Topics on Low-Dimensional Manifolds

低维流形主题

• 批准号: 0204535
• 负责人: Shuguang Wang
• 金额: --
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204535&HistoricalAwards=false
• 关键词: Topics; Low; Dimensional; Manifolds

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

DMS-0204535王曙光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的结构。该项目的另一个方面是研究具有开锥奇点的流形上的相关分析问题。真实的代数几何关注的是在许多实际应用中自然出现的真实的多项式的真实的解。尽管这是最古老的数学分支之一，但迄今为止，我们在理解它方面的进展甚微。我们的项目将通过使用唐纳森、弗洛尔、陶伯斯等的开创性工作的新机制来部分填补差距。通过量子力学，有可能统一强、弱力和电磁力。然而，与第四种力--引力--的统一，却让我们这个时代一些最伟大的思想家望尘莫及。如今，人们普遍认为，弦理论提供了实现这最后一种统一的最佳途径。具有真实的结构的复流形在弦论中称为定向折叠。这些都是很受欢迎的，但对物理学家来说仍然是知之甚少的对象。该项目的目的之一是为Gromov-Witten奠定严格的数学基础