Problems 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 the study of shapes in dimensions one through four. These dimensions are special from an anthropic perspective, since they model our everyday perception of the physical world. They are also special 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. The research component of the project explores a collection of important problems from across low-dimensional topology. A concrete example is a famous old problem which asks whether every continuous closed curve in the plane contains the vertices of a square. A unifying thread through the research is the use of modern methods from nearby fields, such as combinatorics (the mathematics of discrete structures) and symplectic geometry (the geometry of classical mechanics). Alongside the research, the PI proposes education and training initiatives reaching audiences from high schoolers to professional mathematicians. The PI will continue his active involvement with mathematics enrichment at the high school level through the Hampshire College Summer Studies in Mathematics and through Mathematical Staircase, Inc. The PI is in the process of editing a book based on a popular graduate summer school in low-dimensional topology that he ran. Moreover, the PI currently advises three PhD students. The award provides graduate student support and travel support for students and postdoctoral researchers. The PI proposes to study a collection of problems in low-dimensional topology, in continuation of an established program. The main themes are peg problems, using symplectic methods; exceptional Dehn surgery, using graphs of surface intersections; rational homology cobordism, using Floer homology and lattices; and ribbon concordance, using classical topological methods. Combinatorial and symplectic methods have long influenced the field. Amongst the various techniques that come to bear on low-dimensional topology are Floer homology, graphs of surface intersections, and lattice-theoretic methods. Each technique has led to sensational progress on the main problems in low-dimensional topology, and they lend very different perspectives on the subject. This project will more closely bind these techniques and low-dimensional topology. 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.