Ideal plane flows are incompressible inviscid two dimensional fluids, described mathematically by the Euler equations. Infinitely many steady states exist. The stability of these steady states is a very classical problem initiated by Rayleigh in 1880. It is also physically very important since instability is believed to cause the onset of turbulence of a fluid. Nevertheless, progress in its understanding has been very slow. I will discuss several concepts of stability and some linear stability and instability criteria. In some cases nonlinear stability and instability can be showed to follow from linear results. I will also describe some methods and techniques developed recently for stability problems, one of which is to use the geometrical properties of the dynamical system for the particle trajectories.

In classical probability, the Gaussian, Chi-square, and Beta are three of the most studied distributions, with wide applicability. In the last century, matrix equivalents to these three distributions have emerged from nuclear physics (Gaussian ensembles) and multivariate statistics (Wishart and MANOVA ensembles). Their eigenvalue statistics have been studied in depth for three values of a parameter (\beta = 1,2,4) which defines the "threefold way" and can be thought of as a counting tool for their real, complex, or quaternion entries.

The re-examination of the Selberg integral formula, in the late '80s, has brought the advent of general \beta-ensembles, which subsume the classical cases, and for which the Boltzmann parameter \beta acts as an inverse temperature. Their eigenvalue statistics interpolate between the isolated instances 1,2, and 4, offering a "behind the scenes" perspective.

With the discovery of matrix models for the general \beta-ensembles in the early 00's, we have entered a new stage in the understanding of the complex phenomena that lie beneath the threefold way. While the \beta = 1,2,4 cases are and will always be special, we believe that the future of the classical ensembles is written in terms of a continuous \beta>0 parameter.

Public key cryptography is about 25 years old, and relies on number theory. We will discuss Diffie Hellman key exchange and ElGamal encryption, and some recent improvements on them. We show how number theory and algebraic geometry, and in particular the rationality of certain algebraic tori, can be used to give a deeper understanding of these improvements, and to give new cryptosystems.

I will first recall some examples of L-functions and indicate some of the ways they have been important in algebraic number theory. I will then describe what appears to be their intimate connection with Galois theory (eg the Fontaine-Mazur conjectures), as well as touching on their relationship with algebraic geometry and automorphic forms. Finally, I will discuss what can be proved in this direction.