Gauge theory, 3-manifolds, and smooth 4-manifolds

规范理论、3 流形和光滑 4 流形

• 批准号: 0107792
• 负责人: Zoltan Szabo
• 金额: $ 13.45万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107792&HistoricalAwards=false
• 关键词: Gauge; theory; manifolds; smooth

DMS-0107792Zoltan Szabo The main theme of this project is the study of three-manifolds and smooth four-manifolds by using gauge theory and symplectic geometry. The Principal Investigator proposes to study and further develop a recent construction that give invariants for closed oriented 3-manifolds and a related construction that give invariants for smooth closed oriented 4-manifolds. These constructions use handle decompositions and a versionof Lagrangian Floer-homology. The PI will also study applications of these new techniques in 3 and 4-manifold topology. In particular PI will study these invariants for three-manifolds that fiber over a circle and symplectic 4-manifolds.Gauge theory is a very important technique in smooth 4-manifold theory. It involves the theory of Donaldson invariants and Seiberg-Witten invariants. These invariants are constructed by solving certain elliptic partial differential equations inspired by mathematical physics. Both constructions have ramifications for three-manifolds as well, theseare related to solving the partial differential equations over four-manifolds with a three-dimensional boundary. In addition to studying these invariants the Principal Investigator also proposes to study in detail a third construction that involves holomorphic disks and a decomposition of the manifold into elementary pieces.

DMS-0107792 Zoltan Szabo本项目的主题是利用规范理论和辛几何研究三维流形和光滑四维流形。 主要研究者建议研究和进一步发展一个最近的构造，给出封闭定向3-流形的不变量和一个相关的构造，给出光滑封闭定向4-流形的不变量。这些构造使用句柄分解和一个版本的拉格朗日弗洛尔同调。PI还将研究这些新技术在3和4流形拓扑中的应用。特别是PI将研究这些不变量的三个流形，纤维在一个圆和辛四维流形。规范理论是一个非常重要的技术，在光滑四维流形理论。 它涉及到唐纳森不变量和Seiberg-Witten不变量理论。这些不变量是由数学物理启发的椭圆型偏微分方程的求解而构造的。这两种构造对三维流形也有分支，这些分支与求解具有三维边界的四维流形上的偏微分方程有关。除了研究这些不变量的主要研究者还建议详细研究第三个建设，涉及全纯磁盘和分解的流形成基本件。