My work is in the field of applied mathematics and mathematical physics and it is focused on the role of geometric and topological properties in dynamical and energetic aspects of fluid and complex systems. My research interests regard the following thematic areas:
- Classical vortex dynamics and fluid knots;
- Quantum defects in condensates and role of geometric 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 vorticity fields and topologically complex vortex tangles are central in the context of the Euler as well as 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 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, role of invariants and existence and formation of finite time singularity, central aspect in one of the most challenging problems in contemporary mathematics (see the Clay Millennium Prize Problem).
In this context I have contributed with these new results: interpretation of hydrodynamic (soliton) invariants in terms of global geometric properties (1992); self-linking number derivation from the kinetic helicity of a vortex knot (1992); torus knot soliton solutions to 1+1 non-linear Schrödinger equation in closed analytic form (1993); first study of three-dimensional aspects of vortex motion through effects of torsion on vortex dynamics (1994); derivation of Jones and HOMFLYPT knot polynomial invariants from helicity (2012, 2015); proof of writhe conservation under anti-parallel reconnection of fluid structures (2015); winding number effects on the evolution of vortex torus knots (2019); novel interpretation of topological cascade of fluid structures in terms of geodesic flows (2020).
(ii) Quantum defects in condensates and role of geometric and topological phases
In recent years experimental realisation of Bose–Einstein condensates as new states of matter and laboratory production of vortex defects (see the 2001 Physics Nobel Prizes) have stimulated a renovated interest in theoretical, numerical and experimental work in condensed matter physics.
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 the hydrodynamic approach as mean field description given by the Gross-Pitaevskii equation governing condensates. The non-locality aspects, typical of the quantum world, have become a central in the exploitation of geometric and topological properties associated with the formation and production of quantum defects.
In this context I have contributed with these new results: geometric interpretation of linear and angular momentum of defects (2008, 2017); proof of the production of a secondary defect as a Aharonov-Bohm effect (2019); derivation of the complete set of hydrodynamic equations for a defect subject to twist (2020); derivation of a stability criterium for a defect subject to twist (2020); study of physical effects of geometric and topological phases on quantum defects (2020); topological proof of zero helicity for Seifert framed defects (2021).
(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 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 dissipates. In ideal situations (no resistive or dissipative effects present) magnetic knots and links are led to relax to a minimum energy state dictated solely by topology. Lower bounds on energy are thus provided by topological constraints and relations between ground-state energy and complexity are therefore of fundamental importance.
In this context I have contributed with these new results: topological interpretation of magnetic helicity of a flux tube in terms of writhe and twist contributions (1992); proof of inflexional instability of twisted magnetic flux tubes with applications to solar flares (1997, 2005); lower bounds to magnetic energy in terms of topological crossing number (2008); study of winding number effects on the induction of magnetic torus knots (2017).
(iv) Topological transitions of critical energy surfaces
Study of minimum 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 concern the existence of surface solutions as steady states, the study of ideal shapes and the relation between least area surfaces and energy.
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 benefit now 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.In this context I have contributed with these new results: analytical description of topological transition of a soap film surface by a twisted fold singularity (2010); study of twist phase relaxation through the reconnection of quantum defects (2012, 2017).
(v) Hydrodynamic analogue models in cosmology
In recent years 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 of 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 also 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 condensates.In this context I have contributed with these new results: derivation of intrinsic kinematic equations of a vortex defect in (2n+1)-dimensions (1991); extension of hydrodynamics description of Gross-Pitaevskii equation in general Riemannian metric (2021); derivation of a new Einstein field equation for cosmological applications to black-hole analogue gravity models (2021).