Learning is what most adults will do for a living in the 21st century.
by Lewis Perelman

A general approach to the problem of action-angle variables (new version)

Here is the new version of my invited talk

“A general approach to the problem of action-angle variables” (PDF File: AATalk2014, version 26/10/2014)

to be presented et the conference in honour of Charles Michel Marle’s 70th birthday in November 2014 in Paris.

Here is the website of the conference: http://www.imcce.fr/Equipes/ASD/person/albouy/Marle2014.html

 

 

Reduction and integrability of stochastic dynamical systems

Just submitted a joint paper with my student today, “only” 4 months behind schedule:

Reduction and Integrability of stochastic dynamical systems (PDF file)

Abstract:

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction and integrability. In particular, we show that an SDS which is . . . → Read More: Reduction and integrability of stochastic dynamical systems

A general approach to the problem of action-angle variables

These are the slides (preliminary version, to be revised) for a talk that I’ll give in Paris (24/11/2014) in the Conference in honour of the 80th birthday of C-M Marle. It contains  a conceptual approach to the problem of action-angle variables, with some new results (a paper is in preparation based on an old . . . → Read More: A general approach to the problem of action-angle variables

Book chapter on the geometry of integrable non-Hamiltonian systems

Just finished my chapter for a book on integrable systems. In case anyone is interested, here is the PDF file:

IntegrableBarcelona2014

A note on commuting foliations (2012)

03/June/2012

I just modified the text a bit and submitted it on arxiv

PDF File.

08/May/2012

A preliminary working version is available here: PDF File.

10 pages. Wrote down the results in the regular case. Will probably add a section about singularities to make this note more substantial.

07/May/2012

8 pages (9 pages by . . . → Read More: A note on commuting foliations (2012)

Orbital linearization of smooth completely integrable vector fields (2012)

24/Apr/2012 15h20:

The first version of this paper is now available:

Orbital linearization of smooth completely integrable vector fields

(Click on the above link for the PDF file)

Abstract: The main purpose of this note is to prove the smooth local orbital linearization theorem for smooth  vector fields which admit a complete set of . . . → Read More: Orbital linearization of smooth completely integrable vector fields (2012)

Action-angle variables on Dirac manifolds (2012)

18/Apr/2012:

Arxiv: http://arxiv.org/abs/1204.3865

I finally submitted this paper to the journal Geometry & Topology.

Corrections to the first version:

* One of the word “contravariant” should be changed to “covariant”

* The co-affine structures (that are induced from integrable systems on presymplectic manifolds) have been studied in the literature under the name”affine differential geometry”:

. . . → Read More: Action-angle variables on Dirac manifolds (2012)

Geometry of nonsingular integrable systems on 3-manifolds (2012)

28/Apr/2012

This 3D project is not prioritary, so I’ll postpone it until I’m out of other ideas :-)

Need to work on something more exciting, or I’ll feel bored and tired.

Right now maybe I’ll write up some stuff about stability of commuting singular foliations (extenstion of Reeb-Thurston-…-Crainic-Fernandes-…-Scardua-Seade-… to commuting foliations) ?

26/Apr/2012:

I . . . → Read More: Geometry of nonsingular integrable systems on 3-manifolds (2012)

Geometry of integrable systems on 2D surfaces (2012)

The first version of this paper is now finished. Here is the PDF file.

NT Zung & NV Minh, Geometry of integrable systems on 2D surfaces

Abstract: This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence,  of smooth integrable vector fields on  2-dimensional surfaces,  under . . . → Read More: Geometry of integrable systems on 2D surfaces (2012)

Geometry of R^n actions on n-manifolds (2012)

Updated 17/03/2013: this paper has been accepted for publications in J. Math. Soc. Japan, after a small revision.

Apparently it has also been cited in a recent work by Ishida: arxiv.org/pdf/1302.0633

Updated 18/03/2012: 2nd version, which corrects a series of misprints and imprecisions in the 1st version.

After two months of intensive work, we . . . → Read More: Geometry of R^n actions on n-manifolds (2012)

Nondegenerate singularities of integrable dynamical systems (2012)

Last updated: 14/March/2012

I have now updated my preprint “Nondegenerate singularities of integrable dynamical systems”. I’ve also changed the title a bit, from “non-Hamiltonian” to “dynamical”.

PDF file: Nondegenerate_V2_2012.pdf

Preprint on Arxiv: http://arxiv.org/abs/1108.3551

ABSTRACT: We give a natural notion of nondegeneracy for singular points of integrable non-Hamiltonian systems, and show that such nondegenerate singularities . . . → Read More: Nondegenerate singularities of integrable dynamical systems (2012)

New results on the linearization of Nambu structures (2012)

Preprint, version 11/01/2012

Arxiv: http://arxiv.org/abs/1201.2219

Pdf file here

Abstract: In a paper with Jean-Paul Dufour in 1999 \cite{DufourZung-Nambu1999}, we gave a classification of linear Nambu structures, and obtained linearization results for Nambu structures with a nondegenerate linear part. There was a case left open in \cite{DufourZung-Nambu1999}, namely the case of smooth linearization of Nambu . . . → Read More: New results on the linearization of Nambu structures (2012)

Rigidity of Hamiltonian ations on Poisson manifolds (2012)

(with E Miranda and Ph Monnier) Rigidity of Hamiltonian ations on Poisson manifolds , Advances in Mathematics, Volume 229, Issue 2, 30 January 2012, Pages 1136-1179

Arxiv: http://arxiv.org/abs/1102.0175

ScienceDirect: http://www.sciencedirect.com/science/article/pii/S0001870811003574

It took us several years to write up this paper, because although we “always knew” that the rigidity result was true for Hamiltonian actions . . . → Read More: Rigidity of Hamiltonian ations on Poisson manifolds (2012)