Scientific and Pedagogical Considerations and Goals

Introduction

The clue to the hereby planned School should be looked for in the deep ties between complex analysis and perturbation control. We have in mind many technical or scientific domains which are more or less connected to approximation problems or data treatment problems; these domains are just taking wing or enjoying an activity renewal, with a noteworthy connection to modern complex analysis problems. Secondly we observe that the existence of rich interactions between spectral transforms and analyticity questions is considered a classical topic among mathematicians but is very little known among practitioners. Third we specify that the perturbations mentioned above may be certain or random. Introducing probabilities, escorted by assumptions on the noises, may be artificial or oversimplifying. As to the cases where the introduction of probabilities is natural, we want to stress how powerfully alea organize both the theory and the phenomena.

Having laid these three points as foundations, we hereafter elaborate on considerations and goals pertaining to the School and give precisions on the audience we are aiming at. Before going any further we claim our belief concerning the usefulness of encounters of mathematicians with specialists or practitioners from the other fields as the reader probably already grasped we believe that such encounters may have a dramatic rôle on the research of both groups; it is actually a condition to ensure that mathematics may put questions to natural or applied sciences and the reverse.

Scientific Considerations

The assumes rôles of analyticity in particle physics and field theory ;ed the physicists of the XX-th century to an intense activity concerning singularities and resummation methods; but beliefs changed, leading to a decline of this activity. In what appears in retrospect to be a joint event, mathematicians obtained the natural convergence (in capacity) theorems for Padé approximants , drawing this period to a close. At about the same time, the control theory for dynamical systems underwent an important growth due to the working out of the idea to work in the complex plane. This theory became indeed much richer when an extensive use was made of the z-transform or the Fourier-Laplace transform of the transfer function (or matrix), thus allowing for a bridge between the state space description and the frequency domain description. Consequently problems akin to the closed loop stability were mapped back to analyticity domain discussions.

Another period opened with the renewal of interest for the analytic continuation in time of the real solutions of nonlinear differential systems, being integrable, partially integrable, intermittent or chaotic. Many results were obtained using the singularity structure and partial resummation procedures (Padé, -series), but the system classification remained elusive.

Now in the last few years the notion of resonance proved very useful in the study of signals generated by random or chaotic systems. By resonance we mean a singularity of the Fourier transform, specifically a pole. The distribution of the complex resonances allows a characterization of the certain modes and of the random ones. These singularities in turn may be localized using various rational approximants endows with their convergence properties. In conjunction with the latter, the problem of the analyticity in the complexified coupling constant plane was again attacked, using Padé resummation; this allowed some partial but significant progress, on the determination of analyticity domain, the class of Hamiltonian systems under study being much simpler than celestial mechanics, which is he ultimate goal. Finally, in the field of control theory, recent advancements concerning analytic and rational approximation have made one able to treat infinite dimensional systems; the technical point is then to control not only the analyticity domain of a function but also its norm on boundary curves. The general use of such techniques, as well as engineer needs, have led again to the forefront various unsolved problems, for instance in identification and robust control.

Goals of the School

On one hand, the three examples above are linked to three different fields, -- signal, control, and dynamical systems theory -- which are all three faced with a renewal of the rôle of approximation problems being analytic or meromorphic with or without noise, for which the relevant tools are complex analysis and harmonic analysis,as well as probability theory. In mathematics on the other hand, function theory and approximation theory are successfully pursued; recent examples of advances include operator theory, asymptotic error estimation or zeros of random or deterministic orthogonal polynomials. Obviously the contact is lacking between these two communities.

We propose to organize a School which should lead he attending people to master the above and related topics; as a result people will be in a good position to participate to the scientific renewal described in II. Through this teaching a given participant will discover recent developments or the theoretical framework of methods he knew only as a practitioner this in turn will lead to a bending of his way of thinking. On the other hand a given speaker, say a mathematician, will discover how a concept already familiar in his own field may appear under various disguises in other fields. We assign two goals to this school. First to transmit in a pedagogically classical style established ideas and results. Second to elaborate and spread new ideas, techniques and ways of thinking. In order to faster this elaboration, we plan to set aside a real interacting time both at the end of each lecture and in specially devoted round tables. We thus hope to shed light on themes which are on the fore-stage as well as those usually kept in the shades.

As an additional purpose, we want to help young scientists to participate and give them an opportunity to practice the delicate task of scientific communication in a congenial atmosphere.

Finally, among the various applications mentioned in the program, like e.g. control theory or celestial mechanics we want to give a special place to turbulence and to control problems and noise characterization in the interferometric gravitational wave detectors. This concern reflects in the Scientific Committee membership of a representative of the French turbulence community and of a representative of the French-Italian Virgo Collaboration.

Acknowledgments

As to the original idea and the initial phase of the preparation of this School project, we owe very much to discussions with Claude Froeschlé and Elena Lega. Evidently the focus is now on applied mathematics at large, but the present School is to some extent the heir of the series they have dedicated to hamiltonian dynamics.