The Topology of Contact Type Hypersurfaces and Related Topics

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This project, jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR), centers around the geometry and topology of 3- and 4-dimensional spaces, mathematical objects known as symplectic and contact structures, and interactions between these. Symplectic and contact geometries are not just a natural language for some aspects of classical physics, but also naturally arise and find applications in many areas of modern mathematics and mathematical physics. The techniques spring from gauge theory, Floer theory, holomorphic curve techniques, and the theorems and conjectures find applications and connections in several fields, such as: smooth manifold topology, hyperbolic geometry, dynamics, complex analysis in several variables, and complex algebraic geometry. Building on his extensive and collaborative research, the PI aims to study many unique questions and conjectures that sit at the intersection of symplectic/contact topology and smooth manifold topology in low dimensions, and complex analysis. The proposed research and its outcomes will greatly impact our current understanding of geometric topology in low dimensions. As an integral part of this project, the PI will help mentor graduate students and postdoctoral fellows in his research area, maintain an active topology group at the University of Alabama by organizing seminars, workshops and conferences, and devote time to initiate a math circle in Tuscaloosa. The PI will investigate underlying connections between low dimensional smooth manifolds and certain geometric/analytic structures defined on them. The first long-term research objective of this project is to understand symplectic and complex geometric aspects of 3-manifold embedding problem in 4-space, and related symplectic/holomorphic rigidity phenomenon that develops. Specifically, the PI will work towards a complete resolution of Gompf’s conjecture that such embeddings are impossible for non-trivial Brieskorn spheres, determining the topology of contact type hypersurfaces and rationally convex Stein domains with prescribed boundary, and exploring their implications for smooth 4-manifold topology. The second long-term research objective concerns contributing concrete and satisfying connections between gauge theoretical invariants, symplectic/contact geometry and hyperbolic geometry. Towards this latter project, the PI will specifically work on two outstanding problems of existence and classification of tight and fillable contact structures on closed, oriented 3-manifolds. Many special cases of the latter project are understood due to work of the PI with his collaborators and other researchers in the area, but what links them remains to be explored.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.