Title: Airy structures and symplectic geometry of topological recursion,
Abstract: We propose a new approach to the topological recursion of Eynard-Orantin
based on the notion of Airy structure, which we introduce in the paper. We
explain why Airy structure is a more fundamental object than the one of the
spectral curve. We explain how the concept of quantization of Airy structure
leads naturally to the formulas of topological recursion as well as their
generalizations. The notion of spectral curve is also considered in a more
general framework of Poisson surfaces endowed with foliation. We explain how
the deformation theory of spectral curves is related to Airy structures. Few
other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from
the general point of view of Airy structures.