The Topology of Symplectomorphism Groups

摘要奖：DMS-0604769主要研究员：Dusa McDuff辛流形在小尺度上看起来都很相似，这意味着它们有许多辛同构，即保持结构的自映射。这些群形成了非常有趣的群，人们可以尝试从拓扑/代数的观点以及几何的观点来理解这些群。麦克达夫几年来一直在研究它们的结构，并建议继续这项工作。一个项目是将她对四维有理/正则流形的辛同构群的分析(在之前与Abreu的联合工作中)扩展到更高维的类似流形。这里出现了许多新的问题，因为流形不再被伪全纯曲线缠绕。她建议研究辛流形的内部结构(例如嵌入辛球或拉格朗日环面的空间)，看看这如何反映在辛同构群的结构中。她还与霍费尔提出了一个联合项目，该项目将发展一种分支流形理论，该理论可用于新的多折叠理论的等变版本。为了了解我们的物理世界，发展各种测量太空物体的方法是很重要的。最熟悉的是距离和角度的测量，这导致了通常的几何概念。这一建议涉及一种不同的几何，称为辛几何，它是基于对二维物体的测量。正如150多年前汉密尔顿首次发现的那样，许多物理学家感兴趣的空间都有这种结构。例如，它构成了像行星系统这样的能量守恒系统的运动方程，因为在这里，每个位置坐标都与一个动量坐标配对，形成一个基本的二维物体。辛几何与弦理论和镜像对称等现代物理理论也有很大关系。近年来，数学和物理之间进行了一次非常重要的思想交流，这主要是用辛术语来表示的。因此，加深我们对辛几何基本原理的理解是非常重要的。这个项目将发展我们在这个领域的基本知识，特别是集中在理解保持这种几何的运动组的性质上。它将利用来自物理学的新的不变量(如量子上同调和辛场理论)来理解这些群的拓扑结构。除了理论上的兴趣，新的辛几何还导致了计算哈密顿运动的更好方法的发展，例如卫星的轨道。

AbstractAward: DMS-0604769Principal Investigator: Dusa McDuff Symplectic manifolds all look alike at small scales, which meansthat they have many symplectomorphisms, that is, structurepreserving self-mappings. These form very interesting groups,which one can try to understand from a topological/algebraicviewpoint as well as a geometric viewpoint. McDuff has beenstudying their structure for several years, and proposes tocontinue this work. One project is to extend her analysis (inprevious joint work with Abreu) of the symplectomorphism groupfor rational/ruled manifolds of dimension four to similarmanifolds in higher dimension. Many new problems arise herebecause the manifold is no longer fibered by pseudoholomorphiccurves. She proposes to study the internal structure ofsymplectic manifolds (such as spaces of embedded symplectic ballsor Lagrangian tori), to see how this is reflected in thestructure of the group of symplectomorphisms. She also proposesa joint project with Hofer, which would develop a theory ofbranched manifolds that could be used in an equivariant versionof the new polyfold theory. In order to understand our physical world it is important todevelop a variety of ways of measuring objects in space. Themost familiar is the measurement of distances and angles, whichleads to the usual notions of geometry. This proposal concerns adifferent kind of geometry, called symplectic geometry, that isbased on measurements of two-dimensional objects. As firstdiscovered by Hamilton over 150 years ago, many spaces ofinterest to physicists have this kind of structure. For example,it underlies the equations of motion of energy-conserving systemssuch as the planetary system, since here each position coordinateis paired with a momentum coordinate to form a basictwo-dimensional object. Symplectic geometry also has muchrelevance to modern theories of physics such as string theory andmirror symmetry. In recent years there has been a very importantexchange of ideas between mathematics and physics that is largelyexpressed in symplectic terms. Hence it is very important tofurther our understanding of the fundamentals of symplecticgeometry. This project will develop our basic knowledge in thisarea, concentrating particularly on understanding the propertiesof the group of motions that preserve this geometry. It willexploit the new invariants coming from physics (such as quantumcohomology and symplectic field theory) in order to understandthe topological structure of these groups. Besides itstheoretical interest, the new symplectic geometry has led to thedevelopment of better methods for computing Hamiltonian motionssuch as the orbit of a satellite.