Surgery, square integrable cohomology, and quantitative geometry

This project addresses two aspects of topology as it is conventionally understood, through the study of particular cases that seem ripe for attack: The first is that it tends to be a qualitative subject, which typically produces statements that certain kinds of objects or deformations exist. It does not tell us how complicated such an object is, nor how much of some resource (think energy) is necessary to expend in producing the deformation. The second aspect is that for many problems, topology progresses by reduction to algebra -- and the algebraic problems are themselves extremely difficult. In some cases, this reduction can better be thought of as a reformulation of the problem in very different terms, but not necessarily easier ones. The PI will work with and mentor younger researchers on these projects, present his findings at conferences, and work on building bridges to other disciplines. The central attack envisioned in this project is to use analytic methods that are already known to connect to ring theoretic constructions in some cases (square integrable cohomology, and the Betti numbers defined for them using von Neumann algebras, pioneered by Atiyah) to study problems related to the number of handles necessary for manifold representatives of homology classes and how that contrasts with earlier work of the PI on the number of simplices (or volume). This will involve, in the case of lattices, representation theory of semisimple Lie groups. Within pure topology, the PI aims to apply this work in various directions, such as knot theory, open book decompositions and group actions on manifolds. Given the wide use of square integrable cohomology, the range of applications will be much wider. On the applied side, to the extent that this project also advances understanding of quantitative aspects of topology, it will have applications in several other scientific fields, as this is one of the bottlenecks to applications of topology. 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.