Homotopy Theory of Foliations and Diffeomorphism Groups

叶理论是一个大约有50年历史的数学领域，它的研究对象是将流形分解成路径连通的子集，称为叶。局部的叶状结构看起来像是流形的分解，是低维的平行子流形的并集。这样的几何结构在物理学和地质学中自然会出现。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，包括微分几何、同伦理论、非交换几何、遍历理论和动力学系统。PI试图使用同伦理论的新工具来研究叶层和微分同胚群之间的关系。在过去的五十年里，叶层的存在和分类以及这种结构对流形整体拓扑的影响已经得到了广泛的研究。然而，仍然有许多谜团，其中最重要的可能是在叶状同伦理论中的Haefliger猜想。Haefliger问流形上的所有平面场，如果其维度大致小于流形的一半，是否可积至同伦。马瑟和瑟斯顿证明了叶层的同伦理论与离散的微分同胚群的同调不变量有着天然的联系。但作为离散群的微分同胚群的群同调往往很大，人们对它的理解很少。另一方面，人们对具有Whitney拓扑的微分同胚群有了更好的了解，特别是Galatius和Randal-Williams的程序发展了新的工具来研究具有Whitney拓扑的这些群的分类空间。PI的计划是将源自流形的模空间的演化场的新的同伦理论方法与经典的叶层理论相结合来研究离散的微分同胚群的同调不变量。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.