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间的状态由n个位置坐标和n个动量坐标来描述。这2n个坐标指定了2n维相空间中的一个点。相空间具有辛流形的结构，理解辛流形的几何结构对于理解物理系统如何随时间演变至关重要。本项目将探讨辛几何中“空间”和“时间”的密切相关概念。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”概念是里布轨道：它描述了在时间上重复的行为，比如一颗行星绕着一颗恒星公转。为了研究辛嵌入和Reeb轨道，将开发各种形式的接触同源新技术。特别是柱面接触同调及其更广泛定义的对应物正圆等变辛同调的基础工作。这将导致新的辛嵌入障碍。此外，还将构建嵌入式接触同源性（ECH）的新基础结构。特别是，在接触三流形中由横向结决定的ECH上的过滤将被研究，并应用于保面积表面微分同态的动力学。将进一步研究ECH上的“j - 0”指数，并应用于证明在更多情况下存在无限多个Reeb轨道。在这个机制的帮助下，围绕Viterbo猜想的一系列问题，关于凸集的规格化辛能力的唯一性，将被研究。将比较已知的辛容量，并阐述凸性的辛意义。将开发计算辛能力的新算法，以便使计算机辅助搜索维泰博和相关猜想的反例成为可能。其他版本的接触同调在定量辛几何问题中的应用也将被探讨。这些其他版本的接触同调包括有理辛场论，以及可能的高维ECH的类似物。

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.