Applications of Symplectic Geometry

辛几何的应用

• 批准号: 0907110
• 负责人: Jonathan Weitsman
• 金额: $ 1.94万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-10-01 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907110&HistoricalAwards=false
• 关键词: Applications; Symplectic; Geometry

Symplectic geometry, a branch of mathematics having itsorigins in classical dynamics, has become an area of growing importancein contemporary mathematics. In this project we propose severalplans for research that have as their common theme the applicationof ideas from symplectic geometry to other areas of mathematics, andin return the ways in which ideas arising from these applications motivatenew methods in symplectic geometry.One of our projects involves extending ideas from the symplecticgeometry of compact manifolds to the setting of Banach manifoldsappearing in gauge theory and the theory of loop groups. Anotheris a set of applications to combinatorics and real analysis, andpossibly number theory, centered on the generalizations of theclassical Euler-Maclaurin formula. Still another project is anattempt to understand the structure of hyperkahler manifolds byreworking the ideas of Atiyah and Bott on gauge theory over Riemannsurfaces in this context.

辛几何是起源于经典动力学的数学分支，已成为当代数学中日益重要的领域。 在这个项目中，我们提出了几个研究计划，其共同主题是辛几何思想在数学其他领域的应用，以及这些应用产生的思想激发辛几何新方法的方式。我们的项目之一涉及将思想从紧流形的辛几何扩展到规范理论和环群理论中出现的巴纳赫流形的设置。 另一个是组合数学和实分析的一组应用，可能还有数论，以经典欧拉-麦克劳林公式的推广为中心。 还有一个项目是尝试通过在这种背景下重写 Atiyah 和 Bott 关于黎曼曲面规范理论的思想来理解超卡勒流形的结构。