Studying of complexity of infinite aperiodic words, i.e., the number of different factors of the infinite word of a fixed length, is an interesting combinatorial problem. Moreover, investigation of infinite words associated with $\beta$-integers can be interpreted as investigation of one-dimensional quasicrystals. In such a way of interpretation, complexity corresponds to the number of local configurations of atoms.

In this paper we show that the basic external (i.e., not determined by the equations) object in Maxwell vacuum equations is a complex structure. In the three-dimensional standard form of Maxwell equations this complex structure $\mathcal{I}$ participates implicitly in the equations and its presence is responsible for the so called duality invariance. We give a new form of the equations showing explicitly the participation of $\mathcal{I}$. In the four-dimensional formulation the complex structure is extracted directly from the equations, it appears as a linear map $\Phi$ in the space of two-forms on $\mathbb{R}^4$. It is shown also that $\Phi$ may appear through the equivariance properties of the new formulation of the theory. Further we show how this complex structure $\Phi$ combines with the Poincaré isomorphism $\mathfrak{P}$ between the two-forms and two-tensors to generate all well known and used in the theory (pseudo)metric constructions on $\mathbb{R}^4$, and to define the conformal symmetry properties. The equations of Extended Electrodynamics (EED) do not also need these pseudometrics as beforehand necessary structures. A new formulation of the EED equations in terms of a generalized Lie derivative is given.

We consider an integrable model which describes light beams propagating in nonlocal nonlinear media of Cole-Cole type. The model is derived as high frequency limit of both Maxwell equations and the nonlocal nonlinear Schrödinger equation. We demonstrate that for a general form of nonlinearity there exist selfguided light beams. In high frequency limit nonlocal perturbations can be seen as a class of phase deformation along one direction. We study in detail nonlocal perturbations described by the dispersionless Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction method based on symmetry constraints and by the quasiclassical $\bar{\partial}-$ dressing method. Quasiclassical $\bar{\partial}-$dressing method reveals a connection between nonlocal nonlinear geometric optics and the theory of quasiconformal mappings of the plane.

A survey of the interrelationships between matrix models and field theories on the noncommutative torus is presented. The discretization of noncommutative gauge theory by twisted reduced models is described along with a rigorous definition of the large $N$ continuum limit. The regularization of arbitrary noncommutative field theories by means of matrix quantum mechanics and its connection to noncommutative solitons is also discussed.