Mathematical Sciences: Atiyah's Conjectures on Floer Homology

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9626166 Li The problems addressed in this project are in the area of Topology. The main theme is to study conjectures of Sir Michael Atiyah on (homology) three-spheres, using invariants obtained from mathematical physics, especially conformal field theory and string theory, and symplectic topology. The investigator and R. Lee have studied the first conjecture of Sir Michael Atiyah on identifying these invariants of three-spheres, and also extended the invariants to slightly larger classes of three-spheres. The major part of this project is, not only to identify these invariants (the first Atiyah conjecture), but also to identify the way to relate these invariants to each other (the second Atiyah conjecture). Such an identification suggests a "hidden duality" between conformal field theory and four-dimensional Yang-Mills theory. The other part of this project is to study intrinsic properties for the invariants of larger classes of three-spheres. It is a fundamental aim to investigate the change of the new invariants under certain topological operations. The project will lead to a study of relations between Floer theory, which is an invariant from mathematical physics, and other constructions of knot theory, as well as the generalization of the invariants to general three-manifolds. A three-dimensional manifold is a space where a nearsighted person sees a standard three-dimensional space everywhere. (Homology) three-spheres are those three-manifolds that one cannot tell from the standard three-sphere by using the usual topological tools. That such exist indicates the complexity of the world we live in, even ignoring the time dimension, thus making three-spheres of cosmological and physical interest. The topological invariants in the project are intended to distinguish manifolds by systematically studying their properties with respect to several quantum field theories. It is therefore valuable to investigate these new invariants and their structure s for three-spheres and general three-manifolds. This project addresses some of the most fundamental problems in this subject. ***