Algebraic and Syntactic Invariants for Homeomorphism Groups of Manifolds

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

The PI will investigate fundamental problems in geometry and topology, applying ideas from mathematical logic. The research will introduce new tools for studying manifolds, which are central objects in mathematics and physics, and will shed light on some basic unanswered questions about manifolds while simultaneously building bridges between topology and logic. Beyond the scientific investigation and attendant international collaborations, the PI will continue to promote access to education. Activities include organizing a Research Experiences for Undergraduates (REU) both in the United States and in Vietnam, with the goal of attracting global talent to US educational institutions and fostering academic pipelines between Vietnam and the US. The PI will also write a textbook making geometric group theory accessible to a wider range of students. The PI's research involves synthesizing ideas from model theory and applying them to the theory of groups acting on compact manifolds. This work builds on a theory of critical regularity of group actions in one dimension, as developed by the PI and Kim, and on more recent results about first order rigidity of homeomorphism groups of manifolds. One motivating goal is to prove the existence of finitely generated groups with prescribed critical regularities for compact manifolds of dimension two and higher. The methods to be applied here are two-fold: one is dynamical, concerning smooth group actions on manifolds, which has for example been successfully applied to finding groups of critical regularity 1 acting on compact two-manifolds. The second is to understand the interplay between mathematical logic and homeomorphism groups of compact manifolds, in particular the expressivity of first order logic in homeomorphism and diffeomorphism groups, and in countable groups that are elementarily equivalent to homeomorphism and diffeomorphism groups of manifolds. Potential payoffs include examples of torsion-free countable groups that cannot act on given compact manifolds, insight into the smooth Poincare conjecture, and potentially a proof that Gromov random groups cannot act nontrivially on compact manifolds. 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.