Giovanni Sparano

Noncommutative integrability and recursion operators

Abstract Geometric structures underlying commutative and noncommutative integrable dynamics are analyzed. They lead to a new characterization of noncommutative integrability in terms of spectral properties and of Nijenhuis torsion of an invariant (1,1) tensor field. The construction of compatible symplectic structures is also discussed.

A GENERALIZED REDUCTION PROCEDURE FOR DYNAMICAL SYSTEMS

We describe a reduction procedure for dynamical systems. If Γ is a dynamical vector field on a manifold M, a reduced system is obtained by projecting Γ to a manifold Σ/≈ where Σ is a submanifold of M invariant under Γ and ≈ is a suitable equivalence relation.

Eikonal type equations for geometrical singularities of solutions in field theory

Abstract We discuss several aspects of singularities of the solutions of the partial differential equations of Klein-Gordon, Schrodinger and Dirac. In particular we analyze the fold- type singularity, of the first and higher orders, and the related characteristic equations. We also consider the field equations as reduction of homogeneous equations in higher dimensions, and discuss how singularities of the solution are reduced.

DYNAMICAL ASPECTS OF LIE-POISSON STRUCTURES

Quantum groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU(2) and SU(1, 1), as submanifolds of a four-dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some Hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.

On the gravity of light

A solution of the old problem raised by Tolman, Ehrenfest, Podolsky and Wheeler, concerning the lack of attraction of two light pencils ”moving parallel”, is proposed, considering that the light can be source of nonlinear gravitational waves corresponding (in the would be quantum theory of gravity) to spin 1 massless particles.

GEOMETRICAL ASPECTS OF MIXED TORSIONLESS TENSOR FIELDS

Some aspects of the geometry of (1, 1) tensor fields T with vanishing Nijenhuis torsion are discussed; in particular, the cohomolgy of the exterior differential associated with T is analyzed and some geometric properties of the shape operator are derived.

Repulsive Gravity of Light Beams

Communicated by Abraham A. Ungar Abstract. The analysis of some exact solutions of Einstein equations, describing gravitational waves produced by light, shows that there can be repulsion between light beams. This is due only to the spin-1 character of the solutions and not to the introduction of other inputs external to General Relativity. Cosmological relativistic jets give an example of a such phenomenon.