Contacts
Department of Mathematics & Applications
University of Milano - Bicocca
Via Cozzi, 55
20125 Milano, Italy

Building U5, III floor, Room 3065

Email: renzo.ricca@unimib.it
Direct line: (+39) 02 6448 5762
Department: (+39) 02 6448 5755
Fax: (+39) 02 6448 5705

Research Interests

My work is in applied mathematics and mathematical physics of fluid and complex systems, centered on aspects of topological dynamics applied to:

  1. vortex dynamics and tangles;
  2. ideal magnetohydrodynamics and energy relaxation of magnetic fields;
  3. structural complexity and energy-complexity relations;
  4. topological transitions of minimum energy surfaces;
  5. morphological study of elastic filaments and applications to biological systems;

(i) Vortex Dynamics

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Study of dynamical properties of three-dimensional vortex filaments and complex vortex tangles in an ideal fluid governed by the Euler equations. In this context vortex filaments move according to the Biot-Savart law, that is a global, integral, functional of vorticity.

Questions related to global and local aspects involve naturally the geometry and topology of vorticity as well as its functional distribution in space. These questions are intimately related to the integrability of the Euler equations and the existence of finite time singularities in perfect fluids, two aspects of one of the most challenging problems in contemporary mathematics (see the Clay Millennium Prize Problem).

Applications include advanced diagnostics in visiometrics, estimates of dynamical properties and energy distribution in complex vortex flows, superfluid and classical turbulence.

(ii) Ideal Magnetohydrodynamics

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Study of evolution and relaxation of magnetic fields in ideal magnetohydrodynamics. In this context we consider magnetic fields embedded in flux tubes, that evolve according to the associated Lorentz force.

Magnetic energy is led to relax to a minimum energy state, dictated solely by topology. A fundamental open problem here is the study of the groundstate energy spectrum of magnetic knots and links of given topology, a problem at the heart of modern topological (quantum) field theory.

Applications include the study of energy contents in astrophysical flows and, of particular importance to us, in solar coronal magnetic fields and in confined fields for nuclear fusion research.

(iii) Structural Complexity

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Study of structural complexity deals with the analysis of relationships between morphological organization of structures (in a broad sense, from real to abstract objects) and functional properties.

At all scales, from the mass distribution in the Universe to the intricate neural networks in our brain, structures form and evolve. Social networks and language structures (including mathematics!) are other examples. Cooperative behaviour, self-organization and functional purpose (even in more abstract contexts) are often hidden by local "dynamics". One of primary tasks here is to unveil these connection, Of fundamental importance is work on energy-complexity relations and criticality theory.

Applications include a wide range of context from avalanche prediction to solar flares, from stock market econometrics to rare events insurance.

(iv) Minimum Energy Surfaces

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Study of minimum energy surfaces originates with the works of Plateau on soap films and the subject has always attracted mathematicians for its beauty, by coupling the study of ideal shape with that of minimum energy, in the attempt to optimize both.

Fundamental questions raised by Courant in the late 30's about surfaces of minimal area that could be visualized with soap films spanning wire frames of various shapes, including knots and links, have remained mostly unanswered. Recent work on topological dynamics may now contribute to tackle some of these problems.

Applications include froth formation and lipid surfaces in cell biology, nematic surfaces in condensed matter physics and thin shell architecture in engineering.

(v) Elastic Filaments and Applications

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Study of elastic filaments has a long tradition, rooted in the very origin of elasticity theory. Thin rod analysis on instability and buckling has led to the discovery of a wide collection of morphologies of stable and metastable configurations for both closed and open loops.

By a continuous exchange of bending and torsional energy, elastic filaments adjust their shape accordingly in open or confined volumes, by folding, supercoiling and packing to a high degree of complexity. Conformational properties of tertiary structures, spatial network organization and packing mechanisms pose fundamental questions as regards local-to-global information, collective behaviour and functional accessibility and performance.

Applications include DNA genomics, proteomics, viral and bacterial growth in biology, polymeric architecture in chemical physics and structural organization of filamentary structures in mesoscopic physics.