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

规范论描述并利用了自然界的对称性；它是经典电动力学和量子物理学的基础。辛几何描述自然界的动力学；它构成了经典力学的基础。我们的宇宙似乎有四个维度，三个空间和一个时间，它有一些曲率，但我们并不确切地知道全局的图像是什么。我们宇宙中粒子的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这样的周期性轨迹。这个项目的目标是同时使用规范理论和辛几何来探测我们宇宙可能的光滑形状，并探测可能在宇宙中出现的系统的演变，并进一步发展这些数学工具，以便在我们的计算中更有效。特别是，PI将使用量规理论的“Seiberg-Witten单极子”和辛式的“Reeb轨道”来检查某些动力系统是否具有无限多个周期轨道。这种追求的更广泛的影响包括指导学生和推广。光滑4流形的分类和3流形上Reeb向量场的动力学是学界长期追求的目标，并取得了很大进展。该项目将能够通过扩展和利用Seiberg-Witten解与伪全纯曲线和Reeb轨道之间的关系，为新的和改进的方法做出贡献。4流形的Seiberg-Witten不变量可以通过对所述曲线和轨道的适当计数来恢复，使用几乎处处辛的2-形式。PI打算使用这种转录和这些近辛的2-形式来探测SW不变量的结构，并在两个已知在爆炸后变为微分的同胚辛4流形之间寻找微分同态。在三维空间中，PI打算通过使用新开发的swf - floer同调理论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，从而对断言在每个闭合接触3流形上存在周期Reeb轨道的Weinstein猜想进行定量改进。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。

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.