Moment maps and Morse theory

本研究的主要目标是研究群作用在辛几何和其他密切相关的几何中的作用。Tolman和她的合作者计划研究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能深入了解辛流形，还能对原始问题提供新的见解。例如，R. Goldin和Tolman正致力于将舒伯特演算的结果扩展到更一般的辛流形。L. Godinho和Tolman正试图证明Petrie猜想的辛类比。托尔曼正在研究的其他问题旨在通过确定这些空间是否总是像我们通常考虑的自然例子一样表现良好，来探索具有群体行为的辛流形的“地理”。例如，Tolman正在考虑什么时候辛作用是哈密顿的，什么时候这样的流形具有硬Lefshetz性质，以及它们的图必须服从什么限制。她和Karshon还对2n维辛流形上的（n-1）维哈密顿环面作用进行了分类；这将提供一个新的示例来源。许多重要的流形最自然地以辛商的形式出现。虽然这个事实已经被非常成功地用于计算它们的有理上同环，但许多定理并不适用于整数。因此，T. Holm和Tolman正在研究如何计算这类商的整上同环。允许退化的辛形式的推广，如近辛形式和折叠辛形式，最近在四流形的研究中发挥了重要作用。托尔曼计划致力于理解这些几何图形中的角色组行为。最后，Y. Lin和Tolman利用广义Kaehler约简构造了在弦理论中起重要作用的比厄米结构的新例子。在经典物理学中，物理系统的对称性产生了守恒量。例如，太阳系的总角动量是恒定的。从数学的角度来看，研究这些物理系统就相当于研究一种特殊类型的辛流形上的哈密顿群作用——协切束。本研究的潜在目标是更好地理解哪些类型的辛流形允许群行为，以及如何计算它们的不变量。托尔曼和她的合作者计划从多个方面来解决这个问题。她希望这将导致人们对数学和物理学中一个日益重要的领域有更深入的了解。

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.