Activities

Seminars:

Seminars in Rome

Seminar bullettin of the Universities of Rome
Seminar of Analysis and Dynamical Systems (University of "Roma Tre")

- 30 January 2013 h. 11.30. Emanuele Haus: Dynamics on resonant clusters for the quintic non linear Schroedinger equation

Abstract: We use normal form techniques in order to construct solutions to the quintic nonlinear Schroedinger equation on the circle with initial conditions supported on arbitrarily many different resonant clusters. These solutions exhibit a beating effect between modes belonging to the same cluster. As a corollary we obtain the existence of solutions that remain quasi-periodic for long times and for a large set of frequencies, which is a genuinely nonlinear effect. This is a sequel of the work by Benoit Grebert and Laurent Thomann.

-21 March 2013 h. 14.30. Massimiliano Berti: Hamiltonian PDEs

Abstract: We present new existence results about existence of quasi-periodic solutions of autonomous Hamiltonian PDEs.
The approach is based on a combination of Nash-Moser and KAM techniques. We shall emphasize the construction of an approximate inverse for the linearized operator.

-4 November 2013 h. 16.00 organized by the University of "RomaTre". Claudio Procesi: Quasi-periodic orbits: the non linear Schroedinger equation

Abstract: The notion of quasi-periodic orbit is among the fundamental ideas in the theory of dynamical systems and it corresponds to physical phenomena which happens for perturbation of completely integrable systems (like the solar system). This theory has been extended also non non linear PDEs which can be thought of as perturbation of linear equations describing some kind of waves. The case of non linear Schroedinger equation, when the space dimension is >1, presents many problems also from the algebraic, geometric and combinatoric point of view (the rectangles graph) which will be discussed in this talk.

-12 December 2013 h. 11.00. Massimiliano Berti: KAM for quasi-linear KdV

Abstract: We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions for quasi-linear autonomous Hamiltonian and reversible perturbations of KdV and mKdV. The results are based on a Nash-Moser and KAM techniques for the reducibility of the linearized operators along the iteration.

- 25 February 2014 h. 14.30. Marcel Guardia (University Paris 7): Nearly integrable systems with orbits accumulating to KAM tori

Abstract: The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit. This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.

- 5 June 2014 h. 14.30. Zaher Hani (Courant Institute): Energy cascades and wave turbulence for the cubic Schroedinger equation

Abstract: Out-of-equilibrium dynamics are a characteristic feature of the long-time behavior of nonlinear dispersive equations (PDEs) on confined domains. Energy cascades and wave turbulence are main aspects of such out-of-equilibrium dynamics. In this talk, we will start by explaining what all those concepts mean, why they arise, and the mathematical problems involved in studying them. Afterwards, we will describe two main approaches that were adopted to capture out-of-equilibrium phenomena. The first approach is based on a relation between energy cascades and the growth of Sobolev norms of solutions. The second approach is based on deriving "effective equations" for the dynamics by taking various limits of the original system (this is the guiding philosophy of wave turbulence theory). We describe some recent progress on both approaches in joint works with E. Faou, P. Germain, B. Pausader, L. Thomann, N. Tzvetkov, and N. Visciglia.

-September 1-11 2014: Roman Summer School and Workshop KAM Theory and Dispersive PDEs

-30 December 2014 h. 12.00, Aula G
Livia Corsi (McMaster University) On the persistence of resonant tori
Abstract: t is well known that resonant tori persist for quasi-integrable Hamiltonian systems under appropriate non-degeneracy conditions on the perturbation. However a long-standing conjecture states that such tori persist for any perturbation provided that it is sufficiently regular. This conjecture was proved in the case of co-dimension one. We will discuss recent partial results on the subject and discuss a possible strategy.

