Not everything that can be counted counts, and not everything that counts can be counted.
by Albert Einstein (1879-1955)

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)

Entropy of geometric structures (2011)

Entropy of geometric structures (2011)

Nonlinear Schrodinger equation: focusing case (2009)

My only paper on PDEs so far:

(With Th. Kappeler, P. Topalov, P. Lohrmann) Birkhoff coordinates for the focusing NLS equation. Comm. Math. Phys. 285 (2009), no. 3, 1087–1107

The system in question is an infinite-dimensional Hamiltonian system, which admits non-elliptic singularities. But near 0 the singularity is elliptic, and it admits a local . . . → Read More: Nonlinear Schrodinger equation: focusing case (2009)

Obstructions to integrability (2010)

(with M Ayoul) Galoisian obstructions to non-Hmailtonian integrability, Comptes Rendus Mathématiques, Volume 348, Issues 23–24, December 2010, Pages 1323-1326.

Arxiv: http://arxiv.org/abs/0901.4586

This paper, despite being a short paper in CRAS, contains a rather strong result: the differential Galois group of the variational equation of any order along a solution of an analytic integrable dynamical . . . → Read More: Obstructions to integrability (2010)