In this article we present an introduction in the geometrical theory of motion of curves and surfaces in $\mathbb{R}^3$, and its relations with the nonlinear integrable systems. The working frame is the Cartan's theory of moving frames together with Cartan connection. The formalism for the motion of curves is constructed in the Serret-Frenet frames as elements of the bundle of adapted frames. The motion of surfaces is investigated in the Gauss-Weingarten frame. We present the relations between types of motions and nonlinear equations and their soliton solutions.

We discuss cohomological invariants of framed reflexive sheaves on $(n\geq 3)$-folds and show that some numerically admissible values of cohomology do not occur.

We work out the general idea that a composite continuous physical system can be mathematically modelled locally as a completely integrable geometric distribution on a manifold, the time-recognizable subsystems to be modelled by corresponding subdistributions, and any local interaction between two subsystems of the physical system to be described in terms of the nonintegrability of the two subdistributions making use of the corresponding two curvature forms. As an illustration we present the corresponding description of photon-like objects, based on the notion that photon-like objects are real, massless time-stable physical objects with intrinsically compatible translational-rotational dynamical structure. The spatial propagation of the system follows some external/shuffling symmetry of the distribution.

We use the mathematical structure of group algebras and $H^{+}$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are ${\rm SU}(2)$ and its quotient ${\rm SO}(3,\mathbb{R})$. The proposed scheme is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. Secondly, the other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of ${\rm SU}(2)$.