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About Topological Recursion

About Topological Recursion
Topological recursion (TR) is a tool which initially appeared in the context of random matrix models. Today it has far reaching applications in enumerative algebraic geometry, topological string theory, integrable systems, WKB asymptotic analysis and knots invariants, only to name a few. In the resolution of matrix integrals, the data is expressed in terms of a __spectral curve__: an algebraic plane curve together with a heat kernel representing the random spectrum of the matrices. Topological recursion produces a solution from this data, encoded in differential forms, the so called TR _invariants_ of the curve.

At a more abstract level, topological recursion can be thought as an algorithm that has a spectral curve as input and its invariants as output. These invariants (one can think of them as rational numbers) are often solutions to an enumerative problem attached to the matrix model.

It was later realised that the most natural starting point is the spectral curve rather than the matrix model. Incidentally, several mathematical invariants were identified as the TR invariants of new spectral curves, unrelated to the study of matrix models. These includes Gromov-Witten invariants, Hurwitz numbers, volumes of moduli spaces of hyperbolic surfaces, knot polynomials.

Since its introduction in 2007, the subject has attracted the attention of a continuously growing community, including many young researchers. Topological recursion theory has been taught in graduate programs internationally throughout Europe, the United States, Russia, China, Canada, and Australia.

About Topological Recursion