Topics on Low-Dimensional Manifolds

点击翻译按钮获取中文摘要

DMS-0204535Shuguang WangThe project will study several issues in Topology and Geometry.The central theme is that of real structures which can be definedon symplectic and contact manifolds. With the presence of suchstructures, the proposal aims to introduce new versions of Gromov-Wittentheory, Seiberg-Witten-Floer theory, and to apply the resultinginvariants to String Theory as well as Real Algebraic Geometry.Another aspect of the project will study related analytical problemson manifolds that possess open conical singularities.Real Algebraic Geometry concerns the real solutions of real polynomialswhich arise naturally in many real-life applications. Despite thefact that this is one of oldest Mathematical branches, we have so farmade very little progress in understanding it. Our project will attemptto fill the gap partially by using new machinery following from thepioneer work of Donaldson, Floer, Taubes etc. It was Einstein's dreamto unify all four fundamental forces in the nature. Through QuantumMechanics, it is possible to unify the strong force, the weak force andthe electro-magnetic force. However the unification with the fourthforce, the gravity, has eluded some of the greatest thinkers of our time.Nowadays it is commonly believed that String Theory offers the best hopeto achieve this last unification. Complex manifolds with real structureare called orientifolds in String Theory. These are popular butstill poorly-understood objects for Physicists. A purpose in the projectis to lay down a rigorous mathematical foundation for Gromov-Witten