Homotopy Theory of Foliations and Diffeomorphism Groups

叶状结构和微分同胚群的同伦理论

基本信息

批准号：
2113828
负责人：
Sam Nariman
金额：
$ 11.93万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-02-15 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2113828&HistoricalAwards=false
关键词：
Homotopy Theory Foliations Diffeomorphism Groups

项目摘要

Foliation theory is a field of mathematics, which is roughly 50 years old, whose object of study is certain decomposition of manifolds into path-connected subsets, called leaves. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of lower dimensions. Such geometric structures naturally arise in physics and geology. And in mathematics the depth and breadth of foliated objects made mathematicians use tools from many different branches of mathematics including differential geometry, homotopy theory, noncommutative geometry, ergodic theory and dynamical systems. The PI intends to use new tools from homotopy theory to investigate the relation between foliations and diffeomorphism groups.The existence and classification of foliations and the implication of such structures on the global topology of manifolds have been extensively studied in the past five decades. However, there are still many mysteries, perhaps the most important of which in the homotopy theory of foliation is the Haefliger conjecture. Haefliger asked whether all plane fields on a manifold whose dimensions are roughly less than the half of the dimension of the manifold are integrable up to homotopy. It was shown by Mather and Thurston that the homotopy theory of foliations is naturally related to the homological invariants of the diffeomorphism groups made discrete. But the group homologies of diffeomorphism groups as discrete groups tend to be very large and are poorly understood. On the other hand diffeomorphism group with the Whitney topology is better understood, in particular, Galatius and Randal-Williams' program developed new tools to study the classifying space of these groups with the Whitney topology. The PI's plan is to combine the new homotopy theoretical methods that stem from the evolving field of the moduli space of manifolds with the classical foliation theory to study homological invariants of diffeomorphism groups made discrete.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

叶面理论是一个数学领域，大约50岁，其研究对象是将歧管分解为路径连接的子集，称为叶子。叶面在局部看起来像是歧管的分解，是较低维度的平行子手机的结合。这种几何结构自然出现在物理和地质学中。在数学中，叶状物体的深度和广度使数学家使用了许多不同数学分支的工具，包括差异几何，同义理论，非交通性几何学，千古理论和动力学系统。 PI打算使用同义理论中的新工具来研究叶子与差异群体之间的关系。在过去的五十年中，对叶子的存在和分类以及此类结构对多种流形的全球拓扑的影响进行了广泛的研究。但是，仍然有许多谜团，也许在同叶叶子理论中最重要的是haefliger的猜想。 Haefliger询问歧管上的所有平面场是否尺寸大约小于歧管尺寸的一半，这是可以整合到同型的。马瑟（Mather）和瑟斯顿（Thurston）表明，叶子的同质性理论自然与差异群体的同源性不变性有关。但是，作为离散群体的差异群体的组同源物往往很大，并且知之甚少。另一方面，更好地理解了与惠特尼拓扑结构的差异群体，特别是，加拉图斯和兰德尔·威廉姆斯的计划开发了新的工具来研究这些群体与惠特尼拓扑的分类空间。 PI的计划是结合源于歧管的模量空间不断发展的领域的新同质理论方法与经典叶面理论，以研究差异群体的同源性不变性群体，这是分散的。此奖项反映了NSF的法定任务，并被视为值得值得的任务。通过基金会的智力优点和更广泛的影响评估标准通过评估来支持。