Normally hyperbolic smooth trapped sets are structurally
stable and occur in many interesting situations: for instance
for Kerr black hole metrics. We show that the corresponding
semiclassical resonances (e.g. quasinormal modes for Kerr
black holes) are separated from the real axis which has
consequences for decay of waves and other phenomena.

The proof is a simple example of techniques
used in the semiclassical study of quantum resonances
and I hope to present it in a self-contained way.

A dynamical system is chaotic if its behavior is sensitive to a change in the initial data. This is usually associated with instability of trajectories. The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with nonzero Lyapunov exponents.

I will describe main types of hyperbolicity and the still-open problem of whether dynamical systems with nonzero Lyapunov exponents are "typical" in a sense. I will outline some recent results in this direction and relations between this problem and two other important problems in dynamics: whether systems with nonzero Lyapunov exponents exist on any phase space and whether nonzero exponents can coexist with zero exponents in a robust way.

We shall present a solution of a problem which has been open for over twenty years.

In 1992, C. Akemann and G.K. Pedersen described the structure of the norm-closed faces of the unit ball of a C*-algebra A in terms of the compact partial isometries in A**. Three years earlier, C.M. Edwards and G.T.
Ruttimann gave a complete description of the weak*-closed faces of the unit ball of a JBW*-triple, and in particular, in a von Neumann algebra. However, the question whether the norm-closed faces of the unit ball in a JB*-triple E are determined by the compact tripotents in E** has remained open.

We shall survey the positive answer established by
C.M. Edwards, F.J. Fernndez-Polo, C. Hoskin and the speaker in a recent paper.

We consider the the problem of approximating a given object x (say, a function) by a sequence (x_n), whose terms belong to the prescribed family of sets (A_n)$ (for instance, A_n may be
the space of polynomials of degree less than n). For each n, compute the distance E_n from x to A_n. How does the sequence (E_n) behave? What are the connections between its rate of
decrease and the properties of x? Can we discern any patterns in the sequence (E_n)? We attempt to answer these questions for different families (A_n).