Abstract: In this talk we discuss some recent results for a class of nonlinear models in Quantum Mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrodinger equation (NLS) with a periodic potential and a Stark-type perturbation, in the limit of large periodic potential. In the first part of the talk we prove the existence of a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases. In the second part of the talk we prove the validity of the tight binding approximation, i.e. the reduction of the NLS to a discrete NLS. This model has many interesting features: e.g. the measurement of the value of the gravity acceleration g, using ultracold Strontium atoms confined in a vertical optical lattice. 

Abstract: The Canham-Helfrich energy is widely used to describe the elastic properties of biological membranes at the sub-cellular level. The membranes are modeled as surfaces in the 3-dimensional space and their shape minimize the energy which penalizes the curvatures of the surface. The Canham-Helfrich energy can be generalized to the multiphase case in order to model also heterogeneous biological membranes, for instance in the presence of proteins on the membrane. In this seminar first of all I will review some tools of differential geometry and I will introduce the Canham-Helfrich functional. A first step is to look at rotational symmetric shapes, both in one phase and in multiphase: this has been done in 2013 by Choksi, Morandotti and Veneroni. If no symmetry of the minimizers is assumed, the problem requires other tools. I will briefly motivate the necessity of a weaker notion of a surface considering some examples coming from the soap film theory, which is easier to understand. Finally, I will briefly discuss existence of single and multiphase minimizers under area and enclosed volume constraints and regularity of minimizers. This is a joint work with K. Brazda and U. Stefanelli both at the University of Vienna.

Abstract: Roughly speaking, mixing is the property of a dynamical system whereby the dynamics tends to "mix up", or randomize, trajectories. When a finite reference measure exists, the natural formulation of mixing coincides with the decay of correlations (in time) of the observables of the system. In the case of an infinite reference measure, the standard definition--a.k.a. finite mixing--is inapplicable. Finding an effective replacement--infinite mixing--is a fundamental, and debated, question. The (few) definitions that have been attempted in the past use 'local observables', that is, functions that essentially only see finite portions of the phase space. We introduce the concept of 'global observable', a function that represents a certain quantity throughout the phase space. This concept is based on the notion of infinite-volume average, which is inspired by statistical mechanics (which is, after all, the field of mathematical physics that has most successfully dealt with extended systems). Endowed with the notions of global and local observables, we give a few definitions of infinite mixing. These fall in two categories: global-global mixing, which expresses the "decorrelation" of two global observables, and global-local mixing, where a global and a local observable are considered instead. We will discuss these definitions and, time permitting, see how they respond on different kinds of infinite-measure-preserving dynamical systems.

Abstract: In opposition to its analogues in dimension one and three, the Schrödinger equation in dimension two with a concentrated nonlinearity displays a behaviour that is, to our knowledge, unprecedented in similar models and not trivial to interpret. After illustrating some physical motivation of the model, we discuss the features of the dynamics it generates, with some emphasis on the blow phenomenon and on stability of standing waves. This is a joint work with R. Carlone (Napoli), M. Correggi (SNS-Pisa), and L. Tentarelli (Napoli).

Abstract: The novel demands from the science of materials require the conception of materials showing exotic behaviour. In the theory of metamaterials one specifies the mechanical properties of a given material or structure by assigning the action functional which governs it and then he looks for the specific mechanical system whose behaviour is exactly the desired one. We talk about the <<synthesis>> of a specific metamaterial. In some situations homogenisation techniques are needed and the required material is synthesised at a certain length scale by using some smaller scale microstructures. In this context the theory of higher gradient continua needs to be introduced. In this presentation we refer some results recently obtained in the problem of synthesis of second gradient materials, of extremely extensible microstructures and of structures being damage resistant. The synthesis problem is solved by introducing so called <<pantographic>> structures.

Abstract: By iterating a continuous function $f:X \to X$, we obtain a dynamical system on the space $X$. We are interested in the long-term behavior of points in this system. Of particular importance is recurrent and transitive behavior. We will consider some simple notions of recurrence and transitivity (points that are periodic, nonwandering, chain recurrent, strong chain recurrent, or generalized recurrent) and see how they are related to each other, and to the mixing structure of the dynamical system. Finally, we will look at explosions in the generalized recurrent set (joint work with Anna Florio and Olga Bernardi).

Abstract: We develop a hybrid flow simulations method for complex fluids, such as polymer/colloid systems, with a priori unknown relation stress/strain rate. By making use of the scale separation according to the heterogeneous multiscale methods (HMM) we combine the computational efficiency of the continuum description on the macro-level with molecular accuracy of particle-based Molecular Dynamics (MD) simulations on micro-level. The macroscopic solver is based on the governing conservation equations for the momentum and mass of the incompressible flows. These equations are discretized using the discontinuous Galerkin method (dG) with a local high-order polynomial representation of the solution and are solved using the BoSSS simulation package. The rheological properties of a modeled fluid, mimicked by the divergence of the stress tensor in the viscous terms of the momentum equations, depend on the flow velocity field. This information is not available in general, but it can be evaluated by means of the MD.

