Classification problems in foliation theory

叶状结构理论中的分类问题

• 批准号: 0406254
• 负责人: Steven Hurder
• 金额: $ 10.5万
• 依托单位: University of Illinois at Chicago
• 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: 2004-12-01 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406254&HistoricalAwards=false
• 关键词: Classification; problems; foliation; theory

A foliation F of a manifold M is a regular decomposition of the space into connected submanifolds, usually with some assumptions that the decomposition is smooth, and the decomposition is uniform -- either there are no singular strata, or the singular strata themselves have a foliated structure. The breadth of the study of foliations includes all dynamical systems, an immensely broad topic in itself, so truly classifying all foliations is quite impossible. The goal of this project is to investigate three approaches to classification: 1) classify foliations via categorical (in the sense of Lusternik and Schnirelmann) decompositions and other "foliated cell" structures; 2) classify foliations up to homotopy of their transverse Haefliger structure; 3) classify foliations in terms of the ergodic theory and asymptotic properties of their leaves. The PI's study of the relations between the second and third of these topics -- dynamical systems and ergodic theory, and geometric and topological invariants of group actions and foliations -- has been ongoing for more than 20 years with support of the NSF, and resulted in many publications and advances in the field. In the past three years, there have been further decisive advances in each topic. The study of Lusternik and Schnirelmann category of foliations, and its related ideas, is a very new field, with great potential for advances. The fundamental connections between LS category theory and Morse theory, motivate the study of the connections between their foliated versions, and the classification of transverse Haefliger structures. Success with this part of the proposed research would result in truly fundamental new understanding of the structure of foliations.The study of foliations of manifolds began as a subject approximately fifty years ago. The concept arises very naturally in many geometric and topological problems, so it is not surprising that understanding properties of foliations has become a fundamental aspect of the study of many other subjects in geometry, topology, dynamical systems, analysis and various areas of applied and theoretical physics. The intellectual merit of the proposed activity includes advancing our understanding of the structure of foliations, and developing new techniques with broad applications in many fields. Success with this research project will not just be measured by solving the proposed problems, but also in opening the field up with new questions. The broader impact resulting from the proposed activity includes enhanced training of undergraduate and graduate students in mathematics, collaborative research projects with postdoctoral investigators, and international collaborations promoting research in mathematics. The PI intends to continue working with graduate students and postdocs on projects related to this proposal, and to promote the field via conference talks and via the internet.

流形M的叶层F是空间到连通子流形的正则分解，通常假设分解是光滑的，并且分解是一致的--或者没有奇异层，或者奇异层本身具有叶状结构。叶理研究的广度包括所有动力系统，这本身就是一个非常宽泛的主题，因此真正对所有叶理进行分类是非常不可能的。这个项目的目标是研究三种分类方法：1)通过分类(在Lusternik和Schnirelmann意义下)分解和其他“叶状细胞”结构对叶理进行分类；2)将叶理分类到其横向Haefliger结构的同伦；3)根据叶的遍历理论和叶的渐近性质对叶理进行分类。在NSF的支持下，PI对其中第二和第三个主题--动力系统和遍历理论，以及群行动和叶的几何和拓扑不变量--之间的关系的研究已经进行了20多年，并在该领域产生了许多出版物和进展。在过去的三年里，每个主题都取得了进一步的决定性进展。叶理的Lusternik和Schnirelmann范畴及其相关概念的研究是一个非常新的领域，具有巨大的发展潜力。最小二乘范畴理论和莫尔斯理论之间的基本联系促使人们研究它们的分叶版本之间的联系，并对横向Haefliger结构进行分类。这部分拟议研究的成功将带来对叶理结构的真正根本的新理解。流形叶理的研究大约在50年前作为一门学科开始。这个概念在许多几何和拓扑问题中非常自然地出现，因此，理解叶状结构的性质已经成为几何、拓扑、动力系统、分析以及应用和理论物理的各个领域中许多其他学科研究的基本方面也就不足为奇了。拟议活动的智力价值包括增进我们对叶理结构的理解，以及开发在许多领域具有广泛应用的新技术。这一研究项目的成功将不仅仅通过解决提出的问题来衡量，还将以新问题打开领域的大门。拟议活动产生的更广泛影响包括加强对本科生和研究生的数学培训，与博士后研究人员合作开展研究项目，以及促进数学研究的国际合作。国际和平研究所打算继续与研究生和博士后合作，开展与该提案相关的项目，并通过会议讨论和互联网宣传这一领域。