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.