Scientific Interests and Collaborations

Geometry of completely integrable systems. The momentum map the main tool in the theory of dynamical systems with symmetries. The works of Atiyah, Guillemin, Smale, Sternberg drastically changed the understanding of the bifurcation diagram associated to momentum maps. Under the guide of Sergei Petrovich Novikov, I worked on an extension of the convexity theorem of Atiyah, Guillemin-Sternberg to the case of non-Hamiltonian group actions (see [1] and [8]). I then turned my attention to the topological obstructions in this environment. In [2] I described the obstruction theory associated to a completely integrable system borrowed form algebra: the Gelfand-Cetlin system. I proved that the integrals in involution which define this system can be obtained via the introduction of a dynamical system admitting a Lax pair representation.
Later, under the guide of Richard Cushman, I become involved in the investigations of dynamical completely integrable systems and their cohomological obstruction, mostly monodromy. With Holger Dullin, Richard and I studied a special dynamical system, the "swing spring" (or elastic pendulum), whose non-trivial monodromy is somewhat related to a remarkable behavior called "precession of the swing plane". This cooperation produced the article [4] in which we describe the relation of this precession phenomenon with the multivalued rotation angle which is the analytical counterpart of the non-trivial monodromy of the system. With Richard, Boris Zhilinskií and Dimitrií Sadovskií we then described the quantized swing spring, analyzed its quantum spectrum, and scrutinized the relations between the monodromy of this 3 degree of freedom system and the defects in its quantum spectrum. All this can be found in [5]. The interactions of Dimitrií the chemist Marc Joyeux), together with Darryl Holm and Peter Lynch, the original discoverers of the phenomenon of precession of the swing-spring, brought into light a relationships between this dynamical system and the CO₂ molecule described in [6].
My collaboration with Boris and Dimitrií evolved towards the investigation of Boris's idea of fractional monodromy and bidromy. The phenomenon of fractional monodromy was first geometrically described by Nikolaí Nekhoroshev with Boris and Dimitrii, and later analyzed analytically by Konstantinos Efstathiou, Dimitrií and Richard. I gave a contribution to this field finally showing that there is a difference between a homological and a homotopical approach. This fact acquires a meaning in this setup since systems with fractional monodromy are the first example in which the homotopy groups of some fibers are non-commutative, and hence the parallel transport of homotopy cycles differs from that of homology cycles. This investigation is carried on in the article [12].
Also systems with the topological structure of bidromy are typical in physics, and is strictly related to cuspidal singularities. In [10] I produced a complete list of singularities of 2 degree of freedom completely integrable systems, and analyzed their stability. In [15], with Konstantinos, we did show that the interaction of two cuspidal singularities can originate two different topologies: the pleat and the flap. We also explain that this second topological structure can yield non-trivial monodromy and how this is related to lens-spaces embedded in the system.

Non-holonomic "completely integrable" systems. Completely integrable systems are special dynamical systems whose phase space fibrates in tori and whose flow is a winding line in such tori. Classical completely integrable systems are Hamiltonian, but completely integrable behavior is common in a fundamental class of non-Hamiltonian systems: the non-holonomic ones. Francesco Fassò and I did investigate the geometry of a possible, broader definition of complete integrability (see [3]), first introduced by Bogoyavlenskij (Lie?). In [7] Francesco, Nicola Sansonetto, and I did investigate the geometry of a torus fibration and the appearance of a Poisson structure in a very specific non-holonomic system: the homogeneous ball rolling on a surface of revolution. In [9] we described the existence of a flower bifibration --typical in superintegrable systems-- in all dynamical systems with a compact symmetry group and periodic reduced dynamics.
It is well known that symmetries of a dynamical system are related to conserved functions. In [11] we gave a precise meaning to such relation outside of the Hamiltonian setting. In [13] we gave an estimate on the number of Weakly-Nöetherian integrals of motion for a non-holonomic systems. Weakly-Nöetherian integrals of motion are those functions that turn out to be integrals of motion for every non-holonomic system with a give symmetry group and given kinetic form. In [14] we described the relation between weakly-Noetherian constants of motion and horizontal gauge momenta.

