsubjects in Riemannian geometry, where mathematics from severaldisciplines interact; such as differential geometry, analysis andpartial differential equations, transformation group theory andtopology. The PI is pursuing a research program concerning somebasic problems in these areas: 1. Interplay between positivecurvature (with Abelian symmetry) and topology. This project isamplified by the amazing fact that among the manifolds of thesame dimension whose sectional curvature is between two positiveconstants, all but finitely many admit (large) Abelian symmetry.2. The semi-rigidity of the moduli space of non-positively curvedmetrics on a closed manifold.Mathematics is the foundation of natural sciences, anddifferential geometry and Riemannian geometry is one of the mostimportant branches of mathematics. The PI is pursuing solvingsome basic problems in this field which would have a broadintellectual impact. PI will continue to actively pursuecollaborations with other mathematicians in US and abroad and tospeak at several national and international meetings a year.The PI organized Riemannian geometry seminars in summers of2001-2004, which were designed to provide a forum for thedissemination of new ideas, in particular for graduate students.The PI has had three papers with his graduate students andpostdoctoral fellow which were the products of the summerseminars. The PI will continue the Riemannian geometry summerseminars in the next three years. The PI has been writing twoadvance graduate textbooks for three years (titled: ``Theconvergence and collapsing theory in Riemannian geometry'' and``Positive curvature, symmetry and topology''). He wishes tofinish the two books in 2-3 years. This will not only benefitgraduate students and researchers from these areas (due to thelack of graduate textbooks in these topics) but also help toadvertise and disseminate Riemannian Geometry to students andmathematicians in other related fields, by giving them a quickpicture of some of the research directions of Riemanniangeometry, and by showing them how the subject vitally interactswith differential geometry, analysis and PDE, compacttransformation group theory and topology.

摘要：正曲率和非正曲率是黎曼几何中经常出现的主题，是多个学科数学相互作用的领域。如微分几何，分析与偏微分方程，变换群理论和拓扑学。PI正在进行一项关于这些领域的一些基本问题的研究计划：1。正曲率（具有阿贝尔对称）与拓扑之间的相互作用。在截面曲率介于两个正常数之间的相同维数的流形中，除了有限的之外，所有流形都承认（大）阿贝尔对称，这一惊人的事实放大了这个项目。闭流形上非正曲线度量模空间的半刚性。数学是自然科学的基础，而微分几何和黎曼几何是数学的重要分支之一。PI正在寻求解决该领域的一些基本问题，这些问题将产生广泛的智力影响。PI将继续积极寻求与美国和国外其他数学家的合作，并每年在几个国内和国际会议上发言。2001-2004年夏天，PI组织了黎曼几何研讨会，旨在为新思想的传播提供一个论坛，特别是为研究生提供一个论坛。PI与他的研究生和博士后一起发表了三篇论文，这些论文都是夏季研讨会的成果。PI将在未来三年继续举办黎曼几何夏季研讨会。PI用3年时间编写了两本高级研究生教材（题目为《黎曼几何中的收敛和坍缩理论》和《正曲率、对称和拓扑》）。他希望在2-3年内完成这两本书。这不仅有利于这些领域的研究生和研究人员（由于缺乏这些主题的研究生教科书），而且还有助于向其他相关领域的学生和数学家宣传和传播黎曼几何，通过给他们一些黎曼几何研究方向的快速图片，并通过向他们展示该学科如何与微分几何，分析和PDE，紧变换群论和拓扑学相互作用。