Publications and Preprints
by Nguyen Tien Zung

Last updated: 20/Sep/2002

Selected papers:


Levi decomposition of analytic Poisson structures and Lie algebroidsmath.DG/0203023 (2002), submitted to Topology.
[PS] [PDF] [DVI] [ArXiv]
(with Philippe Monnier) Levi decomposition for smooth Poisson structures. math..DG/0209004 (2002), submitted to J. Diff. Geom.
[PS] [PDF] [DVI] [ArXiv]
(with Jean-Paul Dufour) Nondegeneracy of the Lie algebra aff(n). math.SG/0208036 (2002), submitted to Comptes Rendus Acad. Sci. Paris.
[PS] [PDF] [DVI] [ArXiv]
In these series of papers, we show (with P. Monnier) the existence of a local Levi decomposition for analytic and smooth Poisson structures and Lie algebroids, which is a kind of local normal form (or semi-linearization) for these structures based on  semisimple (compact) subalgebras of the Lie algebras corresponding to their linear parts. In particular, we recover J. Conn's linearization theorems for Poisson structures with a semisimple (compact) linear part, and prove the following conjecture of A. Weinstein: any analytic (resp., smooth) Lie algebroid, whose anchor vanishes at a point and whose corresponding Lie algebra at that point is semisimple (resp., compact semisimple), is locally analytically (resp., smoothly) linearizable. Using this Levi decomposition, we show (with Jean-Paul Dufour) that $\mathfrak{aff}(n)$, the Lie algebra of affine transformations of ${\mathbb R}^n,$ is formally and analytically nondegenerate in the sense of A. Weinstein. This means that every analytic (resp., formal) Poisson structure, which vanishes at a point and whose linear part corresponds to $\mathfrak{aff}(n)$, is locally analytically (resp., formally) linearizable.

A la recherche des tores perdus. Document de synthèse pour l'HDR (2001)
[PS]
This mémoire, written in French, is a survey of my results on topological aspects of integrable systems, including some new unpublished results and ideas. Topics discussed include: local normal forms, resonances, torus actions, nondegenerate and degenerate singularities, monodromy, characteristic classes, localization formulas, topological obstructions to integrability, non-hamiltonian integrability, etc.


Convergence versus integrability in Birkhoff normal form. math.DS/0104279 (2001).
[PS] [PDF] [DVI] [ArXiv]
Convergence versus integrability in Poincaré-Dulac normal form. math.DS/0105193 (2001), Math. Res. Lett. 9 (2002), 217-228.
[PS] [PDF] [DVI] [ArXiv]
In these papers, we show that Poincaré-Dulac and Birkhoff normal forms are governed by torus actions. Using this toric characterization,and a new geometric approximation method (in contrast to the fast convergence method), we show that any analytic integrable Hamiltonian (resp., non-Hamiltonian) vector field admits a local analytic Birkhoff (resp., Poincaré-Dulac) local normalization. Our results solve a long-standing problem, and improve in a significant way previous results obtained by Rüssmann, Vey, Ito, Kappeler, Kodama, Nemethi,  Bruno and Walcher.


A note on focus-focus singularities. math.DS/0110147, Diff. Geom. and Appl., 7 (1997), 123-130
[PS] [PDF] [DVI] [ArXiv]
Another note on focus-focus singularities. math.DS/0110148,  Lett. Math. Phys 60 (2002), No. 1, 87-99.
[PS] [PDF] [DVI] [ArXiv]
These papers classify (nondegenerate and degenerate) focus-focus singularities of integrable Hamiltonian systems topologically, calculate the monodromy around these singularities, show the existence of a torus action, and show how the monodromy formula can be deduced from Duistermaat-Heckman localization formulas with respect to this torus action.


Decomposition of nondegenerate singularities of integrable Hamiltonian systems. Letters in Math. Physics, 33 (1995), 187-193.
[PDF]
Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities. Compositio Math., 101 (1996), 179-215
[PS] [PDF] [DVI] [ArXiv]
A note on degenerate corank-1 singularities of integrable Hamiltonian systems. Comment. Math. Helv., 75 (2000), 271-283.
[PDF]
Symplectic topology of integrable Hamiltonian systems, II: Topological classification, math.DG/0010181 (2000), Compositio Math. (to appear).
[PS] [PDF] [DVI] [ArXiv]
These papers (together with the notes on focus-focus singularities), contain my main results on topological aspects of integrable Hamiltonian systems. These include: existence of torus actions near singularities, topological decomposition and classification of nondegenerate singularities, classification of degenerate corank 1 singularities, stratified affine structure of the base space, affine monodromy sheaf, Chern and  Lagrangian classes for systems with generic singularities, integrable surgery, topological and symplectic classification theorems.


