Instanton Homology in Low-Dimensional Topology

Harvard University, Cambridge MA

Investigators

Abstract

This award provides funding for a project that will develop tools to study questions about the geometry of intersecting surfaces, using novel tools arising from areas of mathematics that have not previously seen application in this area. Since antiquity, mathematicians have studied surfaces such as spheres, ellipsoids, paraboloids and hyperboloids in three-dimensional Euclidean space. These surfaces, familiar to the Greeks, can all be described in Cartesian geometry by equations of the second degree. Modern algebraic geometry, as developed primarily in the 20th and 21st centuries, provides tools to study surfaces defined by equations of higher degree. While a single equation of degree five (for example) may define a smooth surface in three-space, a pair of such equations will define a pair of surfaces, and the intersection of the two surfaces will be a curve in space. The project will seek to answer long-standing questions about the possible singularities of a curve arising in this way, using tools that first arose in the description of the fundamental forces of nature at the atomic and nuclear scale. These same tools will also be used in addressing questions about network flows. At the same time, the project will train graduate students and disseminate results to researchers in the area. The project activity will be in the following specific areas. In collaboration with T. S. Mrowka, the PI will develop properties of an instanton homology for spatial trivalent graphs and for knots in general three-manifolds. In particular, tools will be developed that will enable the calculation of instanton homology more generally than is currently possible. This instanton homology was constructed in previous work using a gauge theory related to representations of the fundamental group of complement of the knot or graph in the group of rotations, SO(3). When defined using a local coefficient system, instanton homology of knots and links yields new constraints on the topology of embedded surfaces whose boundary is a given knot or link. Specifically, it yields information about the possible genus of such surfaces and the number of their singularities. The final goal is to develop these tools to the point where they will answer long-standing questions in algebraic geometry concerning the topology of algebraic curves. For example, the PI will seek a negative answer to the question of whether two smooth quintic surfaces can intersect in an irreducible singular curve of genus zero. The goal of developing tools for the calculation of instanton homology will be applicable also to another goal, which is to use instanton homology to provide a new proof of the four-color theorem, which is the statement that the regions of any planar map can be colored using only four colors. The four-color theorem has been proved previously only with computer assistance, and it is hoped that this project might therefore lead the way to the first human-readable proof. 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.