Floer Homology, Concordance, and Complex Curves

Michigan State University, East Lansing MI

Investigators

Abstract

This project will study topological and geometric objects, using a broad array of tools from modern mathematics. A theme central to the project is deepening our understanding of 4-dimensional aspects of "knotting," an area which attempts to understand and quantify the method by which strings become tangled and knotted in space. In addition to a fundamental role that knot theory plays in topology, its has deep interactions with many seemingly unconnected areas of mathematics and physics, and even to areas such as polymer science and biology. One of the primary aims of the project is to understand both how knots evolve over time and the surfaces that they bound in 4-dimensional space. 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 "cobordism," 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 this arithmetic turns out to have deep implications for the study of 4-dimensional space, one of the most difficult and least understood areas of mathematics. Primary goals in this vein include understanding knotted curves whose cobordisms arise from complex polynomials and probing the aforementioned algebraic structures using tools inspired by mathematical physics. In addition to its mathematical goals, the project will contribute to the organization of conferences and workshops, and provide critical support for students. Knot theory and its 4-dimensional aspects play a fundamental role throughout the project and can be viewed as its primary focus. Specific goals include determining the faithfulness of geometric maps between concordance and cobordism groups, topological characterizations of knots that bound complex curves in Stein domains, and the development of algebraic tools to facilitate computations of subtle invariant of knots, tangles, and 3-manifolds defined by way of Lagrangian Floer homology and non-abelian gauge theory. The project crosses boundaries between topology, analysis, algebra, and combinatorics, and uses techniques from homological algebra, gauge theory, knot theory, contact geometry, and the theory of pseudo-holomorphic curves in symplectic manifolds.