The main idea behind this project is the bringing together of two different areas of mathematics (singularity theory and Poisson geometry) hopefully with profit to both. It is well known (since1983 when A. Weinstein stated the Splitting Theorem for Poisson manifolds) that the local structure of Poisson manifolds is not interesting but at singular points, i.e., points of rank zero. The most important works (by A. Weinstein, J. F. Conn and J. P. Dufour) on the local structure of Poisson manifolds concern the possibility of bringing to a linear normal form the original structure. More recently J. P. Dufour has worked on the problem of quadratization, i.e., the possibility of reducing the original structure to its quadratic part. With this project one intends to move towards a classification of Poisson structure around singular points. The concern will be more of obtaining a normal form, rather than linearizability or quadratizability. Singularity theory concentrates on the classification of differentiable maps around degenerate points, and therefore seems to be a good source of ideas to start this classification. In particular, the three-dimensional case seems to reinforce this relation between classification of functions around critical points and the classification of some special Poisson structures. Such Poisson structures will be called "locally exact" and they will be the main object of study in this project. This notion of "locally exact" admits a natural generalization to higher dimensions, so the project can proceed with the study of locally exact Poisson structures in dimension four. Depending on the evolution of the problem one can move to higher dimensions. It must be said, however, that the situation seems of high complexity already in dimension five, and is therefore probably out of the scope of this project.
Saturday, 1 January, 2000