Classification problems in foliation theory

流形M的叶状结构F是将空间正则分解为连通的子流形，通常假设分解是光滑的，分解是均匀的——要么没有奇异层，要么奇异层本身具有叶状结构。叶理研究的广度包括所有的动力系统，这本身就是一个非常广泛的话题，所以真正对所有的叶理进行分类是不可能的。该项目的目标是研究三种分类方法：1)通过分类（在Lusternik和Schnirelmann的意义上）分解和其他“叶状细胞”结构对叶状进行分类；2)将叶理分类到其横向Haefliger结构的同伦；3)根据遍历理论和叶的渐近性质对叶进行分类。在NSF的支持下，PI对第二和第三个主题（动力系统和遍历理论，以及群作用和叶化的几何和拓扑不变量）之间关系的研究已经持续了20多年，并在该领域发表了许多出版物和进展。在过去三年中，每个专题都取得了进一步的决定性进展。Lusternik和Schnirelmann类叶及其相关思想的研究是一个极具发展潜力的新领域。LS范畴论和Morse理论之间的基本联系，激发了对它们的叶状版本之间的联系的研究，以及横向Haefliger结构的分类。这部分研究的成功将导致对叶理结构的真正基本的新理解。流形叶状的研究大约在五十年前开始作为一门学科。这个概念很自然地出现在许多几何和拓扑问题中，因此理解叶的性质已经成为几何、拓扑、动力系统、分析和各种应用和理论物理领域的许多其他学科研究的一个基本方面也就不足为奇了。所提议的活动的智力价值包括促进我们对叶状结构的理解，并开发在许多领域具有广泛应用的新技术。这个研究项目的成功不仅取决于解决了提出的问题，还取决于在这个领域开辟了新的问题。提议的活动所产生的更广泛的影响包括加强对数学本科生和研究生的培训，与博士后研究者的合作研究项目，以及促进数学研究的国际合作。PI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，并通过会议演讲和互联网促进该领域的发展。

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.