Symplectomorphism Groups and Symplectic Topology

辛几何以其哈密顿公式为经典力学提供了数学框架。它也是量子理论和广义相对论的现代方法的基础。在过去的25年里，对经典和量子系统之间相互作用的研究，以及强大的几何和分析技术的引入，导致了辛物体特有的新结构和现象的发现，这些结构和现象在经典微分几何和拓扑学中没有对应。辛几何的一个中心问题是理解辛空间的对称性，也称为辛形态。给定空间的所有辛对称的集合本身是一个无限维的几何对象，称为空间的辛形态群。这个组是一个非常丰富的几何对象，它编码了底层空间的所有属性。从物理的角度来看，相空间的连续对称族对应于守恒量，如能量、动量、角动量等。此外，经典物理系统的时间演化对应于相空间的辛形态群上的连续路径。因此，辛变换的一般性质对应于经典物理系统的一般性质。此外，通过“量子化”过程，我们经常能够深入了解量子系统的特性。相反，量子系统的特性有时会通过一种称为“半经典近似”的过程导致对经典系统的惊人预测。在这项研究中，我对辛同构群的同伦理论和几何性质特别感兴趣，以期在低维几何和数学物理中的应用。我计划进一步研究有理数4流形的辛形态群的同伦类型，确定辛嵌入和拉格朗日嵌入的一些空间的同伦类型，研究辛群作用，并探索（很少被理解的）现代辛拓扑与量子系统之间的联系，这些联系是基于辛变换的内在性质的新的量化过程给出的。

Symplectic geometry provides the mathematical framework of classical mechanics in its Hamiltonian formulation. It also underlies modern approaches to quantum theories and to General Relativity. During the past 25 years, the study of the interactions between classical and quantum systems, as well as the introduction of powerful geometrical and analytical techniques, has led to the discovery of new structures and phenomena peculiar to symplectic objects that have no counterparts in classical differential geometry and topology. A central problem in symplectic geometry is to understand the symmetries, also called symplectomorphisms, of symplectic spaces. The set of all symplectic symmetries of a given space is itself an infinite dimensional geometric object called the symplectomorphism group of the space. This group is a very rich geometric object that encodes all the properties of the underlying space. From a physical point of view, continuous families of symmetries of the phase space correspond to conserved quantities like energy, momentum, angular momentum, etc. Moreover, the time evolution of a classical physical system correspond to a continous path on the symplectomorphism group of the phase space. Consequently, general properties of symplectic transformations correspond to general properties of classical physical systems. Moreover, through "quantization" procedures, we often get insights into properties of quantum systems. Conversely, properties of quantum systems sometimes lead to surprizing predictions for classical systems through a procedure called "semi-classical approximation". In this research, I am especially interested in the homotopy-theoretical and geometric properties of symplectomorphism groups, with a view to applications to low dimensional geometry and to mathematical physics. I plan to further study the homotopy type of symplectomorphism groups of rational 4-manifolds, to determine the homotopy type of some spaces of symplectic and Lagrangian embeddings, to study symplectic group actions, and to explore the (very little understood) links between modern symplectic topology and quantum systems given by new quantization procedures that are based on intrinsic properties of symplectic transformations.