-14 May 2015 h. 15:30, aula G
Luca Biasco (Unversità di Roma Tre)
On the measure of invariant tori in hamiltonian systems
We discuss the measure of (Lagrangian) invariant tori in nearly-integrable hamiltonian systems. It is well known in KAM theory that the measure of the complementary of invariant tori (where chaotic motions possibly occur) is smaller than the sqare root of ε, ε being the perturbative parameter. A conjecture by Arnold, Kozlov and Neishtadt claims that such measure is actually of order ε. We prove that such conjecture holds true for 'general' mechanical systems (namely systems of the form: kinetic energy plus potential energy). Joint work with L. Chierchia.

- 26 May 2015 h. 15 A. Maspero (University of Milan) Freezing of energy of a soliton in an external potential.
Abstract: we study the dynamics of a soliton in the generalized NLS with a small external potential \eV of Schwartz class. We prove that there exists an effective mechanical
system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order \e^r. In the rotational
invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order \e^r.

- 2- December 2015 (In Tor Vergata) Vadim Kaloshin (University of Maryland) Stochastic Arnold diffusion in deterministic systems and celestial mechanics
In 1964, V. Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, physicist B. Chirikov coined the term for this phenomenon 'Arnold diffusion', where diffusion refers to stochastic nature of instability. One of the most famous examples of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by astronomer J. Wisdom. During the talk we describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behaviour. Namely, we show that distributions given by deterministic evolution of certain random initial conditions weakly converge to a diffusion process. This result is conceptually different from known mathematical results, where existence of 'diffusing orbits' is shown. This work is based on joint papers with O. Castejon, M. Guardia, J. Zhang, and K. Zhang.

-19 February 2016 Livia Corsi (Mc Master) An Abstract KAM theorem
Abstract: I'll discuss an abstract KAM result on the existence of invariant tori for possibly infinite dimensional dynamical systems.
Differently from the classical Moser's approach, I'll show that in principle there is no need to impose the second Mel'nikov conditions but only to invert (in some appropriate norm) the linearized operator in the normal directions: in particular this means that the serious technical difficulties in small divisors problems are those appearing in forced cases.The latter statement is commonly believed to be true: the main purpose is indeed to prove it under the weakest possible assumptions.
The result is obtained in collaboration with R. Feola and M. Procesi.

-26 April 2016 Michela Procesi (Universita di Roma Tre) Quasi-periodic solutions with beating effects for the NLS on tori
I will discuss the existence and linear stability of classes of solutions for the NLS on a torus. I will concentrate on quasi-periodic solutions which arise from the resonances of the NLS normal form and exhibit a periodic transfer of the Sobolev norm between Fourier modes. This is a joint work in progress with E. Haus.

- 5 May 2016 Emanuele Haus (Napoli Federico II) Growth of Sobolev norms and beating effects for the NLS on tori
We prove existence of solutions to some NLS on tori exhibiting different qualitative behavior. On the one hand we construct solutions which undergo an arbitrarily large growth of the Sobolev norms for analytic NLS equations on T^2, on the other hand we construct quasi-periodic solutions with recurrent transfer of energy between the Fourier modes for the quintic NLS on the circle.

- 18 October 2016 Filippo Giuliani (SISSA) Quasi-periodic solutions for quasilinear generalized KdV equations We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash-Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.

-14 September 2016 Livia Corsi (Georgia Tech) Locally integrable non-Liouville analytic geodesic flows on T^2 A metric on T^2 is said to be ``Liouville" if in some coordinate system it has the form ds^2 = (g_1(q_1) + g_2(q_2)) (dq_1^2 + dq_2^2); a ``folklore conjecture" states that if a metric is locally integrable then it is Liouville. I will present a counterexample to this conjecture.

-29 September 2016 Massimiliano Berti (SISSA) Almost global existence of periodic capillarity-gravity waves We present long time existence results for the solutions of gravity-capillary water waves equations with small initial periodic data. The proof is based on a paradifferential reduction of the equations and a Birkhoff normal form analysis. Joint work with J.M Delort.

