Topics on Low-Dimensional Manifolds

DMS-0204535曙光网该项目将研究拓扑学和几何学中的几个问题。中心主题是可以定义在辛流形和接触流形上的真实结构的主题。有了这样的结构，该提议旨在引入新版本的Gromov-Witten理论，Seiberg-Witten-Floer理论，并将所得到的不变量应用于弦理论和实代数几何。该项目的另一个方面将研究具有开放锥形奇点的流形的相关分析问题。实代数几何涉及在许多现实生活中自然出现的实多项式的实解。尽管这是最古老的数学分支之一，但到目前为止，我们在理解它方面进展甚微。我们的项目将试图通过使用唐纳森、弗洛尔、陶布斯等人的开创性工作中的新机器来部分填补这一空白。统一自然界中的所有四种基本力量是爱因斯坦的梦想。通过量子力学，可以将强力、弱力和电磁力统一起来。然而，我们这个时代的一些最伟大的思想家未能实现与第四种力量-引力的统一。如今，人们普遍认为，弦理论为实现这种最后的统一提供了最好的希望。具有实结构的复流形在弦论中称为方向折叠。对于物理学家来说，这些物体很受欢迎，但对它们的了解仍然很少。该项目的一个目的是为Gromov-Witten奠定严格的数学基础

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