Abstract: We study the dynamics and the invariant sets for a class of dissipative systems, namely conformally symplectic systems. More precisley, we consider flows that do not preserve the symplectic structure but do alter it up to a constant scalar factor. The study of invariant Lagrangian submanifolds for these systems, in particular KAM tori, has been investigated by means of varied techniques in recent years. We will focus on what happens when these invariant Lagrangian submanifolds stop to exist. Inspired by the celebrated Aubry-Mather and Weak KAM theories for Hamiltonian systems, we prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. We describe the structure and the dynamical significance of these sets, their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.

Abstract: We consider the gravitational 2-body problem on 2-dimensional surfaces of constant curvature. For non-zero curvature the problem is no longer integrable and numerical experiments indicate that its dynamics is chaotic. We perform the Poisson reduction of the equations and classify all relative equilibria (RE) with respect to the action of the group of isometries of the corresponding constant curvature space. These RE are the simplest solutions of the problem and have the property that the distance between the bodies remains constant throughout the motion. We also establish the stability of these RE and consider their behaviour as a function of the curvature of the space.

Abstract: Active materials are media for which deformations can occur in absence of loads, given an external stimulus. In the talk I will compare the two main methods used to model such materials, namely active stress and active strain, keeping in mind the important example of skeletal muscle tissue. Considering an incompressible and transversely isotropic material, constitutive relations will be designed so that the two approaches produce the same results
for a uniaxial deformation along the symmetry axis. In a hyperelastic setting, it will be shown that a simple shear produces different stresses in the two approaches, unless some very restrictive conditions on the strain energy density are fulfilled. Hence, active stress and active strain produce contrasting results in shear, even if they both fit uniaxial data.
Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach better capture the activation mechanics. A further study of other deformations, such as simple shears, would be important in order to develop a realistic model of an active material.
(Joint work with Giulia Giantesio and Davide Riccobelli).

Abstract: Since Kurchan’s seminal work (2000), two-time measurement statistics (also known as full counting statistics) has been shown to have an important theoretical role in the context of quantum statistical mechanics, as they allow for an extension of the celebrated fluctuation relations  (Evans-Searls, Gallavotti-Cohen) to the quantum setting. In this contribution, we consider two-time measurement statistics of heat for a locally perturbed system, and we show that the description of heat fluctuation differs considerably from its classical counterpart, in particular a crucial role is played by ultraviolet regularity conditions. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation. This phenomenon has no classical analogue. (Joint work with Tristan Benoist and Renaud Raquépas)

Abstract: In classical diophantine approximations it is natural to consider the set "Bad" of those real numbers which are badly approximable by rationals: it is a set of zero Lebesgue measure and full dimension. Finer metric properties have been investigated in depth, both for the classical case and for several generalizations, which arise from the relation between diophantine approximations and the dynamics on homogeneous spaces (or other moduli spaces). The set Bad admits a natural exhaustion by sub-sets Bad(c), in terms a positive parameter c>0, and the dimension of Bad(c) converges to 1 as c goes to 0. D. Hensley computed the asymptotic for the dimension up to the first order in c, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded.
I will prove a generalization of Hensley's asymptotic formula in the context of Fuchsian groups, considering the set off points in the boundary of the hyperbolic space which are badly approximable by the orbits of a non-uniform lattice G in PSL(2,R), and an exhaustion of such set by subsets Bad(G,c), in terms of a parameter c>0. Bowen and Series introduced a "boundary expansion" which enables to approximate any set Bad(G,c) by a dynamically defined Cantor set, whose dimension can be estimated with great precision by thermodynamic techniques introduced by Ruelle and Bowen. A perturbative analysis of the spectral radius of the transfer operator gives the dimension of Bad(G,c) up to the first order in c.

Abstract: The bundle of quadratic differentials with simple zeroes over the moduli space of Riemann surfaces can be endowed with a natural symplectic structure, which we call "homological symplectic structure" in terms of explicit Darboux coordinates. On the other hand, the vector bundle of quadratic differentials is a model of the cotangent bundle of the same moduli space, and hence it carries the canonical symplectic structure. The two structures coincide in the common domain, and hence we provide Darboux coordinates for the canonical structure as well.
In addition, the affine bundle of projective connections is modelled on the bundle of quadratic differentials. By choosing a holomorphically varying base projective connection, the projective connections and the space of quadratic differentials can be naturally (but not canonically) identified. This allows to induce symplectic structures also on the moduli space of projective connections.
Moreover, a projective connection defines a monodromy map and hence a point in the character variety, i.e., homomorphism of the fundamental group of a Riemann surface into unimodular two by two matrices modulo conjugations. 
Goldman ('86) introduced a Poisson bracket on the character variety which, in this case, is symplectic.
Then the result is that the push forward of the homological symplectic structure to the character variety (by an appropriate choice of base projective connection) coincides with the Goldman bracket.
I hope to define all the necessary objects and be as elementary as possible.
This is joint work with Chaya Norton (Concordia) and Dmitry Korotkin (Concordia/CRM).

Abstract: The celebrated Poincaré-Birkhoff Theorem, with its several applications, attests the relevance of twist as boundary condition for planar Hamiltonian systems. But what becomes of the twist condition when we try to extend this result to higher dimensions? Whereas on the plane the notion of twist is quite intuitive, the same cannot be said about twist in higher dimensions. Indeed, due to the special nature of the plane, the Poincaré-Birkhoff Theorem can be seen as the overlap of several distinct higher dimensional results. In this seminar we will overview the two main approaches to this issue. 

