Singularities in Geometry and Topology

摘要：主要研究者：Jeff Cheeger， Shmuel weinberger本次会议将探讨在现代几何和拓扑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受精。两种中心推力是基于分层思想的拓扑方法和度量方法，以及它们的分析，特别是它们的微分学。其他的基于上同调与增长条件，或解决奇点，交同调，shef理论等。此外，这些研究的应用是多种多样的，从诺维科夫猜想、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（在各种巴纳赫空间中嵌入有限空间）。预计本次会议将在奇异空间解析几何结构、群的拟等距理论和特征类等方向上取得进展。解析学和几何学的许多基本思想都是基于线性近似的思想。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中的amaniffold，它是一个空间，在小区域内，很好地近似于欧几里得空间。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，这次会议将刺激新的综合的发展，并将在完全不同的环境中出现的共同问题带到前台。会议网站是http://www.math.uchicago.edu/~shmuel/Cappelliday。

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.