Symplectic Topology and Hamiltonian Dynamics

辛拓扑和哈密顿动力学

• 批准号: 0305939
• 负责人: Dusa McDuff
• 金额: $ 31.33万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305939&HistoricalAwards=false
• 关键词: Symplectic; Topology; Hamiltonian; Dynamics

Symplectic geometry is an intriguing generalization of Kahler geometry to a wide, and as yet not completely understood, class of smooth manifolds. Though it retains echoes of many of the structural features of the Kahler world (for example Lefschetz pencils, and complex curves),symplectic geometry is much more flexible than Kahler geometry.In particular, every finite dimensional symplectic manifold has an infinite dimensional group of structure-preserving transformations (called symplectomorphisms), while the corresponding group in the Kahler case is necessarily finite dimensional. McDuff proposes to study the topological properties of the whole symplectomorphism group, and in particular its relation to its finite dimensional subgroups. One important question is to develop a criterion for detecting if a given circle in the symplectomorphism group is homotopially trivial. McDuff has been working with Sue Tolman on this question and proposes to continue this collaboration, making a special study of the case of symplectic toric manifolds. In another separate project, she hopes to develop a fuller theory of symplectic characteristic classes to provide homological tools for understanding these questions. Both symplectic and Kahler geometry are very important in modern theoretical physics; string theories often study fields defined over a special kind of six dimensional Kahler space called a Calabi--Yau manifold, while instanton corrections to many equations and functions are often defined in purely symplectic terms. In order to understand a geometry it is essential to understand what kind of transformations preserve it; for example in the standard Euclidean geometry studied in high school the structure (distances and angle measurements) is preserved by rotations and translations. This project studies the properties of high dimensional families of these transformations (for example, the set of all rotations of the plane) rather than of individual transformations. There is a well understood theory (the theory of Lie groups) that works for rigid geometries such as Euclidean or Kahler geometry. One main aim of this project is to see how much of the structure remains in the more flabby and flexible symplectic world, where are infinitely many intrinsically different ways of perturbing space.

辛几何是一个有趣的推广Kahler几何广泛，但尚未完全理解，类光滑流形。 虽然它保留了卡勒世界的许多结构特征（例如莱夫谢茨铅笔和复曲线），但辛几何比卡勒几何灵活得多。特别是，每个有限维辛流形都有一个无限维的保结构变换群（称为辛同构），而卡勒情形中相应的群必然是有限维的。麦克达夫建议研究整个辛同构群的拓扑性质，特别是它与有限维子群的关系。 一个重要的问题是发展一个标准，以检测如果一个给定的圆在辛同胚群是同伦平凡的。 麦克达夫一直与苏托尔曼在这个问题上，并建议继续这种合作，使一个特殊的研究情况下，辛复曲面流形。 在另一个独立的项目中，她希望开发一个更全面的辛特征类理论，为理解这些问题提供同调工具。 辛几何和卡勒几何在现代理论物理中都非常重要;弦理论经常研究在一种特殊的六维卡勒空间（称为卡拉比-丘流形）上定义的场，而许多方程和函数的瞬子修正经常用纯辛术语定义。 为了理解一个几何，必须了解什么样的变换可以保持它;例如，在高中学习的标准欧几里得几何中，结构（距离和角度测量）通过旋转和平移来保持。这个项目研究这些变换的高维族的性质（例如，平面的所有旋转的集合），而不是单个变换的性质。 有一个很好理解的理论（李群理论）适用于刚性几何，如欧几里得几何或卡勒几何。这个项目的一个主要目的是看看有多少结构仍然在更松弛和灵活的辛世界，那里有无限多本质上不同的扰动空间的方式。