Applications of Symplectic Geometry

辛几何是数学的一个分支，起源于经典动力学，在当代数学中已成为一个日益重要的领域。 在这个项目中，我们提出了几个研究计划，其共同主题是将辛几何的思想应用于其他数学领域，并反过来研究这些应用所产生的思想如何激发辛几何的新方法。我们的一个项目涉及将紧流形的辛几何思想扩展到规范理论和圈群理论中出现的Banach流形的设置。 另一个是一套应用组合学和真实的分析，并可能数论，集中在推广的经典欧拉-麦克劳林公式。 还有一个项目是试图了解结构的hyperkahler流形改造的想法阿提亚和博特规范理论的黎曼曲面在这方面。

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.