CAREER: Symplectic and Holomorphic Convexity in 4-dimensions

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Topology is the study of the shape of different spaces, examples of which include a strand of DNA, a large data set, or even the universe. Particularly natural spaces to study in topology are the so called smooth manifolds. An example of a smooth manifold is the universe we live in, which we think of as 3-dimensional, but if we include time is actually 4-dimensional. A fundamental question is: what manifold is our universe? A goal of a geometric topologist is to list possibilities and most importantly to develop tools to understand the structure and distinguish the overall topology of manifolds. Low dimensional topology mainly refers to the study of 3-and 4-dimensional manifolds, where the theory of such spaces, in particular in 4-dimension, is significantly more complicated and exhibits many unique phenomena that are not seen in any other dimension. A particularly successful and rich way to understand the topology of manifolds is to introduce certain complex analytic structures on them, such as Stein structures and/or geometric structures, such as symplectic and contact geometries. The PI will study questions and conjectures that sit at the intersection of symplectic/contact topology and smooth manifold topology in low dimensions, and complex geometry. Alongside the research component, this project also includes activities that integrate the PI’s research program with education and training initiatives for student research, at both graduate and undergraduate levels. To that end, the PI will maintain an active topology group at the University of Alabama by organizing seminars, workshops, REUs and conferences. More broadly, the PI will use funds from this award to organize a week-long research and professional development summer workshop for talented undergraduate students from underrepresented colleges in the state of Alabama. He will also initiate a collaborative, multi-dimensional, long-term educational and research program for geometry-topology graduate students and faculty in the Southeastern Conference (SEC) schools, initially including the University of Alabama, Louisiana State University, the University of Mississippi, and the University of Arkansas. This project involves studying interactions between complex geometry, symplectic geometry and contact geometry, and low dimensional topology with a focus on various notions of convexity. The first long-term research objective will be to consider constraints on the topology of closed 3-manifolds embedded in 4-space from the perspectives of symplectic topology (e.g. contact type hypersurfaces) and complex geometry (e.g. the boundaries of holomorphically/rationally/polynomially convex Stein domains). Specifically, the PI will work towards a complete resolution of Gompf’s conjecture that predicts no Brieskorn sphere bounds a holomorphically convex domain in complex 2-space, determining the topology of contact type hypersurfaces in 4-space and in complex projective space, and exploring their implications for the exotic nature of smooth 4-manifold topology. The second long-term research objective concerns two outstanding problems dealing with the existence and classification of tight and fillable contact structures on closed, oriented 3-manifolds. The PI’s various joint works provide a framework and new constructions to study the existence question of tight and fillable contact structures from the point of view of Heegaard Floer homology. The PI and his collaborators will investigate to what extent invariants coming from Heegaard Floer homology detect tightness completely. The PI will also work to complete the classification problem for tight structures on small Seifert fibered spaces.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.