CAREER: New Directions in Foliation Theory and Diffeomorphism Groups

叶的整体结构已经在数学的各个领域得到了研究，包括微分拓扑、微分几何、动力系统和非交换几何。自20世纪70年代以来，人们已经知道叶形与微分同构群（或流形对称群）之间存在着深刻的关系，并利用这些群的代数性质来研究叶形。本项目旨在利用叶理理论提取关于这些对称群的信息，并应用最近发展起来的围绕微分同构群研究的技术来产生关于叶理结构的新结果。该项目还将支持教育活动，包括每两周一次的高中学生课程，向他们介绍数学思维和数学在日常生活中的应用。本课题将束理论的观点应用于Haefliger和Thurston关于微分同构群上同调的猜想。这一结果将反过来引导围绕平束不变性的许多研究方向。在之前的工作中，PI证明了在分段线性（PL）范畴中该猜想与实数的代数k理论有关，并证明了余维2 PL对偶的猜想。利用Monod，开发了计算某些差分同态群的有界上同调的方法，得到了平面束的某些不变量的有界性结果。本项目旨在进一步发展这些技术来计算微分同构群的有界连续上同调，以更好地理解微分同构群的群上同调。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。

The global structure of foliations has been studied in diverse fields within mathematics including differential topology, differential geometry, dynamical system, and non-commutative geometry. Since the 1970s, it has been known that there is a deep relationship between foliations and diffeomorphism groups (or symmetry groups of manifolds), and algebraic properties of these groups have been used to study foliations. This project aims to use foliation theory to extract information about these symmetry groups and to apply recently developed techniques around the study of diffeomorphism groups to generate new results on the structure of foliations. The project will also support educational initiatives, including a biweekly program for high school students to introduce them to mathematical thinking and the use of math in the daily world around them. This project will apply a bundle theoretic point of view to a conjecture of Haefliger and Thurston on the cohomology of diffeomorphism groups. The results will in turn lead to a multitude of research directions around the invariance of flat bundles. In previous work, the PI provided evidence that in the piecewise linear (PL) category this conjecture is related to the algebraic K-theory of real numbers and proved the conjecture for codimension 2 PL foliations. With Monod, the PI developed techniques to compute the bounded cohomology of certain diffeomorphism groups leading to boundedness results for certain invariants of flat bundles. The project aims to further develop these techniques to compute the bounded and continuous cohomology of diffeomorphism groups with the goal of better understanding the group cohomology of diffeomorphism groups.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.