RESMON · Resurgent and monogenic functions in dynamical systems and number theory

Resurgent and monogenic functions in dynamical systems and number theory

Project aims and objectives This Intra-European fellowship (IEF) was a two-year mobility project for Dr David Sauzin, an experienced French CNRS (Centre national de la recherché scientifique, France) researcher in pure mathematics, at the Pisa SNS (Scuola Normale Superiore, Italy), to collaborate with Prof. Stefano Marmi who was the supervisor of the project. The mathematical goal of the project was to develop innovative applications of Ecalle's theory of resurgent functions and Borel's theory of monogenic functions in various problems of asymptotic analysis. The project was also intended as an occasion of consolidating the relation between the Pisa mathematical community and the French mathematical community. In these respects, the project was successful beyond all expectations. One of the most striking outcomes of the project is the creation of a new international research unit placed under the joint responsibility of the French CNRS and the Pisa SNS, acting on behalf of the Centro De Giorgi (Centro di Ricerca Matematica Ennio de Giorgi, Italy). Since September 2010, the fellow has been working in close contact with the National Institute for Mathematical Sciences (INSMI) inside CNRS, with a view to creating an International Joint Research Unit in Pisa. He went to Paris, France for several meetings, in some of which the supervisor of the IEF project, Prof. Marmi, and the director of the Centro De Giorgi, Prof. Mariano Giaquinta participated. The result was an agreement between the CNRS and the Pisa SNS to create the 'Laboratory Fibonacci' for a four-year term starting 1 January 2012. http://crm.sns.it/event/group/fibonacci.html This research unit, hosted by the Centro De Giorgi, aims at allowing for better mathematicians' mobility between France and Pisa, especially for medium-sized or longer stays. It will help coordinate the interactions with the Italian community in mathematics, and also in theoretical physics and computer sciences. It will be open to all areas of mathematics and their Interactions, including theoretical physics and computer sciences. As of the date of creation of the Laboratory Fibonacci, it will count five permanent members, among which Prof. Marmi as director, and Dr Sauzin as deputy director. The fellow will thus stay in Pisa to continue the collaborations he had started during the IEF project and to play a role in the supervision of the activities of the Laboratory Fibonacci. He has already contributed to the 2012 programme, with the organisation of the coming of two French colleagues for six months each and his participation to the preparation of an inaugural conference (scheduled for 5-8 March 2012). http://www.crm.sns.it/event/246/ As for the mathematical work accomplished by the fellow, the results obtained in collaboration with Carlo Carminati (Pisa university) and Stefano Marmi must be singled out. They gave rise to a 33-page article entitled 'There is only one KAM curve', which was submitted to one of the top journals in mathematics. The authors consider a well-known example of symplectic dynamical system, the so-called standard family, and they study the corresponding KAM curves. They address what can be considered as the oldest open problem in KAM theory: in 1954, in his International Congress of Mathematics (ICM) conference, Kolmogorov had asked whether the regularity of the solutions of small divisor problems with respect to the frequency could be investigated using appropriate analytical tools, suggesting a connection with the theory of Borel's monogenic functions. This article answers affirmatively by establishing a monogenic regularity result upon a complexified rotation number, which implies a remarkable property of quasi-analyticity, a form of uniqueness of the monogenic continuation, although real frequencies constitute a natural boundary for the analytic continuation from the Weierstrass point of view because of the density of the resonances.

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