Submanifolds and Cobordisms in Contact and Symplectic Topology

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Contact structures are natural objects, born over two centuries ago, in the work of Lie concerning solving differential equations, Gibbs concerning thermodynamics, Huygens concerning geometric optics, and Hamilton concerning classical mechanics. They have been studied by many mathematicians and seem to touch on diverse areas of mathematics and physics, but only in the last few decades have they moved into the foreground of mathematics. This is due to the remarkable breakthroughs in field, resulting in a rich and beautiful theory with many applications both inside mathematics and to science and engineering. In this project the PI will consider a variety of questions about various spaces with contact structures, focusing on objects inside of them, relations between them, and other structures on them. This will not only further our understanding of the field, but also its impacts on other areas of study. The PI will also devote significant time to helping graduate students and postdoctoral scholars become productive researchers in the field. The PI will investigate contact and symplectic structures through a variety of techniques, but focusing on their submanifolds and connections to Riemannian metrics. Recall that in dimension 3 understanding Legendrian and transverse knots in a contact manifold has gone hand in hand with advances in our understanding of contact structures and their subtle links with topology. For example the first proof of existence of contact structures came from surgery on transverse knots and the famed tight versus overtwisted dichotomy comes down to the types of Legendrian or transverse knots a contact structure supports. The PI will continue his investigations of such knots in 3 manifolds, focusing on qualitative features of them. Also recall, that many important concepts in contact geometry are expressed in terms of submanifolds of the contact structure (for example, Giroux torsion, open book decompositions, etc). Trying to understand how these various submanifolds interact and how various surgery constructions affect them will be another focus of the PI. The PI will also consider higher dimensional contact manifolds where much less is known. Here, basic questions about the existence and isotopy classification of contact submanifolds (a generalization of transverse knots) and isotropic submanifolds will be considered - as will surgery constructions and how they affect various properties of contact manifolds. Riemannian metrics have long been known to have deep connections with the smooth topology of manifolds and more recently it has been shown that contact structures do as well. The PI will continue to explore relations between these two geometric structures with the goal of seeing key properties of a contact structure (such as tightness) reflected in Riemannian metrics that are adapted to them. This will hopefully lead to a more complete understanding of the general picture of contact structures on 3 manifolds and create new tools for studying higher dimensional contact manifolds. The PI will also explore recent conjectures of Eliashberg about the existence of symplectic structures by explicitly verifying them in some nontrivial cases and exploring inductive approaches to proving them in some general settings. 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.