- 20 December 2016 Livia Corsi (Georgia Tech) Periodic driving at high frequencies of an Impurity in the Isotropic XY chain I'll consider the isotropic XY chain with a transverse magnetic field acting on a single site and analyse the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. I'll show that for high frequencies the state approaches a periodic orbit synchronised with the forcing and provide the explicit rate of convergence to the asymptotics. This is a joint work with G. Genovese (organized by the math. Phys group)

- 5 April 2017 Nikolaos Karaliolios (Imperial College) KAM normal form for quasi-periodic cocycles in TxSU(2) and spectral dichotomy.We will recall the application of the KAM machinery as it was applied by H. Eliasson and R. Krikorian (among others) to the problem of the (almost) reducibility of such systems. We will subsequently present an (almost) complete classification of the KAM regime, based on the KAM normal form, and, if time permits, sketch the proof of spectral dichotomy: a cocycle in the KAM regime has either pure point spectrum or a maximal component of singular continuous spectrum in the fibers.

- 19-07-2017 Alessandra Fuse'
Quasiconvexity of the Hamiltonian for non Harmonic or non Keplerian central potentials

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 Nekhoroshevtype stability result for the perturbed system.
This is a joint work with D. Bambusi and M. Sansottera.

- 28-09-2017 J.P Marco Integrability and complexity of geodesic flows on ellipsoids : from the finite dimensional case to the infinite dimensional one

Starting from the geodesic flow on finite dimensional Euclidean ellipsoids, for which we will recall the classical constructions of complete sets of first integrals and describe their singularities and dynamical complexity, we will examine the possibility of extensions to the infinite dimensional case. This could possibly make a bridge between integrability in the finite dimensional Hamiltonian case and that of Hamiltonian PDE's.

-22-11-2017 - 16:00
Shidi Zhou (Roma tre) An infinite dimensional KAM theorem with application to 2-d completely resonant beam equation

In this talk we shall consider the 2-dimensional completely resonant beam equation with cubic nonlinearity on Tˆ2. We prove the existence of the quasi-periodic solutions, which lie in a special subspace of Lˆ2 (Tˆ2). We view the equation as an infinite dimensional Hamiltonian system, and write the Hamiltonian of the equation as an angle-dependent block-diagonal normal form plus a small perturbation with some regularity. By establishing an abstract KAM theorem, we prove the existence of a class of invariant tori of this system, which implies the existence of a class of small-amplitude quasi-periodic solutions of the equation. In the KAM iteration, the measure estimate is reached by making use of the regularity of the nonlinearity.

- 6-12-2017 Stefano Pasquali
Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit

We study the the nonlinear Klein-Gordon (NLKG) equation on a manifold M in the nonrelativistic limit (as the speed of light coinfty). We consider a higher-order normalized approximation of NLKG (corresponding to the NLS at order r=1), and prove that when M is a smooth compact manifold or Rd, the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When M=Rd, dgeq2, we prove that for rgeq2 small radiation solutions of the order r normalized equation approximate solutions of the nonlinear NLKG up to times of order cO(c2(r−1)).

-14 dec 2017 Filippo Giuliani Quasi-periodic solutions for Hamiltonian perturbations of the Degasperis-Procesi equation

I present a new result on the existence and the stability of small amplitude quasi-periodic solutions for Hamiltonian perturbations of the Degasperis-Procesi equation under periodic boundary conditions.
These solutions are constructed by a hard Implicit Function Nash-Moser Theorem, based on a Newton-type algorithm. The main issue in implementing this iterative scheme is the inversion of the linearized operator in a neighborhood of the origin. The analysis of the linearized operator requires to solve quasi-periodic transport equations and to exploit some pseudo differential calculus techniques.
To initialize the Nash-Moser scheme we need to perform a bifurcation analysis by means of a Birkhoff normal form procedure.
The Degasperis-Procesi equation presents some non trivial Birkhoff resonances and we show how to overcome this problem by exploiting the integrability of the unperturbed system.
This is a joint work with Roberto Feola and Michela Procesi.

