Starting from a Liouville 1-form on a surface, we have been led to 3-dimensional contact geometry, and convex surfaces. We now go in the other direction.
From Liouville geometry to contact geometry
![From Liouville geometry to contact geometry The standard contact structure on R^3.](https://i0.wp.com/www.danielmathews.info/wp-content/uploads/2018/06/329px-Standard_contact_structure.svg_.png?resize=329%2C149&ssl=1)
(Technical) We’re going to take Liouville structures and move them into 3 dimensions, to obtain contact structures.
A-infinity algebras, strand algebras, and contact categories
In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.