Singularities in Geometry and Topology

几何和拓扑中的奇点

• 批准号: 0706968
• 负责人: Jeff Cheeger
• 金额: --
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706968&HistoricalAwards=false
• 关键词: Singularities; Geometry; Topology

AbstractAward: DMS-0706968Principal Investigator: Jeff Cheeger, Shmuel WeinbergerThis conference will explore differing approaches to non-smoothspaces that arise in modern geometry and topology with an eyetowards cross-fertilization among the various groups ofresearchers. Two of the central thrusts are topological methodsbased on the ideas of a stratification, and metric measurespaces, and their analysis, especially their differentialcalculus. Others are based on cohomology with growth conditions,or resolution of singularities, intersection homology, sheaftheory etc. Moreover the applications of these studies arediverse, from the Novikov conjecture, knot theory, andtransformation groups to Gromov-Hasudorff convergence techniquesin Riemannian geometry and geometric group theory to problems ofinterest to theoretical computer scientists (embedding finitemetric spaces in various Banach spaces). It is anticipated thatas a result of this meeting, progress can be expected in thedirections of producing analytic geometric structures on singularspaces, the quasi-isometric theory of groups, and characteristicclasses.Many of the basic ideas of analysis and geometry are based on theidea of linear approximation. The first example of this iscalculus itself where linear approximation defines the basicnotion of the derivative. A more modern example is the idea of amanifold in topology, which is a space that is, in small regions,well approximated by Euclidean space. Countless furtherdevelopments have shown the need for more general tools to dealwith situations where such approximations are not feasible or maynot exist. By bringing together workers who have dealt withthese issues in varied contexts, this conference will spur thedevelopment of new syntheses and bring to the fore common issuesthat arise in completely different settings. The conference website is http://www.math.uchicago.edu/~shmuel/Cappelliday.

AbstractAward：DMS-0706968首席研究员：Jeff Cheeger，Shmuel Weinberg本次会议将探讨现代几何和拓扑学中出现的非光滑空间的不同方法，并着眼于不同研究小组之间的交叉施肥。 两个中心推力是拓扑方法的思想基础上的分层，度量测量空间，和他们的分析，特别是他们的微分。 另一些是基于增长条件的上同调，或奇点的分解，相交同调，sheaftheory等。此外，这些研究的应用是多样化的，从Novikov猜想，纽结理论，和变换群，到黎曼几何和几何群论中的Gromov-Hasudorff收敛技术，再到理论计算机科学家感兴趣的问题（在各种Banach空间中嵌入有限度量空间）。 可以预见，由于这次会议的结果，可以预期在奇异空间上产生解析几何结构、群的拟等距理论和特征类等方向上取得进展。分析和几何的许多基本思想都是基于线性逼近的思想。 第一个例子是微积分本身，线性近似定义了导数的基本概念。 一个更现代的例子是拓扑学中的流形概念，这是一个在小区域内可以很好地近似于欧几里得空间的空间。 无数的进一步发展表明，需要更通用的工具来处理这种近似不可行或可能不存在的情况。 通过将在不同背景下处理这些问题的工作者聚集在一起，这次会议将促进新的综合研究的发展，并将在完全不同的环境中出现的共同问题放在首位。会议网址是http://www.math.uchicago.edu/~shmuel/Cappelliday。