-03-07-2018 Roberto Feola (SISSA) Birkhoff Normal Form and long time existence for periodic gravity water waves

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof the conjecture of Zakharov and Dyachenko. More precisely, we provide a reduction of the equations to Birkhoff normal form and prove the integrability of the water waves Hamiltonian up to order four. As a consequence, we also obtain a long-time stability result for periodic solutions: perturbations of a flat interface that are of size e in a standard Sobolev space lead to solutions that remain regular and small up to times of the order e−3.
To our knowledge, this is the first such long-time existence result for quasilinear systems in the periodic setting.Indeed, despite the absence of external parameters, our result goes past the natural e−2 scale which one expects for non-quadratically resonant Hamiltonian systems, such as gravity water waves.The main difficulties in the proof are the quasilinear nature of the equations, the presence of small divisors arising from near (trivial) resonances,and of many non-trivial resonant four-way interactions, the so-called Benjamin-Feir resonances.

Seminars in Naples

- 20-23 november 2012 Riccardo Montalto (SISSA, Trieste) cycle of seminars on: KAM theory for quasi-linear Hamiltonian equations of KdV type

- 29 November 2012 h. 11.30. Livia Corsi: An upper bound for the growth of higher Sobolev norms for periodic NLS

Abstract: We shall expose a recent result by Colliander-Kwon-Oh on the growth of higher Sobolev norms for periodic NLS; they prove a bound which is polynomial in time: in particular they improve the exponents with respect to those already known in the literature.

- 9 May 2013 h. 14.00. Emanuele Haus: Dynamics on resonant clusters for the quintic non linear Schroedinger equation

Abstract: We use normal form techniques in order to construct solutions to the quintic nonlinear Schroedinger equation on the circle with initial conditions supported on arbitrarily many different resonant clusters. These solutions exhibit a beating effect between modes belonging to the same cluster. As a corollary we obtain the existence of solutions that remain quasi-periodic for long times and for a large set of frequencies, which is a genuinely nonlinear effect. This is a sequel of the work by Benoit Grebert and Laurent Thomann.

- June 2013 Philippe Bolle (Avignon):

- 26-29 November 2013 Prof. Massimilano Berti (SISSA, Trieste) cycle of seminars on: KAM theory for quasi-linear Hamiltonian equations of KdV type

- 14 May 2014 h. 15. Thierry Paul (Ecole Polytechnique): Quantum singular complete integrability

Abstract: We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schrï¿½dinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant near zero. The unperturbed frequencies satisfy a small divisors condition (including the Diophantine case) and we explicitly estimate how this condition can bereleased when the family tends to the unperturbed one.

PHD level
PhD Course by M. and C. Procesi: KAM theory and Dynamical systems.
starting 26 March 2013 every Tuesday h. 11-13 aula B dept. of math.

Program:
Symplectic formalism and analytical mechanics.
Darboux's theorem.
Classification of quadratic Hamiltonians.
Completely integrable systems.
Liouville-Arnold theorem.
Classic examples.
Near-integrable systems.
Perturbation theory and Birkhoff normal form.
KAM theorem.

n-body problem.
Lower dimensional invariant tori.
Applications to non-linear PDEs.

References:
- Moser-Zehnder, Notes on dynamical systems
- Gallavotti, Meccanica Analitica
- Poeschel, On elliptic lower dimensional tori in Hamiltonian systems. Math. Z. 202 (1989) 559-608

PhD- Master Course by M. Procesi: ODEs
Program:
Local existence and uniqueness theorems.
Systems of linear equations
Periodic solutions Floquet theory
Perturbation theory and Birkhoff normal form.
Reducibilty

Master-PhD Course by M. Procesi Small divisor problems March- July 2017 .
Kam theory for finite dimension. Reducibility. Going to infinite dimensions.

Mini-Course by E. Haus: Growth of Sobolev norms for the NLS.
the course consists of six lectures Wednesdays room B 11-13 starting on April 18, 2013.

