My work concerns the following thematic areas:
- Classical vortex dynamics and fluid knots;
- Quantum defects in condensates and topological phases;
- Energy-complexity relations for magnetic fields;
- Topological transitions of critical energy surfaces;
- Hydrodynamic analogue models in black hole theory.
(i) Classical vortex dynamics and fluid knots
Study of dynamical properties of three-dimensional vortex fields, and of topologically complex vortex tangles, are of central importance in the context of the Euler, as well as of the Navier-Stokes equations. Vortex filament dynamics is governed by the Biot-Savart induction law, that is a global, integral, functional of vorticity.
Questions related to local and global aspects involve geometry and topology of vorticity, as well as of its functional distribution in space, and the formation of knots, links and complex tangles. These questions are intimately related to the integrability of the Euler equations, the role of invariants, and the existence and formation of finite time singularity, that represents a central aspect in one of the most challenging problems in contemporary mathematics (see the Clay Millennium Prize Problem).
(ii) Quantum defects in condensates and role of geometric and topological phases
In recent years experimental realisation of Bose–Einstein condensates as a new state of matter, and laboratory production of vortex defects (see the 2001 Physics Nobel Prizes), have stimulated a great deal of work in this new research field for their potentials for technological application.
The hydrodynamic interpretation of the governing equations rooted in the original work of Madelung (1926), coupled with the extraordinary recent progress in direct numerical simulation of fluid flows, has given further impetus to this hydrodynamic approach by a mean field description given by the Gross-Pitaevskii equation that governs condensates. The non-locality aspects, typical of the quantum world, have become of central interest in the exploitation of geometric and topological properties associated with the formation and production of quantum defects.
(iii) Energy-complexity relations for magnetic fields
Evolution and relaxation of magnetic fields is of fundamental importance for the study of astrophysical flows in solar and stellar physics, and in confined systems in fusion physics. In this context magnetic fields may form complex structures such as those forming braided loops on the Sun, or twisted magnetic fields in tokamaks. These fields are subject to a Lorentz force that govern evolution and energy relaxation.
In general, magnetic energy is bounded from below by helicity, and under resistive effects it converts to kinetic energy. In an ideal situation (no resistive or dissipative effects present) magnetic knots and links are led to relax to a minimum energy state dictated solely by the topology of the system. Lower bounds on energy are thus provided by topological constraints, and relations between ground-state energy and topological complexity are therefore of fundamental importance.
(iv) Topological dynamics of critical energy surfaces
Study of minimal energy surfaces has a long history; it originates with the problem posed by Lagrange and the experimental work by Plateau (see the Wikipedia entry on soap films) and the subject has grown steadily ever since. Mathematical aspects concerning the existence of surface solutions as steady states, the study of ideal shapes, and the relation between least area surfaces and energy are at the heart of current research.
Fundamental questions raised by Courant in the late 30’s about soap film surfaces of least area spanning knots and links have however remained almost unanswered. These problems can now benefit from recent progress in pure mathematics and advanced visiometrics, allowing detailed visualisation of complex Seifert surfaces. This in turn provides new tools to study changes of geometry and topology for physical applications; from soap films to potential surfaces associated with vector fields in classical and quantum fluid systems.
(v) Hydrodynamic analogue models in cosmology
New impetus in cosmological black hole theory emerged thanks to the advances in observational cosmology and progress in the fundamental properties of the recently discovered form of condensed matter in spacetime gravity. The announcement (on April 10, 2019) of the first image of a rotating black hole in Galaxy Messier 87, using a virtual Earth-sized telescope made by the Event Horizon Telescope (EHT) collaboration, represents a groundbreaking achievement in the history of cosmology, and marks the beginning of a new era in black hole theory.
Analogue models based on hydrodynamic properties of condensed matter physics have been proposed to capture fundamental features of black hole cosmology. In this context curved spacetime plays a crucial role, because sound waves in a transonic fluid flow can propagate along geodesics of an acoustic spacetime metric, providing grounds for a stricter analogy between the behaviour of light waves in black hole theory and acoustic waves in condensed matter physics.