The role of group actions in symplectic geometry

托尔曼教授的主要研究目标是研究群作用在辛几何中的作用。她打算把重点放在三个相关领域。首先，她将探索迫使辛作用为哈密顿量的条件；特别是，她计划构造一个紧致的六维辛流形，其非哈密顿辛圆作用有两个不动点。其次，她将研究具有“大”环面作用的辛流形的分类。更具体地说，她将与Y. Karshon一起完成她的项目，研究n-1维环面在2n维流形上的作用，分析六维流形上的哈密顿圆作用，并在这两种情况下找到应用——特别是在法诺流形上。第三，R. Goldin和S. Tolman将把代数组合的思想推广到辛流形上的哈密顿作用的更一般的辛集。除了这三个主要领域，她将与她的合作者在许多其他项目上工作：计算哈密顿环群作用的辛商的上同调，以及计算有限维情况下辛流形及其商的整数上同调。综上所述，这些结果将显著促进我们对这一领域的理解。考虑一个物理系统，比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：所有可能测量的集合称为相空间。对于二体系统，相空间是十二维欧几里德空间。即使对于更复杂的系统，相空间局部看起来像欧几里得空间。此外，一个物体的动量随时间的变化率是由系统的总能量相对于该物体的位置的变化率决定的；反之亦然。最后，许多物理系统都具有对称性。例如，如果我们忽略任何外部引力场，太阳系具有三维旋转对称性。托尔曼教授研究相空间的数学推广，称为“辛流形”。她的研究重点是理解对称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在什么情况下表现得和代数变异体一样好，代数变异体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。

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.