CAREER: New Directions in Foliation Theory and Diffeomorphism Groups

点击翻译按钮获取中文摘要

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.