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; 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.

符号几何形状是数学领域，它扎根于经典力学的数学框架。许多物理系统表现出很大的对称性。例如，在平台上旋转的顶部的动力学不受桌子在房间中的位置的影响。表的位置的变化等于系统的对称性。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关于对称和不变的几何形状的问题。该研究计划在很大程度上是由深入探索示例与推进更通用和抽象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，以至于足以表现出复杂的现象。通过他们的工作，研究人员将在符号几何形状与其他区域（例如代数几何，拓扑结构和数学物理学）之间创建和加强桥梁。继续他们的指导记录，调查人员将为来自世界各地的研究生和博士后研究员提供建议，既是其机构的主管，又是在访问其他机构和他们组织和参加的会议期间的非正式研究环境中。更广泛地说，该奖项支持的活动将为学生和博士后提供新的项目。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未来的职业：他们将学习新的方法，他们的数学观点将扩大，并且将与不同的数学社区建立宝贵的联系。最后，调查人员都参与其中，并计划继续向K-12和大学生进行宣传，并特别关注服务不足的团体的学生。哈密​​顿小组的行动产生了动量图。这允许构建符合性还原，也可以使用几何不变理论来描述代数。此处资助的研究使用动量图来解决有关符合性几何形状中的小组动作和不变的问题，重点是：符号四序列的均等几何；均等理论的应用到符号拓扑；复杂性一个汉密尔顿的圆环动作；完全可以整合的系统；以及动量图的几何形状和拓扑。每个研究者都是出色的沟通者，并已被邀请在同骨几何学和许多相关领域的顶级会议上演讲，包括代数几何，数学物理学和组合学。 他们在这些相关领域的象征性几何形状和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形状与动量形象的组合之间的关系深入了解。支持的活动将提高我们在符合性几何和组合学领域的知识，并应用代数几何，代数拓扑和数学物理学。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子功能和广泛的影响来评估Criteria的智力功能和广泛影响。