Abstract: We show that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered systems in a surprising fashion. For this aim use is made of perturbation theory techniques such as KAM theorem or Nekhoroshev theorem, but they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We cite some results on infinite chains of interacting particles and Hamiltonian partial differential equations, and we explain in details the example of a nonlinear chain of particles with alternating masses, an hyper-simplified model of diatomic solid. In the latter case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, both with analytical and numerical works, we show the advancements with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit, as this is of great relevance for physical implications.

Abstract: In this talk we present a non-homogeneous probabilistic cellular automaton (a "parallel" Markov chain) for spin systems on a square lattice. We describe the stationary measure of the chain and investigate the connection of this dynamics with an equivalent one on the honeycomb lattice. Further we determine a set of values for the parameters of the dynamics that allows us to interpret it as a "microscopic" model for the interaction between lithosphere and mantle. In this context, the transitions of the Markov chain are thought to represent earthquakes and the non-homogeneity of the chain is intended to mimic the effect of tidal forces on the lithosphere.

Abstract: We study the semiclassical inverse spectral problem for periodic Schrödinger operators in connection with the homogenization of the Hamilton-Jacobi equation. This approach, that works without integrability assumptions, provides a way to generalize some results on the spectral limit of quantum integrable operators proved by Pelayo-Polterovich-Ngoc.

Abstract: There are numerous situations in which the mechanics of slender or thin objects manifests itself through significant shape modifications, even in the presence of small stresses. This is due to the fact that small local strains can have a large cumulative effect on the shape of a continuum. By "shape" we indicate geometric features of an object that are invariant under isometries of the three-dimensional ambient space. We name "shape energies" of continua those elastic energy functionals that depend only on such features. 

Abstract: This talk will include a general introduction to the Stokes phenomenon of linear systems of differential equations with singularities, and then explore its applications in Yang-Baxter equations, symplectic geometry/integrable systems  and 2d topological field theories.

Abstract: In this talk we intend to clarify the intimate relations between Lyapunov functions and recurrent sets. The study of this topic --started by Conley in the Seventies-- has had recent developments with the formalism of the weak KAM theory. We first resume the state of the art and then we present some new results, both for homeomorphisms and for flows. Moreover, we pose the following question: when the property of admitting a Lyapunov function —which is not a first integral— is stable under continuous perturbations? We explain how this problem is related to the so-called explosions of recurrent sets and we discuss some conditions which ensure that the above “stability” is verified. 

Abstract: K. Siegel asked if there is an open set of initial conditions for a Three Body Problem which has a dense subset of collision orbits, i.e. initial conditions whose orbits hit a collision. Consider the Restricted Circular Planar 3-Body Problem with mass ratio of the primaries $\mu$. We prove the existence of an open set in phase space where collision orbits form a $O(\mu^{1/20})$ dense set as $\mu$ tends to 0. This is a joint work with V. Kaloshin and J. Zhang.

ABSTRACT: The 3-dimensional Veselova top is a well-known non-holonomic system introduced in the 1980s, which was later generalized to higher dimensions. I will begin by discussing rigid bodies in n dimensions and then describe the general Veslova top. I will then consider certain symmetric versions of this system and show they are integrable.

Abstract: Just as symplectic and Poisson structures naturally arise in Hamiltonian mechanics, Dirac structures were introduced around 1990 by T. Courant and A.Weinstein to provide a geometric framework for mechanical systems with constraints. Dirac structures provide a unified viewpoint to several geometrical structures, and a key role in the theory is played by the so-called Courant brackets. Despite its original motivation in geometric mechanics, recent developments in "Dirac geometry" are related to a broad range of topics in mathematics and mathematical physics, including Lie theory, quantization, generalized complex geometry, group-valued moment maps, etc. The talk will give an introduction to Dirac structures, including their main examples and recent applications.

BSTRACT: Nello studio qualitativo dei sistemi dinamici giocano un ruolo fondamentale gli equilibri e la forma delle soluzioni attorno ad essi. Discuteremo la teoria spettrale soggiacente questo tipo di problema, dando una classificazione di tutti i casi possibili in dimensione 3 e 4 a partire dagli invariati fondamentali della matrice linearizzata. Useremo tale classificazione per studiare un sistema epidemiologico/ comportamentale 3-dimensionale in cui giocano un ruolo informazione e peer-pressure. Per tale sistema mostreremo:
- l’esistenza di equilibri endemici sotto soglia (backward bifurcation),
- la destabilizzazione dell’equilibrio endemico a causa della interazione tra informazione e peer-pressure,
- la nascita di un attrattore periodico.

Abstract: The asymptotic Maslov index gives us the angular asymptotic velocity at which the tangent vectors of a surface rotate, with respect to the linearised dynamics, inherited from a given diffeomorphism. After having defined the asymptotic Maslow index --also called Torsion-- for surface diffeomorphisms, we study the specific case of twist diffeomorphisms over the annulus. We present results concerning the link between properties of the Torsion and the dynamics of twist maps.

