Since the early 1970's, it has been known in both, the mathematical physics and in the physics communities, that propagation of information in quantum spin chains cannot exceed the so-called Lieb-Robinson bound (effectively providing the quantum analog of the light cone from the relativity theory). Typically these bounds depend on the parameters of the model (interaction strength, external field). The recent Hamza-Sims-Stolz result  demonstrates exponential localization (a la Anderson localization) of information propagation in most spin chains (in the sense of a given probability distribution with respect to which interaction and external field couplings are drawn). A natural question arises: what can be said about lower bounds on propagation of information in spin crystals (i.e. the case far from the one in which localization is expected), as well as in the intermediate case--the spin quasicrystals. This problem can be reduced to solving a linear ODE given by a Hermitian matrix, the solutions of which live on finite-dimensional complex spheres. 

In this talk we shall discuss the history, give a general overview of the field, reduce the problem to an ODE problem as mentioned above, and look at some open problems. We shall also present some numerical computations with animations.

We study one-dimensional ergodic operator family with sampling function $\{x\}$ and some its generalizations. The general result by Damanik and Killip implies that they cannot have absolute continuous spectra. We show that for almost all frequencies and all coupling constants these operators have pure point spectrum of positive Lebesgue measure, and that singular continuous spectrum is supported on a closed set of measure zero. In addition, there is no singular continuous spectrum for large coupling. The results are joint with Svetlana Jitomirskaya.

 The dry version of the ten Martini Problem is an interesting problem of Almost Mathieu Operator. In  this talk,

we will discuss  the reducibility    of a  Schrodinger cocycle  by the methods of Localization in the dual model.  As an application, we

will   show  AMO has open gaps with small coupling.