Homotopy Theory of Foliations and Diffeomorphism Groups

叶理论是一个数学领域，大约有50年的历史，其研究对象是将流形分解成路径连通的子集，称为叶。一个叶理在局部上看起来像是流形的分解，作为一个低维平行子流形的并集。这种几何结构自然出现在物理学和地质学中。而在数学中，叶对象的深度和广度使得数学家们使用了许多不同数学分支的工具，包括微分几何、同伦理论、非交换几何、遍历理论和动力系统。PI打算使用同伦理论的新工具来研究叶理和复同态群之间的关系。叶理的存在性和分类以及这种结构对流形的整体拓扑的影响在过去的五十年里得到了广泛的研究。然而，仍然有许多谜团，其中最重要的可能是同伦理论的叶理是Haefliger猜想。Haefliger问是否所有平面领域的流形上的尺寸大致小于一半的维度的流形是可积的同伦。马瑟和瑟斯顿证明了叶理的同伦理论与离散化的同形群的同调不变量自然相关。 但是，作为离散群的单同态群的群同调往往非常大，并且很少被理解。另一方面，具有Whitney拓扑的类同态群得到了更好的理解，特别是Galatius和Randal-Williams的程序开发了新的工具来研究这些具有Whitney拓扑的群的分类空间。PI的计划是联合收割机的新同伦理论的方法，源于不断发展的领域的模空间的流形与经典的叶理理论，研究同调不变量的同形群离散。这一奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.