The lecture will cover the following topics:
1. Global attractor for an autonomous evolution equation. Examples. between the attractor and the family of complete solutions.
2. Fractal dimension of a global attractor. Examples.
3. Nonautonomous evolution equations and corresponding processes. Uniform global attractor of a process.
4. Global attractor of the nonautonomous 2D Navier Stokes system. Translation-compact forcing term. Relation between the uniform attractor and the family of complete solutions. Nonautonomous 2D Navier-Stokes system with a simple attractor.
5. Kolmogorov epsilon-entropy of the global attractor of a nonautonomous equation. Estimates of the epsilon-entropy. Examples.
6. Some open problems.

This talk outlines recent work by Feldman, Ha, and Slemrod on the dynamics of the sheath boundary layer which occurs in a plasma consisting of ions and electrons. The equations for the motion are derived from the classical Euler- Poisson equations. Of particular interest is that the boundary layer interface moves via motion by mean curvature where the acceleration of the front (not the velocity) is proportional to the mean curvature of the front.

In this talk we will consider the spectrum of the Almost Mathieu operator, \[ \left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n\omega + \phi\right)x_n, \] on $l^2(\mathbb{Z})$. We will show that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'', for these values of $b$ and $\omega$.

The proof uses a combination of results on reducibility, localization and duality for the Almost Mathieu operator, and its associated eigenvalue equation, sometimes called the Harper equation.

Finally, we will also show that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

We study the evolutionary game dynamics of a two-strategy game. In infinite populations, the well-known replicator equations describe the deterministic evolutionary dynamics. There are three generic selection scenarios. The dynamics of a finite group of players has received little attention. We provide a framework for studying stochastic evolutionary game dynamics in finite populations. We define a Moran process with frequency dependent fitness. We find that there are eight selection scenarios. And for a given payoff matrix, a number of these sceanrios can occur for different population size. Our results have interesting applications in biology and economics. In particular, we obtain new results on the evolution of cooperation in the classic repeated Prisoner's Dilemma game. We show that a single cooperator using a reciprocal strategy such as Tit-For-Tat can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions for natural selection to favor the emergence of cooperation and derive conditions for evolutionary stability in finite populations.