Abstract; Un semplice teorema generale suggerisce come cercare funzioni di Lyapunov dipendenti dal tempo per sistemi Lagrangiani. Il seminario mostra due degli esempi in cui la ricerca ha successo: i sistemi meccanici con resistenza di mezzo viscosa e le equazioni di Maxwell-Bloch dissipative dei lasers. Tramite queste funzioni di Lyapunov studiamo alcuni aspetti della dinamica.

Abstract: The theory of Poisson Vertex Algebras (introduced by Barakat, De Sole and V. Kac) is a convenient framework to study Hamiltonian and Integrable PDEs: in particular, it provides convenient and fast computational techniques. I will present an extension of the theory to the multidimensional case (namely, for Hamiltonian PDEs with more than one space variables) and show how it can be used to study the dispersive deformations of the Hamiltonian structures of hydrodynamic type. 

ABSTRACT: In this talk I will present our (with A. Buryak) construction of a vast class of quantum integrable systems associated to certain geometric objects from the moduli space of stable curves. The construction produces integrable quantum 1+1 hierarchies with several nice properties, including an universal and previously unknown (even in the classical case) recursion for the Hamiltonians, allowing to recover all integrals of motion. Integrable systems of double ramification type include KdV, ILW, Toda, Gelfand-Dickey, Drinfeld-Sokolov and many other well known classical hierarchies for which we can hence produce a quantization.

Abstract: My talk will focus on the study of the co-orbital resonance in the three-body problem. This domain of particular trajectories, where an asteroid and a planet gravitate around the Sun with the same period possesses a very rich dynamics connected to the famous Lagrangeʼs equilateral configurations $L_4$ and $L_5$, as well as to the Eulerianʼs configurations $L_1$, $L_2$ and $L_3$. A major example in the solar system is given by the Trojan asteroids harboured by Jupiter in the neighbourhood of $L_4$ and $L_5$. A second astonishing configuration is given by the system Saturn- Janus-Epimetheus; this peculiar dynamics is known as ``horseshoeʼʼ. Recently, a new type of dynamics has been highlighted in the context of co-orbital resonance: the quasi-satellites. They correspond to remarkable configurations: in the rotating frame with the planet, the trajectory of the     asteroid seems the one of a retrograde satellite. Some asteroids harboured by Venus, Jupiter and the Earth has been observed in this kind of configuration. The quasi-satellite dynamics possesses great interest not only because it connects the different domains of the co-orbital resonance (see works of Namouni, 1999), but also because it seems to bridge the gap between satellization and heliocentric trajectories. However, despite the term quasi-satellite has become dominant in the celestial mechanics community, some authors rather use the term ``retrograde satellite” which reveals an ambiguity on the definition of these trajectories. In the first part of my talk, I will present a numerical study that clarify the definition of these orbits by revisiting the planar-circular case (planet on circular motion) and describe the idea of an analytic method adapted to explore the quasi-satellite domain. All the previous results involve the averaged problem, an approximation of the Three- Body Problem. KAM type results on co-orbital motions are possible: in the last part of my talk, I will sketch in which frame we intend to give a rigorous proof (and up to our knowledge, the first one) of existence of the ``horseshoe ‘ʼ dynamics over infinite times. This last part is a joint work with Laurent Niederman (Observatoire de Paris, Université Paris XI ``Orsayʼʼ) and Philippe Robutel (Observatoire de Paris).

ABSTRACT: In this talk we study the Hamiltonian of the planar central motion with a real analytic potential. We prove that the corresponding Hamiltonian, when written in action angle variables, is almost everywhere quasiconvex, the only exceptions being the Keplerian and the Harmonic potentials. We underline that this two potentials are the ones discussed by Bertrand’s theorem. We also study the spatial central motion problem and deduce a Nekhoroshev type stability result for the perturbed system. This is a joint work with D. Bambusi and M. Sansottera.

ABSTRACT: In integrable Hamiltonian systems hyperbolic tori form families, parametrised by the actions conjugate to the toral angles. The union over such a family is a normally hyperbolic invariant manifold. Under Diophantine conditions a hyperbolic torus persists a small perturbation away from integrability. Locally around such a torus the normally hyperbolic invariant manifold is the centre manifold of that torus and persists as well.
We are interested in `global' persistence of the normally hyperbolic invariant manifold. An important aspect is how the dynamics behaves at the (topological) boundary. Where the manifold extends to infinity this boundary is empty - this case makes clear that we need the persistence theorem of normally hyperbolic invariant manifolds in the non-compact setting. If the normal hyperbolicity wanes as the boundary is approached we need to ensure that the perturbed dynamics does not come closer to the boundary. This provides the necessary uniform lower bound of normal hyperbolicity to still ascertain persistence under small perturbations. Making use of energy preservation and of Diophantine tori persisting by KAM theory this can be achieved for families of two-dimensional hyperbolic tori.

