The role of group actions in symplectic geometry

托尔曼教授研究的主要目标是研究辛几何中群作用的作用。她打算侧重于三个相关领域。首先，她将探讨条件，迫使辛行动是哈密尔顿;特别是，她计划构建一个紧凑的六维辛流形与非哈密尔顿辛圈行动正好有两个不动点。其次，她将致力于分类辛流形与"大”环面行动。更具体地说，她将与Y完成她的项目。Karshon关于2n维流形上的n-1维环面作用，分析了六维流形上的Hamilton圈作用，并在这两种情况下找到了应用--特别是Fano流形。第三，R. Goldin和S.托尔曼将从代数组合学的思想推广到辛流形上的哈密顿作用的更一般的辛设置。除了这三个主要领域，她将与她的合作者在一些其他项目：计算上同调的辛代数的哈密顿圈群行动，并计算整数上同调的辛流形和他们的代数在有限维的情况下。综合起来，这些结果将大大推进我们对这一领域的理解。考虑一个物理系统，比如一颗围绕星星运行的行星。我们需要跟踪每个物体的位置和动量：所有可能的测量值的集合称为相空间。对于二体系统，相空间是12维欧几里得空间。即使对于更复杂的系统，相空间在局部上看起来也像欧几里得空间。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;匡威亦然。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性在这些空间中的作用。 例如，她正在研究当辛流形的对称性以及表现为代数品种，这是空间削减的解决方案多项式方程。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.