Publications and Preprints
by Nguyen Tien Zung
Last updated: 20/Sep/2002
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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of integrable systems, including some new unpublished results and ideas.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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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) |
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This exposé contains an
account on my first attempts (not very successful ones) to classify integrable
systems topologically using characteristic classes. |
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Unpublished note written
around 1993. |
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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. |
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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. |
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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