This project will continue Mrowka's investigations into the study of three- and four-dimensional manifolds (mathematical models for space-time). These investigations center on relating properties of solutions and spaces of solutions of the various equations of mathematical physics (primarily the Yang-Mills and Sieberg-Witten equations) to the topology of the space-times that carry the equations. This strategy has been very successful in the past twenty years, as evidenced by work of Yau, Schoen, Taubes, Donaldson, and Uhlenbeck. Mrowka hopes in joint work with Kronheimer to resolve a 30-year-old conjecture about the topology of three-dimensional spaces known as the Property P conjecture, an important step towards Poincare's conjecture that the simplest model of a closed three-dimensional space-time is characterized by the property that every closed loop is contractible to a point without leaving the space-time. ***
