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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 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) 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 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 This is work in progress with Ph. Monnier. The nondegeneracy here is in the sense of Weinstein: a real Lie algebra g is called smoothly nondegenerate if any smooth Poisson structure whose linear part is g* is locally smoothly linearizable (i.e. isomorphic to g*) The theorem that we want to prove gives a complete . . . → Read More: Nondegeneracy of simple Lie algebras of real rank 1 last updated: 18/jan/2012 This is a particular case of integrable non-Hamiltonian systems that my student Minh is working on with me for this thesis. We want to study such systems topologically. A real integrable system of type is nothing but a -action (generated by a family of commuting vector fields) on a -dimensional manifold. . . . → Read More: Rn-actions on n-dimensional manifolds This is a particular case of a larger problem of linearization of Lie group and Lie algebra actions. The case of sl(2,R) and SL(2,R) is already non-trivial. Guillemin and Sternberg gave an example of a smooth sl(2,R) action on R3 which is not linearizable. Cairns and Ghys gave an example of a smooth SL(2,R) . . . → Read More: Smooth linearization of sl(2,R) and SL(2,R) actions ? Project with Eva Miranda and Philippe Monnier We have invented a “hammer” called “abstract Nash-Moser normal form theorem” and now are looking for “nails” We used our hammer for the problem of rigidity of Hamiltonian actions on Poisson manifolds (see paper in Advances Math 2012). We don’t know yet to what problems (beyond . . . → Read More: Applications of Nash-Moser normal form theorem ? I’m doing this project about 2nd-order pricing models with a PhD student of mine. The project is quite ambitious. It aims to to be better than known models (Black-Scholes, stochastic volatility, 1st-order jump models, equilibrium, etc.) and be able to explain things that can’t be explained by previous models. Potential applications include investing & . . . → Read More: Second-order models for asset prices This is the research project of a PhD student of mine. Actually the general project is sufficiently large for a number of PhD theses. There are lots of open questions in the non-Hamiltonian case. What my student is doing is to study simplest (mostly low-dimensional) cases, and with only nondegenerate singularities: - Systems of . . . → Read More: Topology of integrable non-Hamiltonian systems Whitehead’s lemma says that and where is a simple Lie algebra and is a linear representation of it. I know an algebraic proof which gives an explicit formula for the homotopy operator. I’m looking for a more geometric proof (using, for example, an averaging formula). Why ? Because the explicit homotopy operator in the . . . → Read More: Geometric proof of Whitehead’s lemma ? Work in progress with Philippe Monnier Conn showed that compact simple Lie algebras are smoothly nondegenerate. Weinstein showed that simple Lie algebras of real rank >= 2 are smoothly degenerate. What about the case of real rank 1 ? There are only 3 series of real rank 1: so(n,1), su(n,1) and sp(n,1) (orthogonal, unitary . . . → Read More: Nondegeneracy of simple Lie algebras of real rank 1 This is a small research project that I intend to give to some master students for their memoir: - Examples of geometric structures of zero entropy. In particular, write down the proof of the fact that linear Poisson structures have zero entropy. - Example of positive entropy. Explicit computations. Comparison with Lyapunov exponents, Godbillon-Vey . . . → Read More: Computation of entropy |
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