(with Jean-Paul Dufour) Linearization of Nambu structures. Compositio Math. 117 (1999), no. 1, 77--98.
[PS] [PDF] [DVI] [ArXiv]
This paper contains a classification of linear Nambu-Poisson structures, and linearization theorems (under some nondegeneracy conditions). The usefulness of Nambu structures in physics is rather dubious. But Nambu structures are dual to integrable differential forms, and as such they are very useful for the theory of singular foliations.

Other papers:


Nonresonant smooth integrable Hamiltonian systems which don't admit a local smooth Birkhoff normal form. math.DS/0207232 (2002).
[PS] [PDF] [DVI] [ArXiv]
We construct examples of nonresonant smooth integrable Hamiltonian systems which don't admit a smooth Birkhoff normal form. This is in contrast to positive results about existence of Birkhoff normal forms, due to Hakan Eliasson in the smooth case under a strong nondegeneracy condition, and to Hidekazu Ito et al. in the analytic case.


A geometric proof of Conn's linearization theorem for analytic Poisson structures. math.DS/0207263 (2002).
[PS] [PDF] [DVI] [ArXiv]
We sketch a geometric proof of Conn's linearizationtheorem for analytic Poisson structures witha semisimple linear part. The proof is based on Reeb stability for singular foliation, Moser trick, and the geometric approximation method developed in our papers on normal forms of vector fields.


Reduction and integrability. math.DS/0201087 (2002).
[PS] [PDF] [DVI] [ArXiv]
This note discusses the relationship between the integrability of a dynamical system invariant under a Lie group action and its reduced integrability, i.e. integrability of the corresponding reduced system.


(with Tit Bau) Singularities of integrable and near-integrable Hamiltonian systems. Journal of Nonlinear Science, 7 (1997), 1-7.  [PDF]
Kolmogorov condition for integrable systems with focus-focus singularities. Physics Letters A, 215 (1996), 40-44. 
[PDF]
These papers  contain a simple effective criterium for checking Kolmogorov's condition of integrable Hamiltonian systems (used in KAM theory), based on the existence of a nondegenerate singularity, generalizing a result of Knörrer. The paper with Tit Bau also contain other results and ideas concerning perturbations of integrable systems.


Singularities of integrable geodesic flows on multi-dimensional torus and sphere. Journal of Geometry and Physics, 18 (1996), 147-162.
[PDF]
This paper contains an attempt to classify all singularities of integrable geodesic flows on multi-dimensionsal ellipsoids by decomposing them into simpler singularities. Unfortunately (or fortunately ?) the paper contains some serious errors in the computation of more complicated singularities (the paper claim that all singularities are nondegenerate, which is perhaps not true)


A topological classification of integrable Hamiltonian systems. Séminaire Gaston Darboux, Université Montpellier II, 1994-1995, 43-54 (1995).
[PDF]
This exposé contains an account on my first attempts (not very successful ones) to classify integrable systems topologically using characteristic classes.


Compatible contact structures for integrable Hamiltonian systems.
[PS]
Unpublished note written around 1993.

Papers from undergraduate period:

(with Lada Polyakova) A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta. J. Nonlinear Sci., 3 (1993), No. 1, 85-108.
GIF Image
(with Lada Polyakova and Elena Selivanova) Topological classification of integrable geodesic flows with an additional integral that is quadratic or linear in the momenta on two-dimensional orientable Riemannian manifolds, Funct. Anal. Appl. 27 (1993), no. 3, 186--196. 
N/A
The first paper does what its title says. The integrable metrics in question are found by Kolokoltsov. The second paper is a combination of the results of the first paper and some related results by Selivanova, and somehow it appeared in the most prestigious Russian journal at that time.


(with Anatoly T. Fomenko) Topological classification of nondegenerate integrable Hamitonian systems on an isoenergy 3-dimensional sphere, Russian Math. Surveys, 45 (1990), No. 6, 91-111
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(with Anatoly T. Fomenko) Topological classification of integrable nondegenerate Hamiltonians on the isoenergy three-dimensional sphere, Topological classification of integrable systems, 267--296, Adv. Soviet Math., 6 (1991), Amer. Math. Soc., Providence, RI.
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This paper does what its title says. An amusing result contained there is the fact that if a system on an isoenergy submanifold diffeomorphic to S^3 contains a periodic orbit which is not a generalized torus knot (a.k.a. zero-entropy knot), then it cannot be integrable.


On the general position property of simple Bott integrals. Russ. Math. Surv., 45 (1990), No. 4, 179-180
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Investigation of generic properties of simple Bott integrals (Russian). Trudy Sem. Vektor. Tenzor. Anal. No. 24, (1991), 133--140
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This announcement + corresponding detailed paper is probably the first instance where the existence of an S^1 action in the neighborhood of a nondegenerate corank-1 singular level set of an integrable system with two degrees of freedom is pointed out. This S^1 action is used in the paper to prove the ``general position property of simple Bott integrals'' by perturbations on the reduced phase space and Moser path method.

Papers/books that cite my work