Collaborative Research: NSF-BSF: Equivariant Symplectic Geometry

合作研究：NSF-BSF：等变辛几何

• 批准号: 2204359
• 负责人: Susan Tolman
• 金额: $ 32.23万
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-09-15 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204359&HistoricalAwards=false
• 关键词: Collaborative; Research; NSF; BSF; Equivariant

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.

辛几何是数学的一个领域，其根源在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，在一张平桌子上旋转的陀螺的动态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性。动量映射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映射来解决辛几何中的对称性和不变量问题。这个研究项目在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驱动的。关键是找到足够简单以易于处理且足够复杂以展示复杂现象的例子。通过他们的工作，研究人员将创建和加强辛几何和其他领域，如代数几何，等变拓扑和数学物理之间的桥梁。继续他们强大的指导记录，研究人员将建议来自世界各地的研究生和博士后研究员，无论是在他们的机构主管，并在非正式的研究环境中访问其他机构，并在他们组织和参加的会议。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生涯：他们将学习新的方法，他们的数学视野将拓宽，他们将与不同的数学社区建立有价值的联系。最后，调查人员一直在参与并计划继续向K-12和大学生推广，特别关注来自服务不足群体的学生。哈密顿群作用产生动量映射。这允许辛约化的构造，其也可以使用几何不变理论来代数地描述。这里资助的研究使用动量映射来解决有关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;复杂性一哈密顿环面作用;完全可积系统;以及动量映射的几何和拓扑。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。 他们之间建立桥梁，他们的工作辛几何和基础问题，在这些相关领域。通过这项研究，霍尔姆，Karshon，Kessler和Tolman将实现更深入的理解之间的关系的几何哈密顿系统和组合数学的动量图像。支持的活动将推进我们在辛几何和组合学领域的知识，并将其应用于代数几何、代数拓扑和数学物理。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。