The connection between random partitions and integrable systems is a fascinating subject, going back to (at least) the famous Baik-Deift-Johansson theorem on the longest increasing subsequence of a random permutation, and the introduction of Schur measures by Okounkov. After reviewing some of the classical results, I will focus on multiplicative averages associated to random partitions obeying the Poissonized Plancherel measure. I will prove that these quantities satisfies a well known set of integrable equations (cylindrical Toda) and describe the asymptotics of the relevant solutions. Time permitting, I will also discuss some applications to cylindrical partitions and random growth models. The results I will present are issued from a collaboration with Giulio Ruzza and a ongoing collaboration with Matteo Mucciconi and Giulio Ruzza.

In the last years an unexpected interplay between the theory of Painlevé equations and supersymmetric quantum field theory (QFT) has been established. We will explain how this link helps in solving Painlevé equations via perturbative susyQFTs and how non-perturbative susyQFTs get solved thanks to the theory of Painlevé equations. The correspondence gets enlighten by the Alday-Gaiotto-Tachikawa (AGT) correspondence which relates the gauge theory computations in terms of CFT2 quantities. A key role is played by the interpretation of the theory of integrable (quantum)-isomonodromic deformations of linear systems as renormalization group equations in susyQFTs on blow-up geometries. We will discuss the above topic in some relevant simple examples (PIII3, PI,Quantum PI,...) to exemplify a more general set of results.

This talk deals with the geodesic flow of the seven-dimensional sphere with respect to four subriemannian structures, which are given by the Hopf fibration, the quaternionic Hopf fibration, and the two distributions spanned by a certain number of canonical vector fields.

Using the method by Thimm that appeared in the beginning of 1980's, one can prove the complete integrability in the sense of Liouville for the geodesic flows associated to each of the four subriemannian structures, by constructing the explicit first integrals.

The differential operators associated with the constructed first integrals are also studied.

The talk is based on a collaboration with Wolfram Bauer and Abdellah Laaroussi.

The skew RSK dynamics is a discrete 2 dimensional deterministic dynamics introduced by Imamura, Sasamoto and myself. It is an integrable system in the sense that it possesses infinitely many conservation laws and a rich set of symmetries. The analysis of its scattering relations produces a nontrivial correspondence which generalizes the celebrated Robinson-Schensted-Knuth bijection. Such correspondence provides a bijective proof of summation identities between special symmetric polynomials, which describe the law of certain interacting particle systems or directed polymer models and allow their asymptotic analysis.

E-functions are power series which solve a linear differential equation, and whose coefficients are algebraic numbers subject to a growth condition of arithmetic nature. The exponential series is an example of an E-function, as well as Bessel functions with integer parameter. Siegel introduced these functions in 1929 with the goal of generalising the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers.

In a joint work with Javier Fresán, we found that certain exponential integrals associated with algebraic varieties give rise to differential equations whose solutions can be expressed in terms of E-functions. I will present some of this work, and talk about the inverse problem of representing a given E-function, or rather the associated differential equation, by an exponential integral, and relating these objects thereby to algebraic geometry.