Topological recursion (TR) is a remarkable universal recursive structure that has been found in many enumerative geometry problems, from combinatorics of maps (discrete surfaces), to random matrices, Gromov-Witten invariants, knot polynomials, conformal blocks, integrable systems... An example of topological recursion is the famous Mirzakhani recursion that determines recursively the hyperbolic volumes of moduli spaces. It is a recursion on the Euler characteristic, whence the name "topological" recursion.
A recursion needs an initial data: a "spectral curve" (which we shall define), and the recursion defines the sequence of "TR-invariants" of that spectral curve.
In this series of lectures, we shall:
- define the topological recursion, spectral curves and their TR-invariants, and illustrated with examples.
- state and prove many important properties, in particular how TR-invariants get deformed under deformations of the spectral curve, and how they are related to intersection numbers of moduli spaces of Riemann surfaces, for example the link to Givental formalism.
- introduce the new algebraic approach by Kontsevich-Soibelman, in terms of quantum Airy structures.
- present the relationship of these invariants to integrable systems, tau functions, quantum curves.
- if time permits, we shall present the conjectured relationship to Jones and Homfly polynomials of knots, as an extension of the volume conjecture.