Multiscale asymptotic analysis is a particularly useful tool for studying nonlinear waves when exact solutions are not available. This is demonstrated in concrete problems: reaction diffusion front speeds in random shear flows, and localized propagating pulses in nonlinear scalar wave equations, both in two space dimensions. Complementary numerical results will also be shown.

A numerical method to simulate interfacial surfactant mechanics within a volume of fluid method has been developed. Two important features of this new method are that it conserves surfactant mass exactly and the form of the equation of state is not restricted, i.e. the relation between surfactant concentration and surface tension can be linear or nonlinear. To conserve surfactant, the surfactant mass and the interfacial surface area are tracked as the interface evolves, and then the surfactant concentration is reconstructed. The algorithm is coupled to an incompressible Navier-Stokes solver that uses a continuum method to incorporate both the normal and tangential components of the surface tension force into the momentum equation.

Numerical simulations demonstrate the effect of surfactant on the dynamics of several problems by comparison to surfactant-free simulations. First, the buoyant rise of a bubble is examined. Next, the evolution of a drop in an extensional flow is studied. Finally, the motion of a drop through a constriction is investigated. In each of these problems surfactant accumulation allows high interface curvature and the formation of small secondary drops or bubbles.

This is a continuation of the learning seminar on
the scattering theory for wave equations.

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.