CAREER: Combinatorial Methods in Low-Dimensional Topology

Boston College, Chestnut Hill MA

Investigators

Abstract

Topology refers broadly to the study of shapes, and low-dimensional topology refers specifically to their study in dimensions one through four. These dimensions are special from an anthropic perspective, since they model our everyday perception of the physical world, and from a mathematical perspective, since the phenomena they exhibit and the collection of techniques used to study them are rather different from those in higher dimensions. Many of these techniques used to study these phenomena involve combinatorics, the study of discrete structures. A central goal of this research project is to advance the combinatorial aspects of these techniques with a view towards concrete problems in the field. Alongside the research component, the PI proposes activities that integrate his research interests with education and training initiatives that reach audiences from the high school level to postdoctoral researchers. For instance, the PI is actively involved with mathematics enrichment at the high school level through the Hampshire College Summer Studies in Mathematics and Mathematical Staircase, Inc. In the context of these programs and in other mentoring activities, he seeks to inspire the discovery process and aid in the exposition of beautiful mathematics. A chief outreach activity in the project is a graduate summer school that will showcase several different perspectives on one central theme in low-dimensional topology, Dehn surgery. Amongst the various techniques that come to bear on low-dimensional topology are graphs of surface intersections, exemplified by the work of Gordon and Luecke, and Heegaard Floer homology, defined and developed by Ozsváth and Szabó. Both techniques have led to sensational progress on the main problems in low-dimensional topology. The two approaches lend very different perspectives on the subject, and they have complementary strengths and weaknesses. The surface intersection techniques are more direct and rely on the development of graph theoretic tools in order to draw topological conclusions. Floer homology methods are less direct but apply heavy machinery to a vast collection of problems. The PI specifically seeks to blend the combinatorial ideas stemming from these techniques and others with a view towards some of the driving problems in low-dimensional topology, spanning topics including the study of knot diagrams, Dehn surgery, and the curve complex.