Ascolta

Description

Mathematical physics has a long history. Initially, its objective was limited to the rigorous formulation of physical theories and the development of increasingly sophisticated mathematical methods for application in physics. However, the areas of application for mathematical physics have enormously expanded, answering the need for mathematically sound methods in applied and theoretical sciences. As a result, a state-of-the-art definition of mathematical physics is that it is a branch of mathematics concerned with the mathematical models of engineering, physics, biology, chemistry, environmental sciences, neuroscience, and other fields. Together with the analysis of such models, the aim is to derive useful information in concrete applications. Mathematical methods commonly employed in mathematical physics include those from mathematical analysis, geometry, probability theory, functional analysis, and operator theory. Additionally, numerical analysis is used to manage the increasing complexity of the models.

Research Topics

At the Engineering Department, there are currently six main research topics in progress:

• Biomathematics: We use models based on reaction-diffusion equations to study the dynamics of degenerative inflammatory pathologies, population dynamics, chemical kinetics, and more.
• Computational fluid dynamics: We focus on the numerical study of stability and transitions to chaotic or turbulent states, as well as singularity formation and separation in boundary layers.
• Hydrodynamics and thermodynamics: Our research includes quantum turbulence and heat transport in superfluid helium, thermal solitons and heat waves in nano-systems, and the mathematical treatment of optical solitons in fiber optics and matter waves in Bose-Einstein condensates.
• Integrability and exact solutions of nonlinear PDEs: We are interested in the theoretical study of nonlinear partial differential equations, especially those of interest in mathematical physics.
• Deformed commutation and anti-commutation relations in quantum mechanics: Our research includes coherent and bi-coherent states, bi-orthogonal bases and D-quasi bases in quantum systems, squeezed states, path integral and generalizations, and their applications to the analysis of certain quantum systems with non self-adjoint Hamiltonians, such as gain/loss systems.
• Modeling of macroscopic systems with operators and ideas arising from quantum mechanics: We consider biological, sociological, and economic systems, studying their dynamics mainly using ladder operators to define a suitable Hamiltonian for the model. We also consider the role of non-commuting operators in decision-making.

Funded Projects

• PRIN 2017 Multiscale phenomena in Continuum Mechanics: singular limits, off-equilibrium and transitions.
• PON Research and Innovation 2014–2020 “Coherent structures in brain dynamics and classification of emotional, physiological and pathological states: neuroimaging techniques and evolutionary models.”
• PNRR: SiciliAn MicronanOTecH Research And Innovation CEnter (SAMOTHRACE)

Keywords

Fluid dynamics; biomathematics; pseudo-Hermitian quantum mechanics; deformed commutation rules; coherent states; Hamiltonian dynamics;exact solutions; quantum turbulence; solitons.