Moment maps and Morse theory

矩图和莫尔斯理论

• 批准号: 0707122
• 负责人: Susan Tolman
• 金额: $ 28.72万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707122&HistoricalAwards=false
• 关键词: Moment; maps; Morse; theory

The main goal of the research described in this proposal is to investigate the role of group actions in symplectic geometry and other closely related geometries. Some of the questions that Tolman and her collaborators plan to study are closely related to important problems in other fields; they hope that these projects will not only give insight into symplectic manifolds but also shed new light on the original problems. For example, R. Goldin and Tolman are working on extending the results of Schubert calculus to more general symplectic manifolds. L. Godinho and Tolman are attempting to prove the symplectic analog of the Petrie conjecture. Other questions that Tolman is working on are designed to explore the "geography" of symplectic manifolds with group actions by determining whether such spaces are always as well behaved as the natural examples which we usually consider. For example, Tolman is considering when symplectic actions are Hamiltonian, when such manifolds possess the hard Lefshetz property, and what restrictions their graphs must obey. She and Karshon are also classifying (n-1)-dimensional Hamiltonian torus actions on 2n-dimensional symplectic manifolds; this will provide a new source of examples. Many important manifolds arise most naturally as symplectic quotients. While this fact has been used extremely successfully to compute their rational cohomology rings, many of the theorems do not hold over the integers. Therefore, T. Holm and Tolman are studying how to compute the integral cohomology ring of such quotients. Generalizations of symplectic forms which allow some degeneration, such as near symplectic forms and folded symplectic forms, have recently played an important role in the study of four-manifolds. Tolman plans to work on understanding the role group actions in these geometries. Finally, Y. Lin and Tolman are using generalized Kaehler reduction to construct new examples of Bihermitian structures, which play an important role in string theory.In classical physics, symmetries of a physical system give rise to conserved quantities.For example, the total angular momentum of the solar system is constant. From a mathematical perspective, studying these physical systems corresponds to studying Hamiltonian group actions on a special type of symplectic manifold - a cotangent bundle. The underlying goal of the proposed research is to gain a better understanding of what types of symplectic manifolds admit group actions, and how to calculate their invariants.Working with her collaborators, Tolman plans to attack this question on a number of fronts. She hopes that this will lead to a greater understanding of an area of increasing importance, both within mathematics and for physics.

本提案中描述的研究的主要目标是研究群体作用在符号几何形状和其他密切相关的几何形状中的作用。托尔曼和她的合作者计划学习的一些问题与其他领域的重要问题密切相关。 他们希望这些项目不仅可以深入了解符合性歧管，而且还对原始问题有了新的启示。例如，R。Goldin和Tolman正在努力将Schubert演算的结果扩展到更一般的符号歧管。 L. Godinho和Tolman试图证明Petrie猜想的符合性类似物。 Tolman正在努力的其他问题旨在通过确定此类空间是否总是像我们通常考虑的自然例子一样好。例如，托尔曼（Tolman）正在考虑何时同时行动是哈密顿（Hamiltonian），何时具有硬质量的lefshetz属性，以及他们的图形必须遵守的限制。 她和卡申（Karshon）也在分类（n-1） - 二维圆环对2n维符号歧管的作用；这将提供一个新的示例来源。许多重要的歧管最自然地以符号人的形式出现。尽管这一事实已被极为成功地计算其理性的共同体学环，但许多定理并不符合整数。因此，T。Holm和Tolman正在研究如何计算此类商的整体共同体学环。允许某种变性的象征性形式的概括，例如近乎共生形式和折叠式形式，最近在四个manifolds的研究中发挥了重要作用。托尔曼计划致力于了解这些几何形状中的角色群体行为。最后，Y. Lin和Tolman正在使用广义的Kaehler还原来构建Bihermitian结构的新示例，这些结构在弦理论中起着重要作用。在经典物理学中，物理系统的对称性产生了保守的数量。例如，太阳能系统的总角度动量是恒定的。从数学的角度来看，研究这些物理系统对应于研究哈密顿小组对特殊类型的符号歧管的行为 - cotangengent束。拟议的研究的基本目标是更好地了解哪种类型的符合性歧管的允许小组行动以及如何计算他们的不变性。与她的合作者一起工作，托尔曼计划在许多方面攻击这个问题。她希望这将在数学和物理学内对越来越重要的领域有更深入的了解。