Abstract: We consider the defocusing cubic NLS on a bidimensional torus. We prove the existence of solutions whose H^s Sobolev norm grows arbitrarily in time.
We refer to the papers by Guardia-Kaloshin and J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao.

Mini-Course by L. Corsi: An abstract Nash-Moser theorem with applications to existence of quasi-periodic solutions for PDEs on compact homogeneous manifolds. the course consists of three lectures room G 11-13 starting July 8, 2013.

Abstract: We will present an abstract Nash-Moser scheme which allows us to find zeros of functionals on sequence spaces. A fundamental tool is a multi-scale method which allows us to give good bounds on the inverse of the functional linearized at an approximate solution.

Mini-Course by L. Corsi: The multiscale Proposition. the course consists of two lectures room C 11-13 starting November 8, 2013.

Abstract: I will present in all the details the multiscale Proposition, which allows to deduce "good bounds" (in high Sobolev norm) of the inverse of a linear operator from bounds on its L^2 norm and the so-called separation of the bad sites.

Organized by the University of "Roma Tre": Mini-Course by V. Kaloshin: Arnol'd Diffusion via Invariant Cylinders and Mather Variational Methods. 16-17 April 2014.

Abstract: The famous ergodic hypothesis claims that a typical Hamiltonian dynamics on a typical energy surface is ergodic. However, KAM theory disproves this. It establishes a persistent set of positive measure of invariant KAM tori. The (weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension n > 2 and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. In the last two decades a variety of powerful techniques to attack this problem were developed. In particular, Mather discovered a large class of invariant sets and a delicate variational technique to shadow them. In a series of preprints: one joint with P. Bernard, K. Zhang and one with K. Zhang and one with M. Guardia we prove strong form of Arnold's conjecture in dimension n=3.

in Naples: Mini-Course by A. Maiocchi: Stochastic partial differential equations and averaging theorem

Lunedi 20 ore 16-18 (aula E)
Martedi 21 ore 15-17 (aula da definire)
Mercoledi 22 ore 11-13 (aula da definire)
Giovedi 23 ore 10-12 (aula da definire)
Venerdi 24 ore 15-17 (aula da definire).

The first part of the course (6 hours) provides an introduction to partial differential equations with stochastic forcing: we present some facts abut the Brownian motion, the convergence of families of measures, the notions of strong and weak solutions for stochastic PDEs and the relation between weak solution and the martingale method.
In the second part (4 hours) we show how to get an averaging theorem for resonant stochastic PDEs, explaining the techniques and the results through some examples.

Brainstormings and study groups:

- 15 January 2013: First meeting of the study group on: Almost-periodic solutions for the NLS.

Descriveremo la letteratura nota (tutta riguardante PDE con parametri esterni), e discuteremo quindi alcune idee e possibili strategie per dimostrare l'esistenza di soluzioni almost-periodiche nel caso di PDE senza parametri esterni come per esempio la NLS.

-15/18 January 2013 Study group (with L. Biasco and M. Berti) Quasi-periodic solutions for KdV. The purpose is to analyze the recent ideas on quasi-linear PDEs and in particular possible applications to the DNLW.

- 2-5 July 2013: Informal group meeting and brainstorming in Sabaudia (LT).

- Sept 2013 (in Naples): Multiscale analysis and quasi-periodic solutions for PDEs. The purpose is to introduce the multiscale methods (as proposed by Berti/Bolle in the Sobole context)

with an eye to generalizations to compact manifolds.

- Oct. 2013 (in Naples): Degenerate Birkoff Normal Forms in Celestial Mechanics. We give an introduction to the problem.

- 24-28 February 2014: Intensive study group (with M. Guardia) on: Growth of Sobolev norms for the NLS. Part I

The project is to generalize the results by the I-Team and Kaloshin Guardia to higher order NLS equations.

- March/April 2014 (with B. Wilson): Normal forms and stability of plane wave solutions for the NLS.

