Gauge Theory and the Topology of Smooth Four-Manifolds

9704174 Feehan The project lies in the interface of geometric topology and mathematical physics. Specifically, the investigator is to establish the equivalence between Donaldson and Seiberg-Witten invariants for smooth four-manifolds. The investigator is to use the approach first suggested by Pidstrigach and Tyurin, which is based on non-quantum ideas. The tools to be used for implementing the Pidstrigach-Tyurin approach include Taubes's gluing maps and nonabelian monopoles. Smooth four-dimensional spaces have received a great deal of attention in recent years due to their applications in relativity theory and due to a recent mathematical discovery allowing their coarse classification. It is notable that instantons and monopoles, two major ingredients in the classification of smooth four-manifolds, both were originally introduced by the physicists studying elementary particles.