Abstract: The term "Arnold's diffusion" is usually referred to a peculiar instability phenomenon that can occur in certain nearly-integrable Hamiltonian systems with more than two degrees of freedom. Named after the Russian mathematician, the "mechanism", proposed in 1964, was able to construct motions in which the action variables (prime integrals of the unperturbed problem) could "drift" by an amount of order one, over a "very large" time (diffusion time), due to the absence of a dimension-related topological entrapment. As a counterpart of Nekhoroshev-type results, the problem of the (lower) bound of the diffusion time has always played a central role in this
context. A first estimate of this instability time was proposed in the comprehensive work "Drift and diffusion in phase space" (1994) by Chierchia and Gallavotti (CG). Having faced the challenging application of the Arnold mechanism to the Earth precession problem, the paper renewed the interest for the subject, stimulating in this way a powerful response of the scientific community. The developments witnessed a curious phenomenon for which the solutions constructed with the "geometrical" approach of (CG) exhibit a drift time far larger than those obtained by using the methods of the emerging "variational school" of U. Bessi, since 1996. In the same period, another geometrical method, due to Alekseev and Easton and known as windows method, was developed and shown to be extremely adequate for this class of problems by J.P. Marco, but the the time estimates produced were substantially different from those obtained in (CG), and they have been shown to be able to "reach", subsequently, the very good (or even optimal) diffusion estimates of the variational methods. Despite this evidence, the reasons for the described discrepancy were somewhat obscure until the work F. (2012), Tesi di Dottorato.  The aim of the talk is to discuss the problem of the diffusion time by means of geometrical methods for a paradigmatic class of systems, addressing some peculiar difficulties of the windows method application to isochronous models. It will be shown how it is possible to construct, also in this case, a set of motions whose drift time matches the one obtained by using variational methods.

ABSTRACT: Understanding the long-term orbital behavior of the population of space debris is a subject of intense current research, since space debris constitute a potentially major contamination of the Earth's near space  environment. The talk will provide a summary of recent results obtained in collaboration with colleagues from an astrodynamical network (A. Celletti, F. Gachet, G. Pucacco), aiming at characterizing the long-term orbital behavior of high area-to-mass ratio in the Earth's Geosynchronous resonance using analytical (normal form) techniques. The problem requires to accurately incorporate perturbations upon space debris caused by the geopotential's multipole harmonics, the Moon's and Sun's gravity, and solar radiation pressure. To this end, we construct an analytical model of secular dynamics based on the computation of a Hamiltonian normal form interpolating the flow of an original system of eight degrees of freedom. One relevant outcome is the  computation of a so-called `forced equilibrium' solution. In practice, this represents a safe disposal orbit for space debris. Mathematically, this is a solution lying on a 5-dimensional invariant torus, which, when mapped from phase space to configuration space, represents an orbit of nearly constant `forced eccentricity' and `forced inclination' (the latter is called the inclination of the invariant Laplace plane in satellite dynamics literature). We stress the benefits from the analytical approach based on normal forms, which allows to determine an invariant low-dimensional torus, whose otherwise determination by numerical means would constitute a hardly tractable task. We finally comment on the use of the normal form technique in defining proper elements for space debris, in analogy with the methods used in the study of asteroidal populations in our solar system.

ABSTRACT: In questo seminario si studia il legame fra lo spettro di alcuni operatori di Schrödinger periodici e l'Hamiltoniana effettiva della teoria KAM debole. Si mostra che l'estensione di quasimodi locali di tipo WKB è legata alla localizzazione dello spettro dell'operatore di Schrödinger. Questo risultato fornisce ulteriori informazioni rispetto alle ben note regole di quantizzazione di Bohr-Sommerfeld, e sotto ipotesi più generali dell'integrabilità o quasi-integrabilità.

ABSTRACT: In this talk we will discuss geometric features of nonholonomic systems before and after reduction. In particular we will characterize the failure of the Jacobi identity of the nonholonomic bracket, of the reduced bracket and of any "gauge related" bracket. We will present several examples that have the reduced dynamics described by a Poisson bracket.  Moreover, in these cases, we will see that first integrals end up being Casimirs of the reduced bracket. That is why we end the talk giving a description of first integrals (induced by the presence of symmetries) using the geometric tools used to characterize the failure of the Jacobi identity.

ABSTRACT: We derive the reduced equations of motion for an articulated n-trailer vehicle that moves under its own inertia on the plane. We show that the energy level surfaces in the reduced space are (n + 1)-tori and we classify the equilibria within them, determining their stability. A thorough description of the dynamics is given in the case n=1. The main results of this work were recently published in Bravo-Doddoli A. and Garcia Naranjo L.C., The dynamics of an articulated $n$-trailer vehicle, Regular and Chaotic Dynamics, 20, 497-517, (2015).

ABSTRACT: We give a recipe to generate ``nonlocal'' constants of motion for ODE Lagrangian systems and we apply the method to find useful constants of motion for dissipative system, for the Lane-Emden equation, and for the Maxwell-Bloch system with RWA.

Abstract: We describe what elements of a variational problem can still be seen in the canonical formalism using the Dirac theory. The talk will be based around two examples, which illustrate the issues involved.  No prior background in the Dirac theory is assumed in order to make the talk as accessible as possible to an audience with only a minimal acquaintance with the Euler-Lagrange equations.