Stability in fluid flows. Recently I became involved in the investigations of the stability of the motionless solutions of some problems of fluid-dynamics. Prompted by Giuseppe Mulone, and in collaboration with Paolo Falsaperla, we worked on the solute Bénard proble, where the evolving fields are velocity, temperature, and solute concentration in an infinite layer of fluid. I With Paolo, we obtained analytic formulas for the boundary of the region of linear instability when the boundary conditions are of reasonably general type [16]. With Giuseppe and Paolo we are also concentrating on a similar problem where the solute is substitute with the presence of a rotational force field [19], and with more general disquisitions of the effects that skew-symmetric or time-dependent terms can have on stability [16].
Giuseppe, Paolo, and I were also involved in a problem of fluid flows in porous media. In [18] we described the dependence of the critical Rayleigh number for a binary fluid in a rotating porous medium with general boundary conditions.

Publications

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[21]	A mathematical model of anorexia and bulimia, Mathematical Methods in the Applied Sciences online (2014). With C. Ciarcià, P. Falsaperla and G. Mulone.
[20]	Stability in the rotating Bénard problem and its optimal lyapunov Functions, Acta Applicanda Mathematica 132, 307-320 (2014). With G. Mulone.
[19]	Some results in the nonlinear stability for rotating Bénard problem with rigid boundary condition, Atti della Accademia Peloritana dei Pericolanti 91, A9 1-10 (2013). With P. Falsaperla and G. Mulone.
[18]	Double diffusion in rotating porous media under general boundary conditions, International Journal of Heat and Mass Transfer 55, 2412-2419 (2012). With P. Falsaperla and G. Mulone.	Padua@research
[17]	Does a linear symmetric operator of a dynamical system help stability?, Acta Applicandae Mathematicae 122, 239-253 (2012). With P. Falsaperla and G. Mulone.	Padua@research
[16]	Marginal regions for the solute Bénard problem with many types of bounday conditions, International Journal of Engineering Science 57, 11-23 (2012). With P. Falsaperla.	Padua@research
[15]	The topology associated to cusp singular points, Nonlinearity 25, 3409-3422 (2012). With K. Efstathiou.	Padua@research
[14]	Linear weakly-Nöetherian constants of motion are horizontal gauge momenta, Journal of Geometric Mechanics 4/2, 129-136 (2012). With F. Fassò and N. Sansonetto.	Padua@research
[13]	On the number of weakly-Nöetherian constants of motion of nonholonomic systems, Journal of Geometric Mechanics 1, 389-416 (2009). With F. Fassò and N. Sansonetto.	Padua@research
[12]	Fractional monodromy: parallel transport of homology cycles, Differential Geometry and its Applications 26, 140-150 (2008).	Padua@research
[11]	Gauge conservation laws and the momentum equation in nonholonomic mechanics, Reports on Mathematical Physics 62, 345-367 (2008). With F. Fassò and N. Sansonetto.	Padua@research
[10]	Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems, Regular and Chaotic Dynamics 12, 717-731 (2007).	Padua@research
[9]	Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, S.I.G.M.A. 3, 1-12 (2007). With F. Fassò.	ArXiv
[8]	Convexity of multi-valued momentum maps, Geometria Dedicata 111, 1-22 (2005).	ArXiv
[7]	Periodic flows, Poisson structures, and nonholonomic mechanics, Regular and Chaotic Dynamics 10, 267-284 (2005). With F. Fassò and N. Sansonetto.	UniPd
[6]	The CO2 molecule as a quantum realization of the 1:1:2 resonant swing spring with monodromy, Physical Review Letters 93, 024302 (2004). With R. Cushman, H. Dullin, D. Holm, M. Joyeaux, P. Linch, D. Sadovskii and B. Zhilinskii.	Igitur
[5]	Monodromy of the quantum 1:1:2 resonant swing spring, Journal of Mathematical Physics 45, 5076-5100 (2004). With R. Cushman, D. Sadovskii and B. Zhilinskii.	Purple
[4]	Monodromy in the resonant swing spring, Physica D 190, 15-37 (2004). With R. Cushman and H. Dullin.	ArXiv
[3]	Geometric structure of "broadly integrable" Hamiltonian systems, Journal of Geometry and Physics 44, 156-170 (2002). With F. Fassò.	UniPd
[2]	Some remarks on the Gelfand-Cetlin system, Journal of Physics A 35, 10591-10605 (2002).	UniPd
[1]	Convexity and multi-valued Hamiltonians, Russian Mathematical Surveys 55, 578-580 (2000).	UniPd

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Updated 10.2012