List of publications and preprints of Nguyen Tien Zung (until 2011)

(together with short abstracts or comments)

*Nondegenerate singularities of integrable non-Hamiltonian systems*, Preprint 2011.- (With Do Duc Thai) Introduction to financial mathematics (in Vietnamese). Book written for Vietnamese students, about 260 pages, 2011. Discusses basic concepts of financial mathematics, including things like: structure of financial markets, interest rate, risk premium, term structure, arbitrage, martingale, stochastic calculus, Ito integral, Cameron-Girsanov theorem, Black-Scholes pricing of options, Markowitz theory, hedging, etc.
*Entropy of geometric structures*, Bull. Bralizian Math. Soc., Vol. 42, No. 4 (2011), 15pp. We define the notion of entropy for general geometric structures, which generalizes the notion of topological entropy of vector fields, and the notion of geometric entropy of foliations. We show some basic properties of entropy, including the*addititivy*, similarly to the physical entropy.- (with E. Miranda and Ph. Monnier)
*Rigidity of Hamiltonian compact group actions on Poisson manifolds*, Advances in Math. (2011). Using an abstract Nash-Moser type normal form theory, we show that Hamiltonian actions of compact semisimple Lie groups on Poisson manifolds are rigid. - (With Do Duc Thai, in Vietnamese) Modern introduction to probability and statistics. (Publishing House of Hanoi University of Education, 2010). A book about probabilities and statistices, written in an intuitive way, with many modern examples, for Vietnamese students.
- (with Michael Ayoul)
*Galoisian obstructions to non-Hamiltonian integrability*, Comptes Rendus Math. (2010), arXiv:0901.4586. We show that the main theorem of Morales–Ramis–Simo about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case. - (with T. Ratiu and A. Weinstein, Editors) Lectures on Poisson geometry — Proceedings of a Summer School in ICTP. Geometry and Topology Monographs (2010)
- (with T. Kappeler, P. Lohrmann and P. Topalov)
*Birkhoff coordinates for the focusing NLS*,*Comm. Math. Phys.*285 (2009), no. 3, 1087–1107. *Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems*,*Regul. Chaotic Dyn.*12 (2007), no. 6, 680–688. The main idea is that if an integrable system admits certain types of singularities then its frequency map must be nondegenerate, due to the singular behavior of the action functions.*Proper groupoids and momentum maps: linearization, affinity, and convexity.*Annales Sci. Ecole Norm. Sup 39 (2006), 841-869.- (with Eva Miranda)
*A note on equivariant normal forms of Poisson structures**.*Math. Res. Lett. 13 (2006), 1000–1012. Equivariant splitting theorem and equivariant linearization theorem for Poisson structures under some conditions. - (with Philippe Monnier)
*Normal forms of vector fields on Poisson manifolds**.*Annales Mathématiques Blaise Pascal 13 (2006), 349–380. Formal and analytic normal forms for Hamiltonian and radial vector fields on Poisson manifold near singular points. *Torus actions and integrable systems.*A. Bolsinov, A. Fomenko and A. Oshemkov eds., Topological Methods in the Theory of Integrable Systems, 2006. A survey paper, which also contains unpublished results about the relations between reduction and integrability.*Convergence versus integrability in Birkhoff normal form.*Annals of Maths. 161 (2005), No.1, 141-156. This paper was written in 2001, before the paper on Poincaré-Dulac normal forms which was published in 2002. Any analytic integrable Hamiltonian system near a singular point admits a convergent Birkhoff normalization, without any additional condition.- (with Jean-Paul Dufour; monograph) Poisson Structures and their Normal Forms. Progress in Mathematics, Vol. 242, 2005, XVI, 324 p., Hardcover, ISBN: 3-7643-7334-2. The aim of this book is twofold: to give a quick, self-contained introduction to Poisson geometry and related subjects, and to present a comprehensive treatment of the normal form problem in Poisson geometry.
- (with Eva Miranda)
*Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems.* - (with Philippe Monnier)
