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This is a work in progress with Christophe Wacheux, a student of San. Christophe went to see me in Toulouse to discuss about his thesis. After several discussions, I gave him 2 problems to work on. The first is about semi-local classification of integrable Hamiltonian systems up to exact isomorphisms (i.e. diffeomorphisms which preserve . . . → Read More: Intrinsic convexity of almost-toric integrable Hamiltonian systems 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 mid-day). Snail speed, 1 page/day. Nothing very exciting, but someone has to do it and . . . → Read More: A note on commuting foliations (2012) Just learned about the following conjecture of Littlewood (1930), which looks very simple and which is apparently still open: Let be two arbitrary real numbers. Denote by . Then This conjecture is related to ergodic theory of It is not difficult to show that the set of pairs of number which donot satisfy . . . → Read More: Littlewood’s conjecture (1930) 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) arXiv:0903.4617 I’m interested in all kinds of decomposition of all kinds of systems. I’m planning to write a paper on the decomposition of dynamical systems into fundamental states. This project is a bit vague. I want to make some ideas/results more precise and easier to apply before writing things up. By a “dynamical system” . . . → Read More: Decomposition of Dynamical Systems 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) 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) 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) I just learned that the mathematician JM Souriau, one of the foundators of symplectic geometry, died two days ago. There is a mathematical conference organized by my colleagues to celebrate his 90th birthday this year, and, sadly, now the conference will be without him. He was born on 03/June/1922. Souriau is known for his . . . → Read More: Jean-Marie Souriau R.I.P. The Journal of Diff. Equations was considered as a good journal in the field of differential equations and dynamical systems, and was given the A* ranking (the highest ranking) by the Australian system. I even considered submitting a paper to that journal. But now I have serious doubts about their quality and the seriousness . . . → Read More: The quality of J. Diff. Equations in doubt Updated: 04/April/2012 This paper is on schedule. It is almost finished now, and has about 30 pages. To be put on arxiv tomorrow. Updated: 19/03/2012 Section: Generic nilpotent singularities (X,F) where F regular, X nilpotent: F =F(y), X = y d/dx + … Since X(F) = 0 –> X = (y + …) d/dx . . . → Read More: Geometry of integrable vector fields on surfaces Updated 18/93/2012: 2nd version, which corrects a series of misprints and imprecisions in the 1st version. After two months of intensive work, we have finished the paper on the geometry of nondegenerate Rn actions on n-manifolds. The pdf file of the article is available here: Rn_Actions_2012.pdf Arxiv: http://arxiv.org/abs/1203.2765 This is a joint article with . . . → Read More: Geometry of R^n actions on n-manifolds (2012) Looks like the boycott against Elsevier is starting to show effects. A letter from Elsevier to mathematicians dated yesterday (probably lots of people, including myself, found this letter in our mailbox) contains the following tidbits: “… Mathematics journals published by Elsevier tend to be larger than those of other publishers. On a price-per-article, or . . . → Read More: Elsevier backing down ?! |
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