Knot theory and low-dimensional topology

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Knot theory is a branch of mathematics that studies an idealized version of a knotted string where the two ends are fused together. The central problem is to distinguish when two knots are the same and when they are different. The early pioneers of the subject developed heuristic methods which they used to tabulate prime knots to 10 crossings. It was only with the introduction of rigorous mathematical techniques that the subject was placed on a solid mathematical footing. With many more sophisticated techniques and tools, mathematicians have now tabulated knots up to 16 crossings.   Low-dimensional topology involves the study of manifolds in dimensions 2,3 and 4. The central problem is to classify 3 and 4-dimensional manifolds. Manifolds are geometric shapes that, near any point, look flat like Euclidean space but whose global structure may be twisted and curved. The surface of a ball and the surface of a doughnut provide concrete examples; imagine a beach ball or an inner tube. Neither is flat, nevertheless, any rupture to the ball or inner tube can be repaired with a small rectangular patch and a bit of glue. Since manifolds are locally indistinguishable, mathematicians search for invariants that reflect the manifold's global curving and twisting. For example, imagine a near-sighted insect living on either the surface of a ball or a flat sheet of paper. How could it tell the two apart? One method would be to compute the Euler characteristic V - E + F, the number of vertices minus the number of edges plus the number of faces in a triangulation, which is independent of the triangulation and is an example of a topological invariant. These two branches of mathematics are intimately related. One reason is that 3 and 4-dimensional manifolds can be constructed as Dehn surgeries along knots or links. Knot theory and low-dimensional topology have numerous applications to physics, chemistry and biology. The applicant proposes to introduce new invariants of 3-dimensional manifolds and knots inside them. He will develop new approaches for computing these invariants. The invariants will be defined and studied using algebraic and geometric methods, including gauge theory and quantum topology. There are many interesting student research projects related to these questions, and the applicant will promote participation from a diverse group of students and postdoctoral fellows. He will foster an inclusive research environment that encourages learning and advancement of women and underrepresented minorities. The long-term benefits of this research program are two-fold: the knowledge gained will help determine to what extent geometric methods can deliver new invariants of knots and 3-manifolds, and the training program will produce highly qualified personnel at all levels, with the necessary analytical and computational skills to take leadership roles in the information-based economy.