CAREER: New Directions in Foliation Theory and Diffeomorphism Groups

职业：叶状理论和微分同胚群的新方向

• 批准号: 2239106
• 负责人: Sam Nariman
• 金额: $ 54.74万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2028-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239106&HistoricalAwards=false
• 关键词: CAREER; New; Directions; Foliation; Theory

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.

叶子的全球结构已在数学内的各个领域进行了研究，包括差异拓扑，差异几何，动力学系统和非共同的几何形状。 自1970年代以来，众所周知，叶子和差异群（或歧管的对称群）之间存在着深厚的关系，这些组的代数特性已用于研究叶子。该项目旨在利用叶面理论来提取有关这些对称群体的信息，并应用围绕差异群体研究的最近开发的技术来产生有关叶子结构的新结果。该项目还将支持教育计划，其中包括为高中生介绍他们在周围的日常世界中介绍数学思维和数学使用的计划。该项目将在Haefliger和Thurston的猜想中应用一个捆绑的理论观点，对差异群体的共同体学。 结果反过来将导致围绕扁平束的不变性的多种研究方向。在先前的工作中，PI提供了证据表明，在分段线性（PL）类别中，该猜想与实数的代数K理论有关，并证明了Condimension 2 PL叶子的猜想。使用monod，PI开发了计算某些差异群体的有界共同体的技术，从而导致某些扁平捆绑包的界限结果。该项目旨在进一步开发这些技术，以计算差异群体的有限和连续的共同体，以更好地理解差异群体的群体共同体学。该奖项反映了NSF的法定任务，并通过使用该基金会的智力来评估，并被认为是值得的。优点和更广泛的影响审查标准。