Conference on Submanifolds, Singular Varieties and Stratified Spaces

子流形、奇异簇和分层空间会议

• 批准号: 0413651
• 负责人: Sylvain Cappell
• 金额: $ 4万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0413651&HistoricalAwards=false
• 关键词: Conference; Submanifolds; Singular; Varieties; Stratified

AbstractAward: DMS-0413651 Principal Investigator: Sylvain CappellThis research conference will bring together workers and studentsin three key and currently active geometrical areas: singularvarieties, submanifolds and knotting, and stratifiedspaces. These three areas of topology have historically enjoyedvigorous and productive scientific interactions, spurringimportant innovations through sharing ideas. Moreover,developments and problems in these subjects in topology areconnected to parallel questions in other branches of geometry,analysis and algebraic geometry, where there are foundationalquestions concerning extending "classical" methods from smooth(nonsingular) settings to more general ones in whichsingularities can arise. Exciting recent developments to bediscussed at this meeting include investigations on howsingularities of embeddings and immersions in four-dimensionalmanifolds are shedding new light on basic questions concerningclassification and structure of four dimensional manifolds; andhow such questions also interact with new approaches to deepinvariants of knots and links in three dimensional manifolds andtheir cobordisms. Also to be discussed are various theories ofcharacteristic clases for singular varieties and foundationalquestions concerning their definitions, calculational methods,and applications, e.g., to current developments in transformationgroups and to geometrical combinatorics of polytopes.This research conference on important, current geometricalapproaches to topology will bring together researchers andstudents who are investigating some mathemtically basic,ubiquitous and related topological phenomena from severaldifferent perspectives. These involve spaces, called varieties,that can have variable geometry at different points. Suchvariations in geometry, called singularities, can be analyzedusing mathematical theories of knots and links (in manydimnensions); to elucidate this, such varieties are decomposedinto strata. Because the natural spaces that arise in manyresearch problems in both mathematics and theoretical physicsdisplay such singularities, investigations of such varieties, oftheir strata and their singularities, and of related questionsabout knots and links are needed. Recent and excitingdevelopments to be analyzed include relations of such questionsto four dimensional geometry, to the symmetries and invariants ofsuch spaces, and to applications to problems in enumeration andcombinatorics that arise in a very wide range of scientificproblems.

摘要奖：DMS-0413651首席研究员：Sylvain Cappe本次研究会议将聚集三个关键和当前活跃的几何领域的工作人员和学生：奇异簇，子流形和纽结，以及分层空间。这三个拓扑学领域历来享受着活跃而富有成效的科学互动，通过分享想法来激发重要的创新。此外，拓扑学中这些学科的发展和问题与几何、分析和代数几何的其他分支中的并行问题有关，其中有一些基本问题涉及将“经典”方法从光滑(非奇异)环境扩展到更一般的可能出现奇点的方法。本次会议将要讨论的令人振奋的最新发展包括研究四维流形中嵌入和浸入的奇点如何为有关四维流形的分类和结构的基本问题提供新的线索；以及这些问题如何与三维流形中的纽结和链环的深度不变量及其余边线的新方法相互作用。还将讨论奇异簇的特征类的各种理论以及有关它们的定义、计算方法和应用的基本问题，例如在变换群和多面体几何组合学的当前发展中。这次关于拓扑学的重要的、当前的几何方法的研究会议将聚集研究人员和学生，他们正在从几个不同的角度研究一些数学上基本的、普遍存在的和相关的拓扑现象。这些空间包括被称为簇的空间，这些空间在不同的点处可以具有可变的几何形状。几何中的这种变化，称为奇点，可以用(在许多维中)结和环的数学理论来分析；为了阐明这一点，这些变化被分解成层。因为在数学和理论物理的许多研究问题中出现的自然空间都表现出这样的奇点，所以需要研究这种变化，它们的层次和奇点，以及有关结和环的相关问题。需要分析的最新和令人兴奋的发展包括这类问题与四维几何的关系，与这种空间的对称性和不变量的关系，以及在非常广泛的科学问题中出现的计数和组合学问题的应用。