One of the main our activity is the study of the problem of influence of inhomogenity on the structure of the general oscillatory regime of gravitational field near cosmological singularity. It was shown that an unavoidable and unlimited development of small-scale exitations in asymptotic vicinity to the singularity make the picture very similar to the highly developed turbulence.
The problems under the current research are: an exact mathematical formulation of the state of gravitational turbulence in terms of the charachteristic functional of the random gravitational field in the vicinity of cosmological singularity and construction of appropriate averaging procedure in order to avoid too detailed description of the complicated small-scale structure of such gravitational turbulent regime.
Another direction of the research is the study of the possible compactification mechanisms in multidimensional inhomogeneous Cosmology.
We developed a detailed study about the influence of different kinds of "matter" on the initial singularity present in the Friedmann-Robertson-Walker, quasi-isotropic and generic inhomogenous solution. In particular we considered the effects of ultrarelativistic matter, particles creation, scalar and vector fields [5].
The most recent results obtained during year 2001-2002 are as follows:

• It has been presented (G. Imponente and G. Montani) a detailed analysis of the stochastic properties characterizing, near the ''Big-Bang'' the Bianchi type VIII and IX cosmological models. In particular we have shown the time independence of their dynamics, as well as of their invariant measure, the choice of a particular temporal gauge [Ref. 2,3,4]; the achievements of these results is based on an Arnowitt-Deser-Misner hamiltonian approach, as written in terms of Misner-Chitrè-like variables. In connection with these subjects, it was provided a straightforward relation between the classical chaoticity of these two Bianchi models and their indeterministic quantum behavior, performed during the Planckian era; The physical ground of this result is based on a WKB approximation, able to reduce the semiclassical quantum probability distribution to the classical microcanonical one [Ref. 5,6]. For a general review on these investigations, see [Ref 7].
• On the base of a standard hamiltonian approach, developed adopting Misner-Chitrè-like variables, is constructed (G. Montani) a solution to the continuity equation for the Bianchi VIII and IX cosmological models, as written in the asymptotic limit to the initial "Big-Bang''; the knowledge of this solutions provides the asymptotic nonstationary corrections to the invariant measure for the system, which result to decay exponentially [Ref. 8].

• On the base of criticism to the nature of the Wheeler-Dewitt approach, it is provided (G. Montani) a reformulation of the quantum geometrodynamics within the (3+1)-slicing representation of the space-time. The fundamental statement of this analysis is the necessity to include the so-called "kinematical action'' even in the case of a quantum space-time, so getting a completely consistent formulation for a "gauge-fixing'' quantization of the 3-geometries [Ref. 9].
• In order to obtain a vacuum theory of the torsion field of the second order, as that one of any other field (including the space-time geometry), it is proposed (G. Aprea, G. Montani, R. Ruffini) a Lagrangian formulation for the space-time geometry, characterized by a non-riemannian connection (we retain only its totally antisymmetric and trace components) which is expressed by a scalar and a tensor potential. The obtained field equations predicts the existence of torsion waves, having interesting implications on the motion of the test particles, as described by the self-parallel lines [Ref. 11].
• By applying, in the line of the idea proposed by I.Prigogine, the theory of open thermodynamical systems to an expanding isotropic universe, is provided (G.Montani) a phenomenolgical approach to the particles creation mechanisms in the very early cosmology; the main result of this work consists in showing how the horizon paradox finds a natural solution upon taking into account the matter creation near the big Bang [Ref. 12].
• It was derived (D.Oriti, R.M. Williams) the the Barrett-Crane spin foam model for Euclidean 4-dimensional quantum gravity from a discretized BF theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown [Ref. 13].
• E.R. Livine and D.Oriti study a generalized action for gravity as a constrained BF theory, and its relationship with the Plebanski action. We analyse the discretization of the constraints and the spin foam quantization of the theory, showing that it leads naturally to the Barrett-Crane spin foam model for quantum gravity. Our analysis holds true in both the Euclidean and Lorentzian formulation [Ref. 14].
• D.Oriti made an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice gauge theory, matrix models, category theory, statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian case. We focus in particular on the Barrett-Crane model for 4-dimensional quantum gravity [Ref. 15].
• D.Oriti extended the lattice gauge theory-type derivation of the Barrett-Crane spin foam model for quantum gravity to other choices of boundary conditions, resulting in different boundary terms, and re-analyze the gluing of 4-simplices in this context. This provides a consistency check of the previous derivation. Moreover we study and discuss some possible alternatives and variations that can be made to it, and the problems they present [Ref. 16].

• It was investigated a model of ballistic ejection effect of matter from spherically symmetric stellar clusters (M. V. Barkov, V.A. Belinski and G.S. Bisnovatyi-Kogan). The problem was solved in newtonian gravity but with cutoff fixing the minimal radius of selfgravitating matter shell by its relativistic gravitational radus. It was shown that during the motion of two initially gravitationally bound spherical shells, consisting of point particles moving along ballistic trajectories, one of the shell may be expelled to infinity at subrelativistic expelling velocity of the order of 0,25c. Also it was shown that the motion of two intersecting shells in the case when they do not runaway shows a chaotic behaviour [Ref. 18].
• It was found (M. V. Barkov, V.A. Belinski and G.S. Bisnovatyi-Kogan) the complete exact solution in the General Relativity for the intersection process of two massive selfgravitating spherically symmetric shells (in general with tangential pressure). It was shown how one can calculate all shell's parameters after intersection in terms of the parameters before the intersection. The result is quite new, the solution of this kind was known only for the massless shells (Dray and t'Hooft, 1985). The solution was applied to the analysis of matter ejection effect from relativistic stellar clusters and to the chaotic motion of the shells in relativistic regime. It was shown that in relativistic case the matter ejection effect is stronger than in newtonian gravity [Ref. 19].

This research line is the further development of the theory of gravitational solitons. Here the research is going in the following three drections. First, it was already established the existence of the gravisolitonic topological charge. However the exact mathematical formulation of this phenomenon (especially the exact expression for the topological current) is still remain to be seen. Second, the possibility of the quantum creation of gravisolitons near the cosmological singularity is under investigation. Third, we are intending to construct the well defined energetics of gravisolitons.
Results during period 2000-2002 regard are based as follows.

• It was accomplished the final version of the book "Gravitational Solitons" (V. Belinski and E. Verdaguer). Here is a self-contained exposition of the theory of gravitational solitons and provides a comprehensive review of exact soliton solutions to Einstein's equations. The text begins with a detailed discussion of the extension of the Inverse Scattering Method to the theory of gravitation, starting with pure gravity and then extending it to the coupling of gravity with the electromagnetic field. There follows a systematic review of the gravitational soliton solutions based on their symmetries. These solutions include some of the most interesting in gravitational physics such as those describing inhomogeneous cosmological models, cylindrical waves, the collision of exact gravity waves,and the Schwarzschild and Kerr black holes [Ref. 20].
• The Alexeev approach for the construction of Electromagnetogravisolitons was elaborated and translated to the conventional Belinski-Zakharov method. The results was included as one of the chapter to the book "Gravitational Solitons" [Ref. 20].

Loop Quantum Gravity is a promising approach for quantum description of the gravitational interaction, being non-perturbative and background independent theory. Being applied to cosmology the theory predicts from the one hand avoidance of cosmological singularities and from the other hand successful mechanism to generate initial conditions for inflation. We study these aspects of Loop Quantum Cosmology within isotropic homogeneous semi-classical framework using methods of qualitative theory of dynamical systems.