Contact Homology and Quantitative Symplectic Geometry

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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.