**PURPOSE AND STRUCTURE**

The Laboratory Fibonacci is an international research unit placed under the joint responsibility of the French CNRS (National Centre for Scientific Research) and the Scuola Normale Superiore di Pisa acting on behalf of the Centro di Ricerca Matematica Ennio de Giorgi. The agreement between CNRS and the SNS is concluded for a four-year term starting 1st January 2012.

The research unit is hosted by the Centro de Giorgi. At CNRS, it is affiliated to the National Institute for Mathematical Sciences (INSMI), under the code "Unité Mixte Internationale n° 3483 - Laboratoire Fibonacci".

The laboratory aims at allowing for better mathematicians’ mobility between France and Pisa, especially for medium-sized or longer stays, with a view to consolidate and structure the scientific exchanges, including at the level of students (through new partnerships for PhD thesis or student exchange programs). The scientific relationships between CNRS, Italian and especially Pisa’s mathematical community follow a long tradition, and it is the vocation of CNRS to maintain and develop its relationships with European research through more structured actions. This research unit will help coordinate the interactions with the Italian community in Mathematics, and also in Theoretical Physics and Computer Sciences.

As of the date of creation of the Laboratory Fibonacci, its permanent members are

- Stefano Marmi (director), professor at the SNS,
- David Sauzin (deputy director), researcher at CNRS,
- Luigi Ambrosio, professor at the SNS,
- Ferruccio Colombini, professor at the Pisa university,
- Mariano Giaquinta, professor at the SNS.

The Laboratory Fibonacci is open to all areas of Mathematics and their Interactions, including Theoretical Physics and Computer Sciences. Among the first themes to be developed in the laboratory, the following ones have been identified in the field of Analysis and Dynamical Systems:

In the recent years, there was a dramatic development of the theory, with applications which range from diffusion equations, possibly on manifolds, to the calculus of variations, or to geometric and functional inequalities. These are topics in which Italy and France have two strong research groups with strong interactions. Themes currently of particular interest are the study of optimal transportation techniques to get synthetic definitions of a Ricci curvature bounded from below for metric measure spaces, and the study of generalised transportation problems in which the geometry is deeply influenced by the concentration and the congestion of the traffic.

Velocity fields with little smoothness, or generated by means of singular integrals, appear in various research areas, for instance in the theory of conservation laws, in fluid dynamics, in kinetic theories. In these contexts, it is important to work out the link between the Eulerian and Lagrangian viewpoints, not only at a conceptual level, but also to be able to define adequate notions of weak solutions, to prove well-posedness results, etc. After the foundational work by DiPerna-Lions, various important results were obtained by Italian and French researchers, but there is still a lot to be understood on the existence and uniqueness of flows in critical cases. In particular, it seems that what is lacking now is an optimal result which would cover the case of any vector field with bounded variation as well as the case of a vector field whose gradient is the convolution of a singular integral and a measure (the latter case is especially relevant in connection with equations of the kind of Euler 2D or Vlasov-Poisson).

The well-posedness results alluded to in the previous paragraph often come with stability results. Recent researches made use of these stability results, even at the very abstract level of flows in spaces of measures, to produce new results about the convergence of the Wigner transforms for the solutions of the linear Schrödinger equation in the semiclassical limit. This allows one to include in the theory irregular potentials (the gradients of which may have jump discontinuities), of the kind encountered in molecular dynamics.

Singular behaviour and concentration phenomena are typical of stationary points but also of minimum points for variational problems (especially when the associated energy functionals are invariant under the action of a large group of transformations); these variational problems may have a geometric origin (search for metrics with prescribed properties for the curvature, harmonic maps, study of curvature, also of higher order curvatures) or they model physical situations or play a fundamental role in physical theories (as for instance in the theory of liquid crystals or, more generally, the study of the so-called complex bodies, but also in non-linear elasticity or when modelling dislocations).

The interest of studying such problems is multifaceted: foundational (as in the case of the macroscopic description of microscopic phenomena), mathematical (allowing for a deep understanding of the origin of singularities and concentration: is their ultimate origin energetic, topological, homological?) and utilitarian (providing a framework for efficient computational analysis). Of course, the previous problems have non-stationary versions in conservative and non-conservative contexts.

In the last 15-20 years, geometric measure theory has provided valid instruments for these studies, especially with the introduction of the notion of cartesian current, which has allowed a homological characterization of weak limits of regular maps or fields. One of the research themes of the laboratory will be precisely this: to cope with specific problems of complex body modelling, to study the singularities and the concentration of the corresponding equilibrium configurations (existence, non-existence, structure), with a view to construct models which suit better the description of the phenomena and to give a more complete understanding of the occurrence of concentration and singularities.

