Symplectic Topology and Hamiltonian Dynamics

辛几何是Kahler几何的一个有趣的推广，推广到一类尚未完全理解的光滑流形上。虽然它保留了Kahler世界的许多结构特征(例如Lefschetz铅笔和复杂曲线)的回声，但辛几何比Kahler几何灵活得多。特别地，每个有限维辛流形都有一个无限维保结构变换群(称为辛同构)，而Kahler情形中的相应群必然是有限维的。McDuff提出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。一个重要的问题是建立一个判定辛同态群中给定的圆是否同伦平凡的准则。McDuff一直在与Sue Tolman在这个问题上合作，并建议继续这种合作，专门研究辛环流形的情况。在另一个单独的项目中，她希望发展一个更全面的辛特征类理论，为理解这些问题提供同源工具。辛几何和Kahler几何在现代理论物理中都是非常重要的；弦理论经常研究定义在一种特殊的六维Kahler空间上的场，这种空间称为Calabi-Yau流形，而许多方程和函数的瞬子修正通常是用纯辛项定义的。为了理解一种几何，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德几何中，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。有一种广为人知的理论(李群理论)适用于刚性几何，如欧几里得或卡勒几何。这个项目的一个主要目的是看看在更加松散和灵活的辛世界中，有多少结构仍然存在，在这个世界里，有无限多本质上不同的方式来扰乱空间。

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.