Several different areas are currently subject of research.
- Quantum Theories and Quantum Relativistic Theories
Research in this area concerns the so-called Modern Mathematical Physics. More specifically, several foundational, algebraic and geometric aspects of Quantum Theories, Quantum Information and Photonics, Theory of Quantized Fields (QFT) also in curved spacetime.
From a mathematical point of view, these research topics involve linear algebra, functional analysis, operator algebra, global analysis. (For further information https://moretti.maths.unitn.it/home.html and http://www.tifpa.infn.it/projects/bell/).
Collaboration about fundamental (theoretical and experimental) aspects of quantum theories is active with members of the group of Stochastic Processes and the one of Geometry, the group of Integrated Photonics of the Department of Physics, and members of the Department of Information Sciences. The group leads and interdisciplinary team named BELL of Trento Institute for Fundamental Physics and Applications (INFN-TIFPA)
- Classical Mechanics and Differential Geometry.
Research activity in this area concerns some applications of Differential Geometry to Analytical Mechanics and Calculus of Variations.
A geometric formulation of Classical Mechanics based on jet-bundles has been developed. In this framework, some well known problems have been examined: the characterization of ideal kinetic constraints, the construction of a dynamical covariant derivative suitable for an analysis of the inverse problem of Lagrangian Mechanics, a geometric interpretation of the Lagrangian gauge and of the Legendre transformation.
Actual main field of research is Calculus of Variations in presence of kinetic or non-holonomic constraints and optimal control theory, focusing the attention to the presence of extremal curves with discontinuous derivative (corners). Extension to this context of the classical theory of second variation of a functional, Jacobi fields and conjugate points, Morse and Maslov theory, including also the possibility of asynchronous variation of corners, have been performed. This research is strongly motivated by applications in Physics, Engineering, Economics, and in models of visual cortex.
Other fields of activity are propagation of acoustic and electromagnetic waves in non-homogeneous media with applications to acoustic microscopy, seismology and optical wave-guides, and historical aspects of Mathematical-Physics.