Abstract: A plane curve is called nondegenerate if it has no inflection points. How many classes of closed nondegenerate curves exist on a sphere? We are going to see how this geometric problem, solved in 1970, reappeared along with its generalizations  in the context of the Korteweg-de Vries and Boussinesq equations. Its discrete version is related to  the 2D pentagram map defined by  R.Schwartz in 1992. We will also describe its generalizations, pentagram maps on polygons in any dimension and discuss their integrability properties. This is a joint work with Fedor Soloviev.

Abstract: In many problems of classical mechanics and theoretical physics dynamics can be described as a slow evolution of some periodic or quasi-periodic processes.  Adiabatic invariants are approximate conservation laws for such systems. Existence of adiabatic invariants makes dynamics close to regular. Destruction of adiabatic invariance leads to chaotic dynamics.  In the talk it is planned to present a review of currently known mechanisms of destruction of adiabatic invariance. It is planned to consider examples of manifestation of these mechanisms in problems related to charged particles dynamics.

Abstract: I will present an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales of Hilbert spaces, modeled for the search of quasi-periodic solutions for PDEs. As an application, I will deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact manifold which is homogeneous with respect to a compact Lie group. (Joint work with M. Berti and M. Procesi). 

Abstract: Qualitatively different systems of molecular energy bands are studied on example of a parametric family of effective Hamiltonians describing rotational structure of triply degenerate vibrational state of a cubic symmetry molecule. The modification of band structure under variation of control parameters is associated with a topological invariant "delta-Chern". This invariant is evaluated by using a local Hamiltonian for the control parameter values assigned at the boundary between adjacent parameter domains which correspond to qualitatively different band structures. 

Abstract: In this talk we discuss the simplest unitary Riemann surface braid group representations, which are constructed geometrically -- via representations of the Weyl-Heisenberg group -- by means of stable bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail with the help of noncommutative geometric tools, and an application of Fourier-Mukai techniques is given as well.

Abstract: The setting for noncommutative integrability for symplectic manifolds leads naturally to a fibration by isotropic tori and to its symplectic orthogonal distribution, which is integrable. Together they build a dual pair of Poisson maps on the symplectic manifold, as explained in [1]. We introduce the notion of dual pair of Jacobi maps on a contact manifold. This helps us to treat similarly the contact noncommutative integrability presented in [2].
[1] F. Fasso, Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87 (2005), 93-121.
[2] B. Jovanovic, Noncommutative integrability and action-angle variables in contact geometry, J. Sympl. Geom. 10 (2012), 536-561.

Abstract: A classical result of Aubry and Mather states that Hamiltonian twist maps have orbits of all rotation numbers. Analogously, one can show that certain ferromagnetic crystal models admit ground states of every possible mean lattice spacing. In this talk, I will show that these ground states generically form Cantor sets, if their mean lattice spacing is an irrational number that is easy to approximate by rational numbers. This is a joint work with Blaz Mramor.

Abstract: Si parlera' del calcolo delle variazioni in presenza di vincoli cinetici (detti anche non integrabili, nel caso lineare) avendo in mente la possibilita' che le estremali possano presentare delle discontinuita' sulle derivate prime (corners). Verra' ricavata la condizione di estremalita' e verra' studiata la variazione seconda del funzionale, considerando variazioni anche asincrone dei corners. La trattazione cerchera' di privilegiare una formulazione geometrico-intrinseca.

Abstract: The existence of an invariant measure for a system of differential equations is a very important property. From the point of view of dynamical systems, it is a key ingredient for the application of ergodic theory. It is also a crucial hypothesis in Jacobi's theorem of the last multiplier that establishes integrability of the system via quadratures. Moreover, the existence of a smooth invariant measure imposes certain restrictions on the qualitative nature of the fixed points of the system; namely, it prohibits the existence of asymptotic equilibria. In this talk I will show how the geometric structure that underlies the equations of motion of a nonholonomic mechanical system with symmetry can be exploited to obtain necessary and sucient conditions for the existence of an invariant volume. As a concrete application, we show that the reduced equations of a rigid body with a planar face that rolls without slipping over a sphere possess an invariant measure if and only if the body is planar, or axially symmetric. (Joint work with Y. Fedorov and JC Marrero).

Abstract: Abstract. Il teorema non-squeezing di Gromov afferma che nessun diffeomorfismo simplettico puo' mandare una palla di raggio R in un cilindro avente per base un disco bidimensionale di raggio inferiore ad R. Questo risultato puo' essere visto come un fenomeno di rigidita' bidimensionale dei  diffeomorfismi simplettici, che si affianca alla ben nota rigidita' 2n-dimensionale espressa dal teorema di Liouville (i diffeomorfismi simplettici conservano il volume). In questo seminario discuteremo una possibile generalizzazione 2k-dimensionale di questo tipo di rigidita'.

