Mathematical Physics and Dynamical Systems

Research topics

Several different areas are currently subject of research.

1. Relativity and Quantum Field Theory.
   Research in this area concerns the so-called Modern Mathematical Physics. More specifically, several algebraic and geometric aspects of Quantum Mechanics, General Relativity and Theory of Quantized Fields (QFT) in Curved Spacetime have been investigated. From a mathematical point of view, these research topics involve functional analysis, global analysis (i.e. functional analysis on Riemannian and Lorentzian manifolds, using the intrinsic geometrical structure), microlocal analysis, differential geometry, spectral geometry. QFT in the presence of a black hole also at finite temperature, axiomatic aspects of QFT in curved background and cosmological applications  have been investigated. (See also http://alpha.science.unitn.it/~moretti/recent.html).
2. Classical Mechanics and Differential Geometry.
   Research in this area concerns some applications of Differential Geometry to Analytical Mechanics. A geometric formulation of Classical Mechanics based on jet-bundles has been developed; in this framework, some well known problems have been examined. Among others, the problem of characterization of ideal kinetic constraints, the construction of a dinamical covariant derivative, the inverse problem of Lagrangian Mechanics, the geometrization of the Lagrangian gauge and the intrinsic formulation of the Legendre transformation have been fruitfully treated in this new scheme. Actual field of research is non-holonomic Lagrangian Mechanics, in connection with variational calculus and optimal control theory. Another (minor) field of activity is propagation of acoustic and electromagnetic waves in non-homogeneous media with applications to acoustic microscopy, seismology and optical wave-guides.
3. Dynamical Systems and Stability.
   Subjects:
   1. The period function of plane centers, in relation to  the existence and/or uniqueness of critical orbits, convexity, monotonicity partitions.
   2. Existence, uniqueness, multiplicity of limit cycles of planar systems.
   3. Perturbation problems for hamiltonian systems, in the line of Poincaré-Birkhoff theorems.

Staff