We shall study the recent paper by Erwan Faou, Ludwig Gauckler, Christian Lubich on the stability of plane waves for the NLS with the purpose of generalizing to higer degree.

- April 2014 (with B. Wilson): Introduction to KAM theory. A brief introduction to the methods of KAM theory thought mainly for team members.

- 10 April 2014: Analytic solutions for the Degasperis-Procesi equation.

The recent results on quasi-periodic solutions for quasi-linear PDEs on the circle are up to now limited to solutions with Sobolev regularity. We propose to study analytic solutions for a quasi linear first order PDE.

- 3-6 June 2014: Intensive study group (with Z. Hani) on: Weak turbulence and unstable KAM tori for the NLS.

We would like to generalize the results of the I-team to prove instability not only close to zero, but also close to other global solutions (say quasi-periodic ones).

- 12-20 July 2014: Informal group meeting and brainstorming in Sabaudia (LT).

- 17-21 November 2014: Intensive study group (with M. Guardia) on: Growth of Sobolev norms for the NLS. Part II

The project is to generalize the results by the I-Team and Kaloshin Guardia to higher order NLS equations.

- December 2014: Study group on: Reducibility, KAM theory and quasi-linear PDEs. Part I

We shall discuss the method proposed by Baldi-Berti and Montalto for existence and stability of quasi-periodic solutions for PDEs. The idea is to try to formalize the method in a more abstract way.

- 25-27 February 2015: Intensive study group on: Reducibility, KAM theory and quasi-linear PDEs. Part II

We shall discuss the method proposed by Baldi-Berti and Montalto for existence and stability of quasi-periodic solutions for PDEs. The idea is to try to formalize the method in a more abstract way.

We wish to construct secondary tori where one can see an exchange of energy between modes.

-9-13 November 2015: Study group on: Growth of Sobolev norms for the NLS near a one-dimensional solution, part I

We would like to generalize the results of the I-team to prove instability not only close to zero, but also close to other global solutions (say quasi-periodic ones). The first step is to work on the finite-gap solutions.

-13-23 December 2015: Study group on: Reducibility, KAM theory and quasi-linear PDEs. Part III (with L. Corsi)

We shall discuss the method proposed by Baldi-Berti and Montalto for existence and stability of quasi-periodic solutions for PDEs. The idea is to try to formalize the method in a more abstract way.

-20-22 January 2016: Study group on: Secondary tori for the quintic NLS. Part II

We wish to construct secondary tori where one can see an exchange of energy between modes.

-8-12 February 2016: Study group on: Reducibiliy of the 2 dimensional NLS at a one dimensional solution (with A. Maspero)

We study resonant BNF for the two dimensional NLS close to a one dimensional solution.

-15-23 February 2016 Study group on: Reducibility, KAM theory and quasi-linear PDEs. Part IV (with L. Corsi and R. Feola)

We shall discuss the method proposed by Baldi-Berti and Montalto for existence and stability of quasi-periodic solutions for PDEs. The idea is to try to formalize the method in a more abstract way.
-2-6 May 2016: Study group on: Secondary tori for the quintic NLS. Part III

We wish to construct secondary tori where one can see an exchange of energy between modes.

-20-24 June 2016: Intensive study group: Growth of Sobolev norms for the NLS near a one-dimensional solution, part III (with M.Guardia, Z. Hani and A. Maspero)

We generalize the results of the I-team to prove instability not only close to zero, but also close to other global solutions (say quasi-periodic ones). The first step is to work on the finite-gap solutions.

-30 Aug.- 02 Sept. 2016: Study group Pseudo differential calculus and reducibility on the line (with R. Montalto)

The method proposed by Baldi-Berti and Montalto for existence and stability of quasi-periodic solutions for PDEs is taylored for approaching PDEs on the circle. We wish to understand possible generalizations to the line.

