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

规范理论描述并利用了自然界的对称性;除其他外，它构成了经典电动力学和量子物理学的基础。辛几何描述了自然界的动力学;除其他外，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，但我们不知道确切的全球图片是什么。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。该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的可能光滑形状，并探测可能出现在所述宇宙中的系统的演化，并进一步开发这些数学工具，以便在我们的计算中更有效。特别是，PI将检查某些动力系统是否有无穷多个周期轨道，使用规范理论的“Seiberg-Witten单极”和辛式的“Reeb轨道”。这一追求的更广泛的影响涉及指导学生和推广。光滑4-流形的分类和3-流形上的Reeb向量场的动力学一直是社区长期寻求的目标，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2-形式来探测SW不变量的结构，并搜索两个同胚辛4-流形之间的同胚，这两个同胚辛4-流形已知在爆破后变成同胚。在三维空间中，PI打算对温斯坦猜想进行定量改进，该猜想断言在每个闭接触三维流形上存在周期性Reeb轨道，利用新发展的SW-流形，重新分析并推广了上述猜想的证明，弗洛尔同源理论。该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查的支持的搜索.

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.