Homotopy Theory of Foliations and Diffeomorphism Groups

叶化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接的子集，称为叶。叶化在局部看起来就像流形分解成低维平行子流形的并。这种几何结构自然出现在物理学和地质学中。在数学中，叶理对象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、遍历理论和动力系统。PI打算利用同伦理论的新工具来研究叶与微分同构群之间的关系。在过去的五十年里，叶状结构的存在和分类以及这种结构对流形整体拓扑的影响得到了广泛的研究。然而，仍有许多未解之谜，其中最重要的可能是叶理同伦理论中的Haefliger猜想。Haefliger提出了一个问题：流形上的所有平面场，其维数大致小于流形维数的一半，是否可积到同伦。Mather和Thurston证明了叶形的同伦理论与离散的微分同态群的同调不变量有天然的联系。但是，作为离散群的微分同构群的群同调往往非常大，而且很少被理解。另一方面，人们对具有Whitney拓扑的微分同构群有了更好的理解，特别是Galatius和Randal-Williams的程序开发了新的工具来研究具有Whitney拓扑的微分同构群的分类空间。PI的计划是将源于流形模空间演化场的新同伦理论方法与经典叶理理论相结合，研究离散化的微分同态群的同调不变量。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.