CAREER: New Directions in Foliation Theory and Diffeomorphism Groups

叶理的整体结构已经在数学的不同领域得到了研究，包括微分拓扑学、微分几何、动力系统和非交换几何。 自20世纪70年代以来，人们已经知道叶理和复同态群（或流形的对称群）之间有着深刻的关系，这些群的代数性质被用来研究叶理。这个项目的目的是使用叶理理论来提取这些对称群的信息，并应用最近开发的技术，围绕研究的单形群，以产生新的结果的结构叶理。该项目还将支持教育举措，包括为高中生提供的双周计划，向他们介绍数学思维和在日常世界中使用数学。这个项目将应用丛理论的观点来猜想Haefliger和Thurston的上同调的同胚群。 这些结果将反过来导致围绕平坦丛的不变性的众多研究方向。在以前的工作中，PI提供的证据表明，在分段线性（PL）范畴这一猜想是相关的代数K-理论的真实的号码，并证明了猜想余维2 PL叶理。与莫诺，PI开发的技术来计算有界上同调的某些同构群导致有界结果的某些不变量的平坦丛。该项目旨在进一步发展这些技术，以更好地理解群同构群的群上同调为目标来计算群同构群的有界和连续上同调。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.