A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determin...

— The sixth Painlevé equation is hiding extremely rich geometric structures behind its outward appearance. In this article, we give a complete picture of its dynamical nature based on the Riemann-Hilbert approach recently developed by the authors and using various techniques from algebraic geometry. A large part of the contents can be extended to Garnier systems, while this article is restricte...