Collaborative Research: NSF-BSF: Equivariant Symplectic Geometry

辛几何是数学的一个领域，它植根于经典力学的数学框架。许多物理系统表现出大量的对称性。例如，在平面桌子上旋转的陀螺的动力学不受桌子在房间中的位置的影响；桌子位置的变化相当于系统的对称。动量图将物理系统的对称性转化为离散数据。该奖项支持的研究使用动量图来解决辛几何中的对称性和不变量问题。这个研究项目很大程度上是由深入探索的例子和推进更一般和抽象的理论之间的相互作用所驱动的。关键是要找到足够简单的例子来处理，足够复杂的例子来展示复杂的现象。通过他们的工作，研究人员将建立和加强辛几何与代数几何、等变拓扑和数学物理等其他领域之间的桥梁。这些研究人员将继续他们出色的指导记录，为来自世界各地的研究生和博士后提供建议，无论是作为他们所在机构的导师，还是在访问其他机构的非正式研究环境中，以及在他们组织和参加的会议上。更广泛地说，该奖项支持的活动将为学生和博士后带来新的项目。在他们的机构之间旅行将丰富学生和博士后的经验，并促进他们未来的职业发展：他们将学习新的方法，他们的数学视角将拓宽，他们将与不同的数学社区建立宝贵的联系。最后，调查人员都参与并计划继续向K-12和大学生推广，特别关注那些来自服务不足群体的学生。哈密顿群体作用产生动量图。这允许构造辛约化，辛约化也可以用几何不变量理论进行代数描述。这里资助的研究使用动量映射来解决关于辛几何中的群作用和不变量的问题，重点是：辛四流形的等变几何；等变理论在辛拓扑中的应用单哈密顿环动作的复杂性；完全可积系统；以及动量图的几何和拓扑。每一位研究者都是杰出的沟通者，并被邀请在辛几何和许多相关领域的顶级会议上演讲，包括代数几何、数学物理和组合学。他们在辛几何的工作和这些相关领域的基础问题之间建立了桥梁。通过这项研究，Holm、Karshon、Kessler和Tolman将对哈密顿系统的几何与动量图像的组合之间的关系有更深的理解。支持的活动将推进我们在辛几何和组合学领域的知识，应用于代数几何，代数拓扑和数学物理。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Symplectic geometry is an area of mathematics that has roots in the mathematical framework for classical mechanics. Many physical systems exhibit a great deal of symmetry. For example, the dynamics of a top that is spinning on a flat table is not affected by the position of the table in the room; changes in the table's position amount to a symmetry of the system. The momentum map translates the symmetries of a physical system into discrete data. The research supported by this award uses the momentum map to address questions about symmetries and invariants in symplectic geometry. This research program is largely driven by the interplay between deeply probing examples and advancing more general and abstract theory. The key is to find examples simple enough to be tractable and complex enough to exhibit complicated phenomena. Through their work, the investigators will create and strengthen bridges between symplectic geometry and other areas such as algebraic geometry, equivariant topology, and mathematical physics. Continuing their strong mentoring record, the investigators will advise graduate students and postdoctoral fellows from all over the world, both as supervisors in their institutions, and in informal research settings during visits to other institutions and at the conferences they organize and attend. More broadly, the activities supported by this award will lead to new projects for students and postdocs. Traveling between their institutions will enrich the students' and postdocs' experiences and advance their future careers: they will learn new methods, their mathematical perspective will broaden, and they will make valuable connections with different mathematical communities. Finally, the investigators have all been engaged in and plan to continue the outreach to K-12 and college students, with particular attention to students from underserved groups. Hamiltonian group actions give rise to the momentum map. This allows for the construction of the symplectic reduction, which can also be described algebraically using geometric invariant theory. The research funded here uses the momentum map to address questions about group actions and invariants in symplectic geometry, focusing on: the equivariant geometry of symplectic four-manifolds; applications of equivariant theory to symplectic topology; complexity one Hamiltonian torus actions; completely integrable systems; and the geometry and topology of momentum maps. Each of the investigators is an outstanding communicator and has been invited to lecture at top conferences in symplectic geometry and many related fields, including algebraic geometry, mathematical physics, and combinatorics. They build bridges between their work in symplectic geometry and foundational questions in these related fields. Through this research, Holm, Karshon, Kessler, and Tolman will achieve a deeper understanding of the relationship between the geometry of a Hamiltonian system and the combinatorics of the momentum image. The supported activities will advance our knowledge in the fields of symplectic geometry and combinatorics, with applications to algebraic geometry, algebraic topology and mathematical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.