We will review some of the results and conjectures on dynamics of the standard map. The talk will serve as a short introduction to the subject accessible for interested graduate students.

An important problem in signal processing is the "cocktail party problem", where several people are speaking at the same time and the objective is to separate the different speakers, typically using several microphones placed in different localities. Numerous techniques had been proposed to solve the cocktail party problem, with various degrees of success. Many of these techniques work very well for artificially mixed speech signals, but when it comes to real recordings, even with two speakers, the success is much more mixed. In this talk, we present a very robust method for solving the cocktail party problem in real recording with two speakers based on time-frequency separation.

A related problem is to suppress background noise so the intended speaker can be heard more clearly. We present a highly effective technique for solving this problem.

Variational principles are at the core of the formulation of mechanical problems. What happens in the presence of symmetry when variables can be eliminated? I will discuss the geometry underlying this reduction process and present the induced constrained variational principle and the associated Euler-Lagrange equations. The rigid body and the Euler equations for ideal fluids are examples of such reduced Euler-Lagrange equations in convective and spatial representations, respectively. This geometric structure permits the introduction of a new class of optimal control problems that have the remarkable property that the control satisfies precisely these reduced Euler-Lagrange equations. As an example, it is shown that geodesic motion for the normal metric can be controlled by geodesics on the symmetry group. In the case of fluids, these optimal control problems yield the classical Clebsch variables and singular solutions for the Camassa-Holm equation. Relaxing the constraint to a quadratic penalty yields associated optimization problems. Time permitting, the equations of metamorphosis dynamics in imaging will be deduced from this optimization problem.