Abstract: Si mostrano dei sistemi Hamiltoniani integrabili in dimensione 4 con equilibrio che e' linearmente stabile ma instabile alla Lyapunov e  senza moti che convergono all'equilibrio per t che tende a meno infinito. Nessun esempio esplicito di questa instabilità debole per sistemi Hamiltoniani pare fosse noto in letteratura. Questi sistemi fanno parte di una classe che comprende anche rari elementi per cui l'equilibrio e' stabile e si hanno oscillazioni periodiche isocrone; si tratta di sistemi superintegrabili. Alcuni sono ritrovati anche tramite il teorema variazionale di Noether. 
Gaetano Zampieri, Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony, to appear in Comm. Math. Phys.
Gianluca Gorni, Gaetano Zampieri, Variational Noether's theorem: the interplay of time, space and gauge. In preparation

Abstract: The class of methods named after Ritz and Galerkin is today one of the most  fundamental instruments for scientific computations. On the occasion of the 100th anniversary of Walther Ritz's death was organized a colloquium in Ritz's birth-place Sion, Switzerland. In two successive talks of that conference, given by Martin Gander and myself, had been retraced the long and interesting path of the discovery of this method, beginning with the Bernoulli brothers, Euler, Lagrange, Gauss, Riemann, Schwarz, Hilbert, Kirchhoff, Ritz, Timoschenko, Bubnov, Galerkin until Richard Courant. The present talk is a compilation of these two talks.

Abstract: Nonholonomic mechanical systems are not Hamiltonian. They are Hamiltonian with respect to an "almost Poisson bracket" of functions that fails to satisfy the Jacobi identity. In this talk we will use this bracket to investigate some features of an integrable version of the nonholonomic Suslov problem that was originally studied by Okuneva. This system concerns the motion of a constrained rigid body moving in a potential and has the remarkable property that the flow takes place in 2 dimensional compact invariant manifolds whose genus may vary from zero to five. This contrasts with the usual Liouville notion of integrability where the generic invariant manifolds are tori.

Abstract: The study of topological invariants of completely integrable Hamiltonian systems is naturally intertwined with the classification of Lagrangian fibrations. Several authors have studied this problem from different perspectives. In this talk I shall give a definition of the topological characteristic classes of Lagrangian fibrations, emphasising the relationship between these invariants and the affine geometry which is intrinsic to such bundles.

Abstract: 

Abstract: 

Abstract:  pdf 

Abstract: The FPU paradox is the persistence of energy localization in the `low--q' Fourier modes of Fermi--Pasta--Ulam nonlinear lattices, preventing
equipartition among all modes at low energies. Two approaches have been recently proposed, for the explanation of this paradox. In the first approach, a low-frequency fraction of the spectrum is initially excited, leading to the formation of the so-called `natural packets', which exhibit exponential stability. In the second, emphasis is placed on the existence of `q--breathers', i.e continuations of the periodic orbits of the linear modes, which are exponentially localized solutions in Fourier space. Following ideas of the latter, we introduce the concept of `q--tori', representing exponentially localized solutions on low--dimensional tori, demonstrate their existence and estimate their harmonic energy profiles in order to provide a more complete explanation of the FPU paradox.

Abstract: The Fermi-Pasta-Ulam experiment is famous for the unexpected recurrent behavior of long waves. Attempts to explain this observation usually involve an approximation by a KdV equation and a non-rigorous KAM argument. A rigorous justification of the KdV approximation for finite times was given only recently, independently by Bambusi-Ponno and Wayne-Schneider. In this talk I will show how to derive the KdV equation as a resonant normal form, and I will present an improvement to this first approximation. This is a step towards the justification of the so-called metastability scenario.

Andrea Boattini
The Univerisity of Arizona

The Catalina Sky Survey: transitioning through NEO survey generations

Abstract: The Catalina Sky Survey (CSS) is a NASA funded program to discover and track asteroids and comets that can approach the Earth closely (Near Earth Objects, NEOs). The CSS has been operating 3 telescopes for the past three years while extending its sky coverage into the far southern hemisphere but also towards lower solar elongations. This effort led to the discovery of more than 1,500 NEOs (almost 70% of the current discovery rate) that allows to compare orbital characteristics, size distributions of NEOs discovered at different solar elongations by the CSS telescopes. Results from a sample of about 1,000 bodies (~400 Amors, ~500 Apollos and ~100 Atens) will be analyzed and discussed in view also of refining observing strategies for the next generation NEO surveys. More than 100 comets were discovered serendipitously during the survey.

Olga Bernardi
Universita` di Padova

Localizzazione degli insiemi di Aubry-Mather tramite tecniche di localizzazione ispirate alle teorie di viscosità

Abstract: Gli insiemi di Aubry-Mather (o cantori) sono insiemi invarianti per sistemi dinamici quasi-integrabili. Tali insiemi sono caratterizzati da una peculiare topologia e non sono rilevabili da tecniche numeriche standard. Si introducono gli insiemi di Aubry-Mather per una classe di mappe quasi-integrabili in 2 dimensioni e si discute la loro localizzazione tramite tecniche di regolarizzazione ispirate alle teorie di viscosita'.

Larry Bates
University of Calgary, Canada

Symmetry, periodic solutions of Hamiltonian systems, and the isotropic symplectic groups

Abstract: Some simple examples will be discussed to show some new invariants exposed by certain natural subgroups of the symplectic group that arise naturally in the study of periodic solutions of Hamiltonian systems, especially those with symmetry.

