CAREER: Floer Homology and Low-Dimensional Topology

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This project studies the topology and geometry of low-dimensional manifolds and the knots and surfaces embedded therein. Specific goals include a classification of knots in the 3-dimensional sphere which produce simple manifolds under surgery, topological characterization of knots which bound complex curves inside complex surfaces, and a deeper understanding of the structure of cobordism groups of knots and 3-manifolds. Primary tools for this study arise in symplectic geometry, gauge theory, and quantum algebra. Many of these tools come in the form of algebraic invariants e.g. Floer homology theories or combinatorial knot invariants such as Khovanov homology. The project also seeks to further our understanding of the invariants themselves, and a central theme is to determine the extent to which they faithfully represent the topological objects which they shadow. The specific mathematical goals of the project are complemented by concrete initiatives targeted at graduate and undergraduate education, together with activities aimed at an international community of topologists and geometers. For instance, a new graduate course will be designed and implemented whose dual focus is on the development of topological breadth and communication skills in a variety of scenarios. A summer school for undergraduates and a broad-interest conference on topology will be organized. Topology and geometry are mathematical fields which study shapes and spaces, and low-dimensional topology focuses on those shapes which are within or just out of reach of our vision. This project makes fundamental contributions to low-dimensional topology. One of the key problems that the project attacks is the mathematical theory of "knotting", an area which attempts to understand and quantify the method by which ideal strings become tangled and knotted in space. In addition to a fundamental role which knotting plays in low-dimensional topology, its theory has deep interactions with many seemingly unconnected areas of mathematics and physics, and even to areas such as polymer science. One of the central aims of the project is to understand how knots evolve over time. Imagine a movie in which a piece of string freely moves in space, becoming more or less tangled over time. Further imagine that at some frames in the movie more drastic phenomena occur such as the appearance or disappearance of a new loop of string or the gluing of two segments of the string together. Such a movie is called a "concordance", and using this idea one can treat knotted pieces of string in much the same way that we treat numbers; namely, one can add and subtract knots in an algebraic way. Understanding the arithmetic of knots turns out to have deep implications for the study of 4-dimensional space, one of the most difficult and least understood areas of modern mathematics. In addition to its specific mathematical aims, the project will also contribute in a significant way to graduate and undergraduate education and to a global research community through design and organization of innovative courses, workshops, and conferences.