Geometry and groups: Structure and complexity

Purdue University, West Lafayette IN

Investigators

Abstract

The Principal Investigator will investigate different notions of complexity and structure in mathematics. These topics are relevant to several active mathematical research areas. Specifically, the goals of this proposal are connecting complexity with structure and the characterization/recognition of highly symmetric spaces from from natural data associated to the spaces. Aside from the direct appeal of these questions, this research hopes to establish new connections between areas in mathematics in an effort to foster cross-discipline interactions. Such interactions have been central to the progress of mathematics and more broadly science. This research project investigates the interplay between geometry, topology, and group theory in three broad, distinct projects. First, complexity functions associated to decision problems on groups. The main purpose is the interplay between the behavior of the complexity functions and the structure of the group. One specific focus for this project is on linear representations as both a tool and a conclusion. There are direct ties to the algorithmic complexity for decision problems on groups that has the potential to connect to areas beyond mathematics. Second, a group theoretic take on some classical work of Thom on representing homology classes. This leads to the study of the possible homological dimensions of fundamental groups of smooth manifolds. Part of this work displays prominently the interaction between geometry, group theory, dynamics, analysis, and topology. The potential results are quite exciting as are these rich connections. Third, the analogy between primitive geodesics on a negatively curved manifold and prime ideals in the ring of integers of a number field. The investigation centers around arithmetic progressions and a weaker version of progressions in the set of primitive geodesic lengths. This project has two main possible results, one that characterizes arithmetic manifolds, another that resolves an old conjecture in spectral rigidity problems. The weak notion of progressions has the potential to have impact in areas beyond geodesic geometry. These projects have the potential to have impact beyond the subjects they directly address. Indeed, part of the motivation for this work is the production of rich connections between distinct mathematical fields.