Current Projects

**Nonlinear analysis**

*Non-convex minimization and applications to material
sciences*: non-linear partial differential equations are to be
solved through the variational approach for those cases where
the standard methods of minimization for convex functionals fail;
presently my focus is on the case vector fields; the class of nonlinear
PDE's which are considered are modeled on those of material sciences
(nonlinear elasticity, plasticity). The proofs of the core results are
contained in my article in the Questiones Mathematicae 29,
2006. There one finds also references to the solution of
concrete problems.

**Mathematical
physics**

*Quantum field theories with a fundamental length*
(ongoing collaboration with Prof. S. Nagamachi, Japan): the challenge
is to reconcile the apparently conflicting conditions coming from the
existence of a fundamental length (as suggested 50 years ago by W.
Heisenberg and predicted by string theory) with the established
physical conditions of Poincaré covariance, locality and physical
energy momentum spectrum in a precise mathematical setting; first
encouraging results in the framework of ultra-hyperfunctions have been
published (Outline of the basic theory) . A solution of the
Heisenberg-Okubo equations in this framework has been
obtained. These investiagtions lead us to consider the edge of the
wedge theorem for tempered
ultrahyperfunctions.

**Quantum
information theory**

The mathematical and conceptual foundations of quantum
information theory, in particular cryptography. Investigations into the
mathematical structure of density matrices and their time evolution
(with F. Petruccione; a first paper has been published). Currently we investigate the general structure of positive maps, i.e., maps which
take density matrices into density matrices.

A project on continuous variables quantum teleportation is completed and
an article on this subject has been published too (with S. Nagamachi). Our approach is based on the use of the holomorphic representation of the canonical
commutation relations and allows an accurate formulation of translations of states.

Present project: include localization of such states in the formulation of teleportation (as part
of a South Africa - Japan scientic collaboration, funded by NRF and JSPS). Envisaged project: Apply the same framework to the theory of lasers.

See also Centre for Quantum Technology.