We review some recent results concerning the open BCS model of superconductivity as originally proposed by Buffet and Martin. We also briefly analyze some possible generalizations.

We use a method developed earlier to construct new matrix solvable models. We apply this method to the model of the $B_2$ type and obtain new solvable model of the system with matrix potential.

We study magnetic Schrödinger operators with random or almost periodic electric potentials on the hyperbolic plane, motivated by the quantum Hall effect (QHE) in which the hyperbolic geometry provides an effective Hamiltonian. In addition we add some refinements to earlier results. We derive an analogue of the Connes-Kubo formula for the Hall conductance via the quantum adiabatic theorem, identifying it as a geometric invariant associated to an algebra of observables that turns out to be a crossed product algebra. We modify the Fredholm modules defined in [4] in order to prove the integrality of the Hall conductance in this case.

The two-dimensional noncommutative field theory on the light cone is studied from the usual Moyal product formalism. The Noether currents and the components of the energy-momentum tensor are explicitly computed.

We consider the Pauli grading of the Lie algebra $\frak{sl}(3,\mathbb{C})$ and use a concept of graded contractions to construct non-isomorphic Lie algebras of dimension eight. We overview methods used to distinguish the results and show how associated algebras, uniquely determined by the original algebra, simplify this task. We present a short overview of resulting Lie algebras.

With Lie-algebraic methods we obtain the set of orthogonal harmonic polynomials on the Poincaré dodecahedral three-manifold. The expansion in these polynomials of temperature fluctuations of the cosmic micro wave background tests this manifold as a candidate for cosmic topology.

We define a stochastic diffeology over the n-fold loop-space of a n-connected manifold. We perform a stochastic cohomology in the sense of Chen and Souriau and we show that the stochastic cohomology groups are equal to the classical de Rham cohomology groups of the n-fold loop space. We give a stochastic analogue of the classical fact that the homology groups of the n-fold loop space are an algebra for the little cube operad.

This is a review of one of the approaches to unify Quantum Mechanics and the theory of General Relativity. Starting from the pioneer work of Regge and Penrose, other scientists have constructed state sum models, as Feymann path integrals, that are topological invariant on the triangulated Riemannian surfaces, and that become the Hilbert-Einstein action in the continuous limit.

In this note we show that the momentum and position operators of $\mu$-deformed quantum mechanics for $\mu \gt 0$ are not Accardi complementary in a sense that we will define. We conjecture that this is also true if $-1/2 \lt \mu \lt 0$.

The purpose of this contribution is to display an outlook of a certain approach that enables one to study various spectral perturbation phenomena such as perturbation of eigenvalues and resonances (scattering poles) from the general viewpoint. Some applications of this elaborated technique are presented as well.

Nonlinear quantum mechanics at the Planck scale can produce non-local effects contributing to resolution of singularities, to cosmic acceleration, and modified black-hole dynamics, while avoiding the usual causality issues.