Renzo L. Ricca, Ph.D. (Cantab)

University of Milano-Bicocca (UniMiB)
Department of Mathematics and Applications

Professional Profile

I am a Professor of Mathematical Physics at the University of Milano-Bicocca and a Guest Professor of the Beijing University of Technology (BJUT). My principal research interests are in classical field theorydynamical systems (classical and quantum vortex dynamics and magnetohydrodynamics) and geometric and topological aspects of structural complexity. I contributed to the field of geometric and topological fluid dynamics with on-going work on kinetic and magnetic helicity, and applications of physical knot theory.


I was born in Casale Monferrato and educated in Turin and Cambridge (UK). I  attended the Liceo Scientifico Palli, before reading engineering and applied mathematics at the Politecnico di Torino. After a long thesis work, I was awarded a prestigious scholarship by the Association for the Scientific and Technological Development of Piedmont (ASP, Turin) and admitted to Trinity College of the University of Cambridge (UK) to read Mathematics. There I conducted doctoral research under the guidance of Professor H. Keith Moffatt on the subject of topological fluid dynamics. In 1991, while completing my doctoral studies, I was awarded the J.T. Knight’s Prize in Mathematics for work on the geometric interpretation of soliton conserved quantities. I was then awarded the Ph.D. in Applied Mathematics for work on geometric and topological aspects of vortex dynamics and magnetohydrodynamics.

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Scientific Career

After a post-doctoral period spent at the Institute for Advanced Study (Princeton) and at the Institute for Theoretical Physics (UC Santa Barbara). in 1992 I returned to Europe joining the faculty of the Mathematics Department of University College London as a Research Fellow. From 1993 to 1995 I also held a joint position at the Politecnico di Torino as a University Researcher. I stayed at UCL as a Senior Research Fellow and part-time Lecturer till 2003, when I moved to the Department of Mathematics and Applications of University of Milano-Bicocca as Associate Professor of Mathematical Physics. From 1996 to 2004 I held several visiting positions in various institutions worldwide, including Kyushu University (Japan),  Geneva University, and École Normale Supérieure in Paris. In recent years I have been an Erasmus Professor at Crete University and Côte d’Azur University. From 2016 I am a Guest Professor (and from 2023 a Distinguished Guest Professor) of  Beijing University of Technology (BJUT), and from 2023 an Affiliate Member of the World Premier Institute SKCM2 of Hiroshima University.

Research and scientific contributions

My main research interests lie in applications of classical field theory to ideal fluid dynamics, particularly as regards geometric and topological aspects of vortex flows and magnetic fields in the presence of knots, links and braids. Aspects of potential theory of knotted fields, structural complexity and energy of filament tangles are also at the heart of my research.

Geometric aspects of dynamical systems

In the context of classical vortex dynamics my main contributions concern the geometric interpretation of certain conserved quantities associated with soliton solutions of integrable systems and the study of three-dimensional effects of torsion on vortex filament dynamics. In ideal magnetohydrodynamics I have shown how the effect of inflexional instability of twisted magnetic flux tube may trigger instabilities and braid formation in solar coronal loops. In more recent years I have became interested in the study of minimal Seifert surfaces spanning knots and links, providing analytical description of the topological transition of a soap film surface by the emergence of a twisted fold (cusp) singularity. My current work aims to establish connections between iso-phase minimal surfaces spanning defects in Bose-Einstein condensates and critical energy states.

Topological fluid dynamics

By relying on earlier work by Berger and Field (1984), in a joint work with Moffatt (1992) I established a deep connection between the topology of knotted fields and fundamental aspects of classical field theory, providing a topological interpretation of hydrodynamical helicity by a rigorous derivation of the self-linking number in terms of writhe and twist. In the same year I also derived explicit torus knot solution to integrable equations of hydrodynamic type, and contributed to determine new relations between energy of knotted fields and topological information in terms of crossing and winding number information. Years later, in collaboration with Xin Liu, I derived the Jones and HOMFLYPT knot polynomial invariants from the helicity of fluid flows (2012. 2015), by extending the initial work on helicity to a complex networks of filament structures in space. This work allows to quantify topological complexity of natural processes by new methods. In the years 2019-2022 I have investigated several aspects associated with the dynamics  and stability of vortex defects in Bose-Einstein condensates, showing in particular  how  the production of new vortex defects can be triggered by the superposition of a twist phase (as in a Aharonov-Bohm effect), and provided a topological proof of zero helicity for Seifert framed defects.

Dynamical models in high-dimensional manifolds

In 1991,  I derived the intrinsic equations of motion of a string in Riemannian manifolds, as a model for the then emerging string theory of high-energy physics,  showing the connection between the hierarchy of integrable equations of hydrodynamic type and the general setting of intrinsic kinematics of one-dimensional objects in (2n+1)-dimensional manifolds. More recently, in a joint work with my PhD student Alice Roitberg, I contributed to extend the hydrodynamic description of the Gross-Pitaevskii equation to general Riemannian manifolds, with possible applications to analog models of black holes in cosmology.

Origin and developments of mathematical ideas

In a series of papers (1992-1996) I contributed to publicise and review some lost works by Tullio Levi-Civita and his student Luigi Sante Da Rios, devoted to the development of the asymptotic potential theory of thin tubes and vortex filament dynamics. I also provided evidence for Karl Friedrich Gauss‘ unknown derivation of the linking number formula, and a reconstruction of the independent derivation of the very same formula by James Clerk Maxwell.

MATEMATICA - Renzo Ricca, Nodi e applicazioni - Accademia dei Lincei e Scuola Normale - 17/02/ 2017 - YouTube

Research-related activities

In the past 25 years I have organised and directed several international programmes and advanced schools. Among these, a 4-month research programme on geometry and topology of fluid flows at the Newton Institute for Mathematical Sciences in 2000, a CIME Summer School on topological fluid mechanics (under the auspices of the Italian Mathematical Union) in 2001, a 3-month programme on knots and applications at the Ennio De Giorgi Mathematical Research Centre of the Scuola Normale Superiore in Pisa (2011), an IUTAM Symposium on helicity in Venice (2016), the first Chinese programme on topological aspects of knotted fields  at the Beijing University of Technology (2019), and an international workshop on topological methods in mathematical physics held at the Majorana Centre in Erice in 2022.

I am a founding member of GEOTOP-A, an international web-seminar series launched in 2018 to promote applications of geometry and topology in sciences, and a founding member of The Association for Mathematical Research, a non-profit organisation launched in 2021 to support mathematical research and scholarship through a broad spectrum of services to the mathematical community.