Probing Near-Symplectic 4-Manifolds and Contact 3-Manifolds with Seiberg-Witten Theory

用 Seiberg-Witten 理论探测近辛 4 流形和接触 3 流形

• 批准号: 2105445
• 负责人: Chris Gerig
• 金额: $ 15.52万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2021-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105445&HistoricalAwards=false
• 关键词: Probing; Near; Symplectic; Manifolds; Contact

Gauge theory describes and exploits the symmetries of nature; among other things it forms the foundation of classical electrodynamics and quantum physics. Symplectic geometry describes the dynamics of nature; among other things it forms the foundation of classical mechanics. Our universe seemingly has four dimensions, three spatial and one time, and it has some curvature, but we do not know exactly what the global picture is. Motions of particles in our universe can be cyclical, but we do not always know how many such periodic trajectories these particles can have. The goal of this project is to use both gauge theory and symplectic geometry in tandem to detect the possible smooth shapes of our universe and to probe the evolution of systems that can arise in said universe, and to develop these mathematical tools further in order to be more efficient in our calculations. Especially, the PI will check whether certain dynamical systems have infinitely many periodic orbits, using the gauge-theoretic “Seiberg-Witten monopoles” and the symplectic-style “Reeb orbits”. The broader impacts of this pursuit involve mentoring students and outreach.The classification of smooth 4-manifolds and dynamics of Reeb vector fields on 3-manifolds have been long-sought out goals in the community, with much progress. This project will be able to contribute with new and refined methods, by extending and exploiting the relations between Seiberg-Witten solutions and pseudoholomorphic curves and Reeb orbits. The Seiberg-Witten invariants of 4-manifolds may be recovered by suitable counts of said curves and orbits, using 2-forms that are symplectic almost everywhere. The PI intends to use this transcription and these near-symplectic 2-forms to probe the structure of the SW invariants, and to search for diffeomorphisms between two homeomorphic symplectic 4-manifolds that are known to become diffeomorphic after a blow-up. In 3 dimensions the PI intends to give quantitative refinements to the Weinstein conjecture that asserts the existence of periodic Reeb orbits on every closed contact 3-manifold, by re-analyzing and extending the proof of said conjecture to the case of certain non-closed contact 3-manifolds using a newly developed SW-Floer homology theory.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.

规范理论描述并利用了自然的对称性；辛几何描述了自然的动力学；我们的宇宙似乎有四个维度，三个空间和一个时间，并且它具有一定的曲率，但我们并不确切地知道宇宙中的粒子运动可以是周期性的，但我们并不总是知道这些粒子可以有多少个这样的周期性轨迹。该项目的目标是同时使用规范理论和辛几何来检测我们宇宙可能的平滑形状，并探索所述宇宙中可能出现的系统的演化，并进一步开发这些数学工具。特别是，PI 将使用规范理论“Seiberg-Witten 单极子”和辛式“Reeb 轨道”来检查某些动力系统是否具有无限多个周期轨道。这一追求的更广泛影响包括指导学生和推广。光滑 4 流形的分类和 Reeb 矢量场在 3 流形上的动力学一直是社区长期寻求的目标，该项目将能够取得很大进展。通过扩展和利用 Seiberg-Witten 解与伪全纯曲线和 4 流形的 Seiberg-Witten 不变量之间的关系，贡献了新的和改进的方法。可以通过使用几乎处处辛的2-形式对所述曲线和轨道进行适当的计数来恢复。PI打算使用该转录和这些近辛的2-形式来探测SW不变量的结构，并寻找。两个同胚辛 4 流形之间的微分同胚，已知在 3 维中会变得微分同胚，PI 打算对其进行定量细化。韦恩斯坦猜想，通过使用新开发的 SW-Floer 同调重新分析并将该猜想的证明扩展到某些非闭合接触 3 流形的情况，断言每个闭合接触 3 流形上存在周期性 Reeb 轨道该奖项反映了 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。