Painlevé equations were discovered at the beginning of XX-th century by Paul Painlevé for purely mathematical reasons. Their solutions, the Painlevé transcendents, are, in general, “new transcendental functions” and, as the classical special functions, they appear in many problems of mathematics and physics. Applications “exploded” at the end of XX-th century: Einstein metrics, Frobenius manifolds, correlation function of the -dimensional Ising model, reduction of integrable PDEs, reduction of self-dual Yang-Mills equations, random matrix theory, -dimensional CFT (conformal blocks), non perturbative effects in strings theory (d quantum gravity)…

These lectures are part off the 9th IST Lectures on Algebraic Geometry and Physics — 2020