Elena Celledoni
Universita` di Trondheim

Energy-preserving methods and B-series

Abstract: pdf

Armando Bazzani
Universita` di Bologna

Emergent properties in models for pedestrian dynamics

Abstract: The application of dynamical systems theory to social sciences using the complexity key of lecture, has been recently considered, by various research groups.  Understanding the macroscopic emergent properties of particle systems that perform a cognitive behavior, is one of the main goal of complexity science. We have studied automata gas models for pedestrian dynamics whose microscopic "particles" have iteractions based on a local vision mechanism and are able to take simple decisions in presence of multiple choices.  We discuss the statistical properties of the models and the transitions to self-organized states. The problem of comparison with expemental observations will be illustrated using video films of crowd dynamics during the Venezia Carnival 2007.

Massimo Villarini 
Universita' di Modena e Reggio Emilia

Classification of perturbations of the hydrogen atom in small static electric and magnetic fields: resonance zone structure

Abstract: Si discutera' il problema dell'esistenza e della molteplicita' delle soluzioni periodiche su un fissato livello di energia di una Hamiltoniana naturale, i.e. del tipo "energia potenziale + energia cinetica". Verra' presentato un risultato di molteplicita' che, sotto una ipotesi geometrica di "pinching", risponde affermativamente ad una congettura di Seifert (1948). Si accennera' alla possibilita' di utilizzo di analoghi metodi variazionali per provare esistenza e molteplicita' di tori invarianti.

Antonio Siconolfi
Roma La Sapienza

Omogeneizzazione di  Hamiltoniane in ambiente stazionario ergodico

Abstract: Si vogliono illustrare alcune estensioni della ricca teoria qualitativa inerente il problema cella o equazione critica dal caso periodico a quello stazionario ergodico. Si noti che i risultati conosciuti  di  omogenizzazione stocastica per equazioni di Hamilton-Jacobi  (Souganidis, Rezakhanlou e Tarver) sono stati ottenuti lavorando direttamente su formule di rappresentazione dele soluzioni,  e usando  il teorema subadditivo di Kingman. Quindi prescindendo dal problema cella e dall'esistenza di correttori esatti o approssimati.

Dmitrii Sadovskii
Universite' du Littoral

Classification of perturbations of the hydrogen atom in small static electric and magnetic fields: resonance zone structure

Abstract: We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalising with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1:1 resonance corresponds to the orthogonal field limit, studied  earlier by Cushman & Sadovskii in 2000.  We describe the structure of the 1:1 zone, where the system may have monodromy of different kinds, and consider briefly the 1:2 zone. See the paper by K. Efstathiou, D.A. Sadovskii and B.I. Zhilinskii in Proc. R. Soc. A Vol. 463, 2007, pp.1771-1790.

James Montaldi
University of Manchester

Bifurcations in Hamiltonian systems with symmetry: the role of zero momentum

Abstract: The bifurcations that occur in dynamical systems with symmetry have been studied a great deal and much is known. However, the role of conserved quantities is not so well-understood, and in this talk I will show how the geometry of symplectic reduction governs the bifurcations that arise, and illustrate the results with some examples.

Anna Maria Cherubini
Universita` del Salento

Comportamento asintotico di una 'pallina che rimbalza'

Abstract: I modelli piu' usati di particelle che collidono, diffusi per esempio nello studio dei fluidi granulari, presentano una patologia nota come 'collasso anelastico': gruppi di particelle sono soggette ad un numero infinito di collisioni in un tempo finito. Una caratteristica comune a questi modelli e' l'uso di un coefficiente di restituzione per modellizzare la dissipazione di energia al momento dell'impatto. Abbiamo studiato un semplice modello unidimensionale per un palla che rimbalza contro un pavimento orizzontale, tenendo conto della sua deformabilita', attribuendo ad essa la dissipazione, ed evitando cosi' di usare un coefficiente di restituzione: dimostriamo con stime asintotiche rigorose che questo modello non e' soggetto a collasso anelastico per nessuna condizione iniziale. 

Alessandra Celletti
Universita` di Roma 2

Mercurio e gli attrattori quasi-periodici

Abstract: Consideriamo un modello di interazione spin-orbita costituito da un satellite non rigido ruotante attorno ad un asse interno e orbitante attorno ad un corpo centrale. La forza di attrazione mareale e' introdotta utilizzando la formulazione di MacDonald. Tale problema e' descritto da un sistema dinamico quasi-integrabile debolmente dissipativo. Discutiamo sotto quali condizioni esistono attrattori quasi-periodici e utilizziamo i risultati per giustificare l'attuale condizione di risonanza spin-orbita di Mercurio.

Andrea Giacobbe
Universita` di Padova

Singolarita' di sistemi completamente integrabili a 2 gradi di liberta`

Abstract: Il teorema di Liouville-Arnol'd fornisce una descrizione esauriente delle fibre regolari di un sistema completamente integrabile. D'altra parte la geometria globale di un sistema dinamico completamente integrabile e' contributo unico delle sue fibre singolari. Quali e quante sono le singolarita' "importanti" nei casi semplici (2D)? Che topologia originano?

Carlangelo Liverani
Universita` di Roma "Tor Vergata"

Fourier Law And Random Walks In Dynamical Environment

Abstract: I will discuss a program to investigate the Fourier Law in a lattice of  weakly coupled strongly chaotic dynamical systems. I will then illustrate the small part of such a program that has been carried out as of today.