Higher-dimensional contact topology

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Contact structures occupy a central role in modern geometry and topology and have close connections with physics. They appear naturally as the boundary of space-time in mathematical physics and play an essential role in the mathematical study of three- and four-dimensional spaces and the knotting of DNA. In dimension three, contact structures are comparatively well-understood due to dramatic advances in the previous decades. The goal of this research project is to further develop higher-dimensional contact geometry, which is still quite mysterious and in its infancy despite significant advances in recent years. As part of this project, the Principal Investigator will also promote the training of future mathematicians. This research project on higher-dimensional contact geometry and its Floer-theoretic invariants has two parts: The first is to continue the systematic study of convex hypersurface theory - a technique to decompose a contact manifold into easier-to-analyze pieces - in arbitrary dimensions initiated by the Principal Investigator and his collaborator Dr. Huang and further develop the techniques that were used. The second is to develop a higher-dimensional generalization of Heegaard Floer homology. The goal is to obtain invariants that are effective at distinguishing contact and symplectic manifolds and their diffeomorphisms, as well as three- and four-manifold invariants. 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.