Conference on Submanifolds, Singular Varieties and Stratified Spaces

本次研究会议将汇集三个关键和当前活跃的几何领域的工作人员和学生：奇异品种，子流形和结，以及分层空间。这三个拓扑学领域在历史上有着积极而富有成效的科学互动，通过分享思想激发了重要的创新。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平行问题有关，在这些分支中存在有关将“经典”方法从光滑（非奇异）设置扩展到可能出现奇点的更一般设置的基础问题。本次会议将讨论令人兴奋的最新进展，包括对四维流形中嵌入和浸入的奇点如何揭示有关四维流形分类和结构的基本问题的研究；以及这些问题如何与三维流形中结点和连杆的深度不变量及其协坐标的新方法相互作用。还将讨论奇异变异体特征类的各种理论，以及有关其定义、计算方法和应用的基本问题，例如变换群和多面体几何组合的最新发展。这次关于重要的、当前的拓扑几何方法的研究会议将汇集研究人员和学生，他们正在从几个不同的角度研究一些数学基础、普遍存在的和相关的拓扑现象。这些空间被称为变种，在不同的点有不同的几何形状。这种几何上的变化，称为奇点，可以用结点和链接的数学理论来分析（在多维空间中）；为了说明这一点，这些变种被分解成地层。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表现出这样的奇点，所以需要研究这样的变种、它们的地层和奇点，以及关于结和链接的相关问题。要分析的最新令人兴奋的发展包括这些问题与四维几何的关系，与这些空间的对称性和不变性的关系，以及在枚举和组合问题中的应用，这些问题出现在非常广泛的科学问题中。

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.