In the first part, the theory of bounded symmetric domains is presented along two main approaches: as special cases of Riemannian symmetric spaces of the noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand. In the second part, an invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in $\mathbb{C}^2$ and also the triple Maslov index on the Lagrangian manifold.

In this survey we describe a general method to deal with the variational problem associated to the generalized elastic curves, paying special attention to closed critical curves in real space forms due to its special geometric significance. We illustrate the method by studying particular choices of this energy in some more detail. Finally, we will review also some interesting applications of generalized elasticae to other higher dimensional variational problems in Physics, Biophysics and the Theory of Submanifolds.

We outline several specific issues concerning the theory of multicomponent nonlinear Schrödinger equations with vanishing and constant boundary conditions. We start with the spectral properties of the Lax operator $L$ for vanishing boundary conditions. We introduce the fundamental analytic solutions (FAS) and demonstrate their importance for relating the scattering problem to a Riemann-Hilbert problem, and for the construction of the resolvent of $L$. Then we generalize this procedure to constant boundary conditions case. We start with the structure of the class of allowed potentials $M$ and give a recipe of how FAS can be constructed on each of the leafs of the relevant Riemannian surface. This allows us to relate the scattering problem to a Riemann-Hilbert problem posed on a Riemannian surface. Next we use these FAS to construct the resolvent of $L$ and study its spectral properties. We also introduce the minimal set of scattering data on the continuous spectrum of $L$ which generically has varying multiplicity. The general construction is illustrated by three representative examples related to A.III, C.II and D.III symmetric spaces. Finally we consider regularized Wronskian relations which allow us to analyze the mapping between the potential of $L$ and the scattering data.

Two reduced standard $k$-symplectic Poisson manifolds with respect to the action of a Lie group $G$ are considered, and the relation between the corresponding Poisson structures is established.

Spacelike constant mean curvature (CMC) surfaces in Minkowski three-space $\mathbb{L}^{3}$ have an infinite dimensional generalized Weierstrass representation. This is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU(2) with SU(1,1). The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. The construction is described here, with an emphasis on the difference from the Euclidean case.

It is shown that in five dimensional Einstein-Gauss-Bonnet theories exact vacuum solutions with non-vanishing torsion can be constructed. Some possible observational effects related to neutrino oscillations are pointed out. In the theory of continuum media (in which suitable defects can be described by localized non vanishing torsion) “the gravitational intuition” is a rather useful tool to describe the physical effects of such defects. A possible astrophysical application is shortly described.

We have generalized the fermionic coherent states to pseudo-fermion oscillator system. The system of coherent states constructed consist of two subsets, which are bi-normalized and bi-overcomplete. The two subsets are built up as eigenstates of two annihilation operators $b$ and $\tilde{b} = \eta b \eta^{-1}$ of respectively $H$ and $H^+$ where $\eta$ is the Hermitian and invertible operator that ensures the pseudo-Hermiticity of the Hamiltonian $H = \eta^{-1} H^+ \eta$

Solutions of the free elastica are deformed in order to provide the analytical part of the corresponding mechanical construction which produce the shapes of Euler elastica by bending appropriately the rectangular one.

We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.

We give one parameter deformation of level $k$ free field realization of the screening current of the elliptic algebra $U_{q,p} (\widehat{\mathfrak{sl}}_{2})$. By means of these free field realizations, we construct infinitely many commutative operators, which are called the nonlocal integrals of motion associated with the elliptic algebra $U_{q,p} (\widehat{\mathfrak{sl}}_{2})$ for level $k$. They are given as integrals involving a product of the screening current and elliptic theta functions. This paper give level $k$ generalization of the nonlocal integrals of motion given in [1].

The special interest is devoted to such situations when the mate- rial space of our object with affine degrees of freedom has generally lower dimension than the one of the physical space. In other words when we have $m$-dimensional affinely-rigid body moving in the $n$-dimensional physical space, $m \lt n$. We mainly concentrate on the physical situation $m = 2, n = 3$ when “thickness” of flat bodies performs one-dimensional oscillations orthogonal to the two-dimensional central plane of the body. For the isotropic case in two “flat” dimensions some special solutions, namely, the stationary ellipses, which are analogous to the ellipsoidal figures of equilibrium well known in astro- and geophysics, e.g., in the theory of the Earth’s shape, are obtained.

Dynamical Systems theory is a valuable tool in the study of Cosmology, indeed qualitative methods allow to characterize cosmological solutions on the basis of their relevant physical features (e.g., stability and asymptotic behaviour). Here we briefly review some well established results for cosmological models in the framework of General Relativity. Then we consider a family of modified cosmological models which are gaining more and more relevance, namely Loop Quantum Cosmologies. In particular we analyze the geometrical structure and dynamical properties of the model presented in [13].

In this paper we study some properties of a Lie algebroid and its prolongation over the vector bundle projection of the dual bundle. We generalize some results on Poisson manifolds to the level of a Lie algebroid. The notions of canonical Poisson bivector and horizontal lift are studied and their compatibility conditions are pointed out.

We study noncommutative (NC) instantons and vortexes. At first, we construct instanton solutions which are deformations of instanton solutions on commutative Euclidean four-space. We show that the instanton numbers of these NC instanton solutions coincide with the commutative solutions. Next, we also deform vortex solutions similarly and we show that their vortex numbers are unchanged under the NC deformation.

The presence of defects in material continua is known to produce internal permanent strained states. Extending the theory of defects to four dimensions and allowing for the appropriate signature, it is possible to apply these concepts to space-time. In this case a defect would induce a non-trivial metric tensor, which can be interpreted as a gravitational field. The image of a defect in space-time can be applied to the description of the Big Bang. A review of the four-dimensional generalisation of defects and an application to the expansion of the universe will be presented.

We present a generalized formulation of Poisson dynamics suitable to describe the n-bodies interactions. Examples are given of physical systems endowed with such a general structure.