Contact Homology and Quantitative Symplectic Geometry

辛几何是经典力学的基础几何。在经典力学中，物理系统在任何给定时刻的状态由n个位置坐标和n个动量坐标来描述。这2n个坐标指定2n维相空间中的一个点。相空间具有辛流形的结构，理解这个辛流形的几何对于理解物理系统如何在时间上演化至关重要。这个项目将研究辛几何中密切相关的“空间”和“时间”概念。要研究的“空间”概念是辛嵌入的概念：这是一种坐标的变化，它保留了辛结构，但可能使运动方程更容易求解。要研究的“时间”概念是Reeb轨道：它描述了在时间上重复的行为，例如一颗行星围绕恒星运行。为了研究辛嵌入和Reeb轨道，将开发各种形式的接触同源的新技术。特别是，将对柱面接触同调及其更一般定义的对偶--正圆等变辛同调进行基础性工作。这将导致新的辛嵌入障碍。此外，还将建设嵌入接触同源(ECH)的新基础设施。特别地，我们将研究由接触三流形中的横向纽结决定的ECH上的滤子，并将其应用于保面积曲面微分同态的动力学。ECH上的“J-零”指标将被进一步研究，并在更多的情况下应用于证明无穷多个Reeb轨道的存在性。在这个机器的帮助下，围绕Viterbo猜想的一系列问题将被研究，即关于凸集的正规化辛容的唯一性。将比较已知的辛容量，并阐述凸性的辛意义。将开发计算辛容量的新算法，以便能够通过计算机辅助搜索Viterbo猜想和相关猜想的反例。我们还将探索接触同调的其他形式在定量辛几何问题中的应用。这些接触同调的其他版本包括有理辛场理论，以及可能的高维ECH的类似。

Symplectic geometry is the fundamental geometry underlying classical mechanics. In classical mechanics, the state of a physical system at any given time is described by n position coordinates and n momentum coordinates. These 2n coordinates specify a point in a 2n-dimensional phase space. The phase space has the structure of a symplectic manifold, and understanding the geometry of this symplectic manifold is crucial to understanding how the physical system evolves in time. This project will investigate closely related notions of "space" and "time" in symplectic geometry. The "space" notion to be studied is that of a symplectic embedding: this is a change of coordinates which preserves the symplectic structure but may make the equations of motion easier to solve. The "time" notion to be investigated is that of a Reeb orbit: this describes behavior which repeats in time, such as a planet orbiting around a star.In order to study symplectic embeddings and Reeb orbits, new technology in various forms of contact homology will be developed. In particular, foundational work will be carried out on cylindrical contact homology and its more generally defined counterpart, positive circle-equivariant symplectic homology. This will lead to new symplectic embedding obstructions. Also, new infrastructure in embedded contact homology (ECH) will be constructed. In particular, a filtration on ECH determined by a transverse knot in a contact three-manifold will be studied, with applications to the dynamics of area-preserving surface diffeomorphisms. The "J-zero" index on ECH will be further studied with applications to proving the existence of infinitely many Reeb orbits in more cases. With the help of this machinery, a circle of questions surrounding Viterbo's conjecture, on the uniqueness of normalized symplectic capacities for convex sets, will be investigated. Known symplectic capacities will be compared, and the symplectic significance of convexity will be elaborated. New algorithms for computing symplectic capacities will be developed, in order to enable computer-assisted searches for counterexamples to Viterbo's and related conjectures. Applications of other versions of contact homology to questions in quantitative symplectic geometry will also be explored. These other versions of contact homology include rational symplectic field theory, and possible analogues of ECH in higher dimensions.