We will review the recent (and not so recent) results on dynamics of piecewise isometries (especially piecewise translations), both in one and in higher dimensional case. Some interesting results (by Suzuki, Goetz, Zhuravlev, Boshernitzan, Bruin, Troubetzkoy, Buzzi) are known, but most of natural questions are still open. The main goal of the talk is to expose these open questions to potential researchers.

We study discrete Schr ̈odinger operators with trigonomet-
ric potentials. In particular, we are interested in the connection be-
tween the absolutely continuous spectrum in the almost periodic case
and the spectra in the periodic case. We prove a weak form of a precise
conjecture relating the two.
We also bound the measure of the spectrum in the periodic case in
terms of the Lyapunov exponent in the almost periodic case.
In the proofs, we use a partial generalization of Chambers formula.
As an additional application of this generalization, we provide a new
proof of Hermans lower bound for the Lyapunov exponent.

In scientific and engineering computing, one major computational bottleneck is the solution of large scale linear algebraic systems resulted from the discretization of various partial differential equations (PDEs). These systems are still often solved by traditional methods such as Gaussian elimination in many practical applications. Mathematically optimal methods, such as multigrid methods, have been developed for decades but they are still not much used in practice. In this talk, I will report some recent advances in the development of optimal multilevel iterative methods that can be applied to various practical problems in a user-friendly fashion. Starting from some basic ideas of designing efficient iterative methods such as multigrid and domain decomposition methods, I will give a brief description of a general framework known as the Fast Auxiliary Space Preconditioning (FASP) Methods and report some applications in various problems including Newtonian and non-Newtonian models, Maxwell equations, Magnetohydrodymics and battery and reservoir (porous media) simulations.