- December 2016: Reducibility, KAM theory and quasi-linear PDEs. Part V (with L. Corsi and R. Feola)

We shall discuss the method proposed by Baldi-Berti and Montalto for existence and stability of quasi-periodic solutions for PDEs.

We formalize the method in a more abstract way and hence prove existence of analytic solutions. We also discuss the connection between formal and converging BNF in the context of quasi-linear PDEs.

-7-14 May 2017: Growth of Sobolev norms for the NLS near a one-dimensional solution, part IV (with Z. Hani and E. Haus)

-10-14 July 2017 Reducibiliy of the 2 dimensional NLS at a one dimensional solution, part II (with A. Maspero)

We study resonant BNF for the two dimensional NLS close to a one dimensional solution.

-27 sett - 2 ott 2017 Liouville integrability in infinite dimensions (with J.P. Marco).

We discuss the concept of integrability and of infinite dimensional tori from a more geometric viewpoint.

-24-27 ott 2018 the DP equation (with R. Feola and F. Giuliani). Study group on the construction of quasi-periodic solutions for the DP equation. We are finally finishing the paper!

-28-30 ott 2018 Quasi-toplitz matrices and pseudo-differential operators in 2d tori (with A. Maspero). Up to now there are very few results on reducibility on higher dimensional tori. We discuss this problem and revisit quasi-Toplitz matrices in this context.

-31/1 to 26/2 2015 McMaster Unversity (collaboration with former team member L. Corsi)

discussions over the problem of persistence of Diophantine tori in the context of diffeomorphisms, in any dimension. Part I

This brought to the generalization in any dimension of a seminal result of Russmann (the theorem of the translated curve, that, so far, was stated and known only in dimension 2) together with a version,

new in this discrete time frame of the counter-term theorem of Moser (1967).

- 7 July - 5 August 2016 IMCCE, Observatoire de Paris (collaboration with ASD group)

discussions over the problem of persistence of Diophantine tori in the context of diffeomorphisms, in any dimension. part II

collaboration over the problem of persistence of invariant tori in non necessarily Hamiltonian systems under a very general dissipative effects

(this shall bring to the generalization of known results where both a Hamiltonian structure and particular dissipative terms are taken into account) Part 1

We discussed the relation between formal BNF and the convergent one. We started an ongoing project on billiards.

- Zurich ITS-ETH 25-27 April 2017 (seminar and collaboration with R. Montalto and V. Kaloshin)

We discussed recent results on the nonlinear stability of finite-gap solutions in the 2D cubic NLS.

-Université Nice Sophia Antipolis - April 2018 Séminaire de Systèmes Dynamiques.

-InstitutMathématique de Jussieu PRG - Université Paris 7 et 6, 19-23 March 2018 Séminaire de Systèmes Dynamiques.

J. Massetti held a seminar on the topic of almost-periodic solutions for NLS + collaboration with B. Fayad over the persistence of normally hiperbolic tori part 2

- Università degli Studi di Roma Torvergata, december 2017 Seminario di equazioni differenziali

- Imperial College, London, March 2017 invited talk at Aspects of Dynamical Systems conference

J. Massetti held a seminar on the topic of generalization of Russmann translated curve theorem.

Attended conferences (both as participants or invited speakers):

- "Dynamique et EDP", Marseille (France), November 2012.

- Winter school "Dynamics and PDEs", St. Etienne de Tinee (France), February 2013.

- "Conference on Dynamics of Differential Equations", Atlanta (Georgia, USA), March 2013.

- "Conference HANDDY 2013 - Hamiltonian and Dispersive Equations", Marseille (France), June 2013.

- "Planetary motion, satellite dynamics and Spaceship orbits", Montreal (Quebec, Canada), July 2013.

- "16th General Meeting of the European Women in Mathematics", Bonn (Germany), September 2013.

- "CELMEC VI - The Sixth International Meeting on Celestial Mechanics", Viterbo (Italy), September 2013.

- "Finite and infinite-dimensional Hamiltonian systems", Rome (Italy), October 2013.

