Positively and Non-Positively Curved Manifolds

AbstractAward：DMS-0504534主要研究者：荣晓春正曲率和非正曲率是黎曼几何中经常出现的主题，在黎曼几何中，来自多个学科的数学相互作用;例如微分几何，分析和偏微分方程，变换群理论和拓扑学。PI正在进行一项研究计划，涉及以下领域的一些基本问题：1.正曲率（具有阿贝尔对称性）与拓扑之间的相互作用。这个项目被一个惊人的事实放大了，即在截面曲率在两个正常数之间的相同维度的流形中，几乎所有的流形都承认（大的）阿贝尔流形。闭流形上非正曲度量的模空间的半刚性数学是自然科学的基础，微分几何和黎曼几何是数学中最重要的分支之一。PI正在寻求解决这一领域的一些基本问题，这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，并在每年的几次国家和国际会议上发言。PI在2001 -2004年夏季组织了黎曼几何研讨会，旨在为传播新思想提供一个论坛。特别是研究生。PI与他的研究生和博士后研究员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。三年来，PI一直在编写两本高级研究生教科书（标题为：“黎曼几何中的收敛和坍缩理论”和“正曲率，对称性和拓扑学”）。他希望在两三年内完成这两本书。这不仅有利于这些领域的研究生和研究人员，（由于缺乏研究生教科书在这些主题），但也有助于宣传和传播黎曼几何的学生和数学家在其他相关领域，给他们一个快速图片的一些研究方向的黎曼几何，并向他们展示如何主题vitally相互作用与微分几何，分析和偏微分方程，紧变换群理论和拓扑学。

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.