*Levi decomposition of smooth Poisson structures.*J. Diff. Geom. 68 (2004), 347-395. Shows the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids. An abstract normal form theorem of Nash-Moser type is included. *Levi decomposition of analytic Poisson structures and Lie algebroids.*Topology 42 (2003), No.6, 1403-1420. Shows the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids, generalizing a linearization theorem of Conn.*Actions toriques et groupes d’automorphismes de singularités de systèmes dynamiques intégrables.*C. R. Acad. Sci. Paris 336 (2003), No. 12, 1015-1020.*Symplectic topology of integrable Hamiltonian systems, II.*Compositio Math. 138 (2003), No. 2, 125-156. Classifies integrable Hamiltonian systems in terms of singularities, monodromy, stratified integral affine structures, and characteristic classes. Introduces integrable surgery.- (with Jean-Paul Dufour)
*Nondegeneracy of the Lie algebra aff(n)*. C. R. Acad. Sci. Paris (2002). Using Levi decomposition, we show that the Lie algebra aff(n) is formally and analytically nondegenerate in the sense of Weinstein. *Another note on focus-focus singularities.*Lett. Math. Phys 60 (2002), No. 1, 87-99. Topological classification of (non)degenerate focus-focus singularities of (non-)Hamiltonian integrable systems. Existence of T^1 action. Monodromy via Duistermaat-Heckman.*Convergence versus integrability in Poincaré-Dulac normal form.*Math. Res. Lett. 9 (2002), 217-228. Extends the results of the paper “Convergence versus integrability in Birkhoff normal forms” to non-Hamiltonian systems.*Reduction and integrability.*math.DS/0201087 (never published in a journal). Relationship between the integrability of a dynamical system invariant under a Lie group action and its reduced integrability.*A la recherche des tores perdus.*Document de synthèse pour l’HDR (2001). This mémoire, written in French, is a survey of my results on topological aspects of integrable systems.*A note on degenerate corank-1 singularities of integrable Hamiltonian systems*. Comment. Math. Helv., 75 (2000), 271-283.- (with Jean-Paul Dufour)
*Linearization of Nambu structures.*Compositio Math. 117 (1999), no. 1, 77–98. Nambu structures are dual to integrable differential forms and are very useful for the study of singular foliations. *A note on focus-focus singularities.*Diff. Geom. and Appl., 7 (1997), 123-130. Topological classification of nondegenerate focus-focus singularities of integrable Hamiltonian systems, existence of T^1 actions, and the monodromy formula.- (with Tit Bau)
*Singularities of integrable and near-integrable Hamiltonian systems.*Journal of Nonlinear Science, 7 (1997), 1-7. *Kolmogorov condition for integrable systems with focus-focus singularities*. Physics Letters A, 215 (1996), 40-44. This paper and the one with Tit Bau ( Tit Bau is a fictive author — Tit was the nickname of my 3-year old son at that time) 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.*Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities.*Compositio Math., 101 (1996), 179-215. Results include: existence of torus actions, partial action-angle coordinates, and a topological decomposition theorem for nondegenerate singularity.*Singularities of integrable geodesic flows on multi-dimensional torus and sphere*. Journal of Geometry and Physics, 18 (1996), 147-162. This paper contains some errors in the computation of more complicated singularities (claims that all singularities are nondegenerate, which is not true)*Decomposition of nondegenerate singularities of integrable Hamiltonian systems.*Letters in Math. Physics, 33 (1995), 187-193. This is a short announcement of the paper “Arnold-Liouville with singularities”.*A topological classification of integrable Hamiltonian systems*. Séminaire Gaston Darboux, Université Montpellier II, 1994-1995, 43-54 (1995). 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*. Unpublished note written around 1993.- (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. This paper does what its title says. The integrable metrics in question are found by Kolokoltsov. - (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. This paper is a combination of the results of the paper with Polyakova with 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; and*Topological classification of integrable systems*, 267–296, Adv. Soviet Math., 6 (1991). This paper (published twice) 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;*and Investigation of generic properties of simple Bott integrals*(Russian). Trudy Sem. Vektor. Tenzor. Anal. No. 24, (1991), 133–140. This announcement + corresponding detailed paper is probably the first instance where the existence of an T^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.

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