- "Conference on Hamiltonian PDEs: Analysis, Computations and Applications", Toronto (Ontario, Canada), January 2014.

- Winter school "Dynamics and PDEs", St. Etienne de Tinee (France), February 2014.

- SPT2014 "Symmetry and Perturbation Theory", Cala Gonone (Italy), May 2014.

- JISD2014 "Jornades d'Interaccio entre Sistemes Dinamicos i Equacions en Derivades Parcials", Barcelona (Spain), June 2014.

- Geometric and Analytic Aspects of Integrable and nearly-Integrable Hamiltonian Systems, University of Milano-Bicocca (Italy), 18-20 June 2014.

- 10th AIMS conference on Dynamical Systems, differential equations and applications. Madrid, spain. July, 7--11.

- International Congess of Mathematicians, Seoul (Korea) Aug. 2014 (team member G. Pinzari is among the invited speakers!).

- Symplectic Techniques in Topology and Dynamics. Colonia, Germany. September, 22-- 26

-Symposium on Mathematical Physics. University of Z\"uric. nov. 10--11.

- Workshop "Dynamics and PDEs", Cargese (Corsica, France) 11-14 November 2014.

- KAM and Dispersive Methods in PDEs, Milano (Italy) 1-5 December 2014.

- Two-day meeting in honor of Antonio Ambrosetti, Venezia (Italy) 14-15 December 2014.

- The Conference on Hamiltonian Dynamical Systems, Fudan University in Shanghai (China), 4-10 January 2015.

- "Sixth Itinerant Meeting in PDEs" Trieste, 14-16 January 2015.

- Winter School "Dynamics and PDEs", Saint Etienne de Tinee (France), 2-6 February 2015.

- Summer school "Normal forms and large time behavior for nonlinear PDE", Nantes (France), 22 June-3 July 2015.

-Minisymposium on "Celestial Mechanics". Equadiff conference. Lyon, July 6-10, 2015.

-St Petersburg Hamiltonian systems and their applications June 3-8, 2015 A. Maspero was a young invited speaker.

-Conference on Dynamical Systems
at ICTP, Trieste, July 27 - August 07, 2015.

-European Women in Mathematics. Cortona. August, 31- sept., 4 2015

-Convegno Umi. Siena, 1Stt. 2015

-Hamiltonian systems and celestial mechanics, Oaxaca, Mexico September 6-11, 2015
BIRS, workshops.

- Summer school "Normal forms and large time behavior for nonlinear PDE", Nantes (France), 22 June-3 July 2015.

-SIAM Converence Analysis and PDEs, Scottsdale (USA), 7-10 Dec 2015

- The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Orlando, Florida, USA, July 1 - July 5, 2016

- Hamiltonian Dynamics PDEs and Waves on the Amalfi coast, Maiori, Italy 5-11 Sept. 2016

- Double resonances in Arnold's diffusion, Mini course by V. Kaloshin and J-P. Marco at IHP, Paris, 12-16 december, 2016

- Winter School in Conservative Dynamics, Engelberg, Swizerland, 5-12 February, 2017

-ETH-CSF Conference on Hamiltonian Dynamics, November 2017 Conference in memory of John N. Mather. Ascona, Swizerland

-Recent advances in Hamiltonian dynamics and symplectic topology, February 2018 Padova.

- Bekam International Meeting 2018, from May 07th to May 11th 2018 in Cargèse (France).

- Symmetry and Perturbations theory 2018, from June 03 to June 10 in Pula (Sardinia, Italy).

-16TH SCHOOL ON INTERACTIONS BETWEEN DYNAMICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS, from June 25 to June 29 2018, CRM Barcelona (Spain). M. Procesi held one of the courses.

- Workshop INdAM 2018, Linear and Nonlinear Wave Phenomena: Stability, propagation of regularity and Turbulence, from September 9 to September 14 in Cortona (Tuscany, Italy).