The role of group actions in symplectic geometry

群作用在辛几何中的作用

• 批准号: 1206365
• 负责人: Susan Tolman
• 金额: $ 19.16万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206365&HistoricalAwards=false
• 关键词: role; group; actions; symplectic; geometry

The main goal of Prof. Tolman's research is to investigate the role of group actions in symplectic geometry. She intends to focus on three related areas. First, she will explore the conditions that force a symplectic action to be Hamiltonian; in particular, she plans to construct a compact six-dimensional symplectic manifold with a non-Hamiltonian symplectic circle action with exactly two fixed points. Second, she will work on classifying symplectic manifolds with ``large" torus actions. More specifically, she will finish her project with Y. Karshon on n-1 dimensional torus actions on 2n dimensional manifolds, analyze Hamiltonian circle actions on six-manifolds, and find applications in both cases -- especially to Fano manifolds. Third, R. Goldin and S. Tolman will generalize ideas from algebraic combinatorics to the more general symplectic setting of Hamiltonian actions on symplectic manifolds. In addition to these three main areas, she will work with her collaborators on a number of other projects: calculating the cohomology of the symplectic quotients of Hamiltonian loop group actions, and computing the integer cohomology of symplectic manifolds and their quotients in the finite dimensional case. Taken together, these results will significantly advance our understanding of this field.Consider a physical system, such as a planet orbiting around a star. We need to keep track of the position and momentum of each object: The set of all possible measurements is called phase space. For the two-body system, the phase space is twelve dimensional Euclidean space. Even for more complicated systems, the phase space looks locally like Euclidean space. Moreover, the rate of change over time of the momentum of an object is determined by the rate of change of the total energy of the system with respect to the position of that object; the converse also holds. Finally, many physical systems have symmetries. For example, if we ignore any external gravitational field, the solar system has three-dimensional rotational symmetry. Prof. Tolman studies a mathematical generalization of phase space, called "symplectic manifolds." Her research focuses on understanding the role of symmetries on these spaces. For example, she is studying when symplectic manifolds with symmetry are as well behaved as algebraic varieties, which are the spaces cut out by solutions to polynomial equations. Working with Prof. Karshon, she is trying to describe all symplectic manifolds with extremely large amounts of symmetry.

托尔曼教授研究的主要目标是研究群作用在辛几何中的作用。她打算专注于三个相关领域。首先，她将探索迫使辛作用成为哈密顿作用的条件；特别是，她计划构造一个紧致的六维辛流形，其中具有恰好两个不动点的非哈密尔顿辛圆作用。其次，她将致力于对具有“大”环面作用的辛流形进行分类。更具体地说，她将与Y.Karshon一起完成关于2n维流形上n-1维环面作用的项目，分析六维流形上的哈密尔顿圆作用，并在这两种情况下找到应用--特别是Fano流形。第三，R.Goldin和S.Tolman将把代数组合学的思想推广到辛流形上哈密顿作用的更一般的辛集上。除了这三个主要领域外，她还将与她的合作者合作一些其他项目：计算哈密尔顿环群作用的辛商的上同调，以及计算有限维情形下辛流形及其商的整数上同调。综上所述，这些结果将极大地促进我们对这一领域的理解。考虑一个物理系统，例如一颗围绕恒星运行的行星。我们需要跟踪每个物体的位置和动量：所有可能的测量的集合被称为相空间。对于二体系统，相空间是十二维欧几里得空间。即使对于更复杂的系统，相空间也局部地看起来像欧几里德空间。此外，物体动量随时间的变化率由系统总能量相对于该物体位置的变化率决定；反之亦然。最后，许多物理系统具有对称性。例如，如果我们忽略任何外部引力场，太阳系就具有三维旋转对称性。托尔曼教授研究相空间的一种数学推广，称为“辛流形”。她的研究重点是了解对称性在这些空间中的作用。例如，她正在研究具有对称性的辛流形何时表现得像代数簇一样好，代数簇是由多项式方程的解切割出来的空间。与Karshon教授合作，她试图描述所有具有极大对称性的辛流形。