So long as men worship dictators, Caesars and Napoleons will arise to make them miserable.
by Aldous Huxley (1894- 1963)

Normal forms of vector fields on Poisson manifolds (2006)

(With Ph. Monnier) Normal forms of vector fields on Poisson manifolds. Ann. Math. Blaise Pascal 13 (2006), no. 2, 349–380.

Cited in:

Khesin, Boris ; Tabachnikov, Serge . Pseudo-Riemannian geodesics and billiards. Adv. Math. 221 (2009), no. 4, 1364–1396. ^ Miranda, Eva ; Zung, Nguyen Tien . A note on equivariant normal forms of . . . → Read More: Normal forms of vector fields on Poisson manifolds (2006)

Equivariant splitting theorem for Poisson structures (2006)

(With Eva Miranda) A note on equivariant normal forms of Poisson structures. Math. Res. Lett. 13 (2006), no. 5-6, 1001–1012.

This small paper estabishes the equivariant version of Weinstein’s splitting theorem for Poisson structures, under an additional “tameness” condition. This tameness condition can be dropped if the symmetry in question is Hamiltonian (see our . . . → Read More: Equivariant splitting theorem for Poisson structures (2006)

Existence of torus actions near singular orbits of integrable systems (2000 & 2003)

1) A note on degenerate corank-one singularities of integrable Hamiltonian systems. Comment. Math. Helv. 75 (2000), no. 2, 271–283.

2) Actions toriques et groupes d’automorphismes de singularités des systèmes dynamiques intégrables. C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1015–1020.

The second pape (2003), despite being a short paper in CRAS, actually . . . → Read More: Existence of torus actions near singular orbits of integrable systems (2000 & 2003)

Torus actions and integrable systems (2006)

Torus actions and integrable systems. Topological methods in the theory of integrable systems, 289–328, Camb. Sci. Publ., Cambridge, 2006.

Arxiv: http://arxiv.org/abs/math/0407455

The above paper, which is mostly a review paper, also contains original results on the theme “reduction commutes with integrability”: The question is, what is the relation between the reduced integrability (i.e. integrability . . . → Read More: Torus actions and integrable systems (2006)

Semi-local normal form of integrable systems (2004)

(With Eva Miranda) Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 819–839.

This paper establishes a smooth symplectic semi-local normal form for nondegenerate singularities of integrable Hamiltonian systems, enhancing a result of Hakan Eliasson.

Cited in:

Alvaro Pelayo & San . . . → Read More: Semi-local normal form of integrable systems (2004)

Topology of integrable geodesic flows (1993-1996)

Some old papers, written before my PhD, about the topology of integrable geodesic flows on (multi-dimensional) sphere and torus:

1) (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.

2) (With Lada Polyakova and . . . → Read More: Topology of integrable geodesic flows (1993-1996)

Kolmogorov condition in KAM theory (1996 & 2007)

1) Kolmogorov condition for integrable systems with focus-focus singularities. Phys. Lett. A 215 (1996), no. 1-2, 40–44.

2) Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems. Regul. Chaotic Dyn. 12 (2007), no. 6, 680–688.

Kolmogorov condition is a nondegeneracy condition needed as a hypothesis in the K.A.M. theory about the persistence of quasi-periodic . . . → Read More: Kolmogorov condition in KAM theory (1996 & 2007)

Linearization of Nambu structures (1999)

(With Jean-Paul Dufour) Linearization of Nambu structures, Composition Math. 117 (1999), No. 1, 77-98 .

Arxiv: http://arxiv.org/abs/dg-ga/9707006

Link to paper: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=308690

A Nambu structure is nothing but a singular foliation plus a contravariant volume form on the leaves. In this paper we give a classification of linear Nambu structures, and some linearization theorems for . . . → Read More: Linearization of Nambu structures (1999)

Levi decomposition (2002-2004)

1) Levi decomposition of analytic Poisson structures and Lie algebroids. Topology 42 (2003), no. 6, 1403–1420.

2) (With Ph. Monnier) Levi decomposition for smooth Poisson structures. J. Differential Geom. 68 (2004), no. 2, 347–395.

3) (With J-P Dufour) Nondegeneracy of the Lie algebra aff(n). C. R. Math. Acad. Sci. Paris 335 (2002), no. 12, . . . → Read More: Levi decomposition (2002-2004)

Focus-focus singularities (1997 & 2002)

1) A note on focus-focus singularities. Differential Geom. Appl. 7 (1997), no. 2, 123–130.

Arxiv: http://arxiv.org/abs/math/0110147

2) Another note on focus-focus singularities. Lett. Math. Phys. 60 (2002), no. 1, 87–99.

Arxiv: http://arxiv.org/abs/math/0110148

My first paper on focus-focus singularities (written in 1994 and accepted for publication soon after that but published only in 1997 due . . . → Read More: Focus-focus singularities (1997 & 2002)

Analytic normal forms of integrable vector fields (2002 & 2005)

1) Convergence versus integrability in Birkhoff normal form. Ann. of Math. (2) 161 (2005), no. 1, 141–156.

2) Convergence versus integrability in Poincaré-Dulac normal form. Math. Res. Lett. 9 (2002), no. 2-3, 217–228.

These two papers form a mini-series. The first paper (Ann. Math.) was actually written in 2001, before the 2nd paper (Math . . . → Read More: Analytic normal forms of integrable vector fields (2002 & 2005)

Integrable Hamiltonian systems on S3 (1990)

A.T. Fomenko & Nguyen Tien Zung, Topological classification of integrable nondegenerate Hamiltonians on a constant energy three-dimensional sphere. (Russian) Uspekhi Mat. Nauk 45 (1990), no. 6(276), 91–111, 189; translation in Russian Math. Surveys 45 (1990), no. 6, 109–135

A. T. Fomenko & Nguyen Tien Zung, Topological classification of integrable nondegenerate Hamiltonians on the isoenergy . . . → Read More: Integrable Hamiltonian systems on S3 (1990)

Proper groupoids and momentum maps : Linearization, affinity, and convexity (2006)

Nguyen Tien Zung, Proper groupoids and momentum maps: linearization, affinity, and convexity. Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 5, 841–869.

Preprint on arxiv: http://arxiv.org/abs/math/0407208

Paper at Science Direct: http://www.sciencedirect.com/science/article/pii/S0012959306000413

The main results of this paper are the following:

- Local smooth linearization theorem for proper Lie groupoids

- An analogous theorem . . . → Read More: Proper groupoids and momentum maps : Linearization, affinity, and convexity (2006)