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.

- 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.

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

Courses:


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


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.

 Master-PhD Course AM550 Small divisor problems March- July 2017 35 hours


Brainstormings and study groups:

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

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

- Sept 2013 (in Naples): Multiscale analysis and quasi-periodic solutions for PDEs

- Oct. 2013 (in Naples):  Degenerate Birkoff Normal Forms in Celestial Mechanics

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

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

- April 2014 (with B. Wilson): Introduction to KAM theory.

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

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

- 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

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

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

- 27-30 April 2015: Study group on: Secondary tori for the quintic NLS. Part I

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

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

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

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

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

-2-6 May 2016: Study group on: Secondary tori for the quintic NLS. Part III

-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)

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

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

-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)

Collaborations and invited talks:

- 20-30 June 2016 IMCCE, Observatoire de Paris  (collaboration with ASD group)

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

- 23-27 Jan 2017 Paris 7 University Denis Diderot (collaboration with B. Fayad) 

- 10/17 April 2017 Georgia Tech  (seminar and collaboration with L. Corsi)

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

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.

-Conference on Dynamical Systems
at ICTP, Trieste, which will take place on July 27 - August 07, 2015.

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

-Convegno Umi. Siena, 1-6 Sett. 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

- Dynamics of Evolution Equations, CIRM -Luminy (France) March 21-25, 2016

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

- NonLinear Waves 2016 Summer school, 18-29 Luglio 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 

- Aspects of Dynamical Systems, Imperial College, London, 16-18 March, 2017 

 - Dynamics and PDEs St Etienne de Tinee 30 Jan. 3 Feb. 2017

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