Topics on Low-Dimensional Manifolds

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

DMS-0204535 Shuguang Wang The project will study several issues in Topology and Geometry. The central theme is that of real structures which can be defined on symplectic and contact manifolds. With the presence of such structures, the proposal aims to introduce new versions of Gromov-Witten theory, Seiberg-Witten-Floer theory, and to apply the resulting invariants to String Theory as well as Real Algebraic Geometry. Another aspect of the project will study related analytical problems on manifolds that possess open conical singularities. Real Algebraic Geometry concerns the real solutions of real polynomials which arise naturally in many real-life applications. Despite the fact that this is one of oldest Mathematical branches, we have so far made very little progress in understanding it. Our project will attempt to fill the gap partially by using new machinery following from the pioneer work of Donaldson, Floer, Taubes etc. It was Einstein's dream to unify all four fundamental forces in the nature. Through Quantum Mechanics, it is possible to unify the strong force, the weak force and the electro-magnetic force. However the unification with the fourth force, the gravity, has eluded some of the greatest thinkers of our time. Nowadays it is commonly believed that String Theory offers the best hope to achieve this last unification. Complex manifolds with real structure are called orientifolds in String Theory. These are popular but still poorly-understood objects for Physicists. A purpose in the project is to lay down a rigorous mathematical foundation for Gromov-Witten