Partial Differential Equations, PDE, is a well established field of research in Mathematical Analysis. It grew over a couple of centuries to be one of the richest domains of Mathematics. Major progresses have been accomplished in the analysis of PDE's during the last twenty years, both from theoretical and applied points of view. Many of them are based on the development of powerful tools microlocal analysis. The idea, at the crossing point of harmonic analysis, functional analysis, quantum mechanics and algebraic analysis, is that many phenomena depend jointly on position and frequency and therefore must be understood and described in the phase space. The introduction of phase space analysis in the seventies put in evidence a number of unifying factors between traditionally diverse sectors, like e.g. hyperbolic and (degenerate) elliptic equations; moreover the theory has been geometrized, putting in evidence its invariance character. The basic tools that have been developed concern the theories of pseudodifferential calculus, Fourier-Bros-Iagolnizter transform, Fourier integral operators, Weyl's calculus and other quantizations and refinements based on Littlewood-Paley decompositions; extension to non smooth coefficients and nonlinear equations have been made through para-differential calculi.

Many problems which are central to this part of the proposal originate from physics and other applications. Just to cite a few: Euler, Navier-Stokes and related equations from fluid mechanics, both in the incompressible case (global solutions, singularities, rotating fluids, water waves, etc) and for compressible fluids (low Mach number flows, shocks, etc). Maxwell's equations from electromagnetism and field equations are basic models for the theory of hyperbolic equations, including now nonlinear problems (global solutions, decay, stability, shock formation, etc). Nonlinear optics leads to various models (geometric optics, non linear Schrödinger equations, Zakharov systems and other dispersive equations). The control of systems governed by PDE's (observability, stabilization, exact controllability) is an important and very active domain; strictly related is the study of fine properties of solutions of PDE with non-smooth coefficients (existence of solutions to the Cauchy problem, unique continuation, observability), a topic with important and immediate applications in inverse problems and control theory. Local solvability and its dual problem, unique continuation, strongly profits from a microlocal point of view. Furthermore its microlocal aspect is made quite evident when local solvability for operators of nonprincipal type is considered. A particularly interesting problem in Microlocal Analysis and in the applications to PDE's is the one of finding lower bounds for pseudodifferential operators whose symbol is non-negative.

In mathematics or in physics, less general theories are often obtained as limits of more general theories, as a parameter gets small. Examples are the passage from quantum to classical mechanics as Planck's constant tends to zero; the passage from statistical mechanics to thermodynamics as the number of particle tends to infinity; the passage from hyperbolic dynamics to parabolic dynamics as some parameters of a dynamical system makes its spectrum become resonant. So, reductions of one theory to another are problems of asymptotics. In most cases the limits are non-trivial because they are singular. The singularities give rise to rich physics or mathematics in the borderland between theories. Attempts to express more general theories as power series, the first terms of which describe less general theories, almost invariably lead to divergent series and call for appropriate summation techniques.

The mathematical theory of dynamical systems, particularly with Poincaré’s researches in celestial mechanics, always considered small divisor problems as a fundamental issue: near-resonances made questionable the convergence of the series used by astronomers in their perturbative methods, for instance, and Poincaré could prove that resonances themselves were responsible for the divergence of other series they used. Thorough studies were subsequently devoted to the convergence of the solutions under suitable arithmetical conditions in certain mathematical models. The dependence of the solutions with respect to parameters was much less investigated.

Borel’s theory of monogenic functions, which is a somewhat neglected chapter of the mathematical theory of functions, and the much more recent theory of Gevrey asymptotics and resurgent functions are sophisticated tools which will be used to tackle these problems. Recent Franco-Italian collaborations led to new approaches to the monogenic dependence on parameters and the quasianalyticity issue in the context of small divisor problems. Various problems in KAM theory for near-integrable Hamiltonian systems or symplectic maps, but also in ergodic theory with the study of the dependence of SRB measures on parameters, are on the agenda.

In the last ten years, spaces of smooth Gevrey functions were used in a new manner in non-linear dynamics. The classical KAM and Nekhoroshev theorems were extended to Gevrey near-integrable Hamiltonian systems, and original examples were constructed to test the optimality of the stability results. A lot of questions, more general and geometric, remain open: in particular, geometric dynamically meaningful quantities which vanish for integrable systems, viewed as a measurement of non-integrability (size of separatrix splitting, angles of Green bundles, topological and metric entropies, volume or capacity of wandering domains) deserve a systematic study.

Ergodic aspects, related with the possibility of embedding random walks in near-integrable systems and the geometric structure of simply-resonant and doubly-resonant surfaces, are also part of the picture; the statistical properties of dynamics along simple resonances lend themselves to analytic approaches, as well as numeric studies in the spirit of the ones performed in the framework of celestial mechanics.

Resurgent functions constitute algebras of formal series on which act the so-called alien derivations (operators measuring the divergence of the resurgent formal series). This theory began to be developed by the French mathematician Jean Écalle around 30 years ago, initially in the context of holomorphic dynamics in one or several variables, to deal with difficult classification problems which involve Gevrey asymptotics, multisummability, non-linear Stokes phenomena, where it allows one to build sectorial normalisation diffeormorphims from divergent series. The so-called mould calculus appeared at the same time, as an algebraic apparatus to deal with the infinite-dimensional free associative algebras generated by the alien derivations; by its ability to handle multiply indexed series of operators, it soon proved to be a flexible tool to construct formal solutions of dynamical problems via a systematic reformulation of the equations in terms of operators, and even to reach analytic solutions with the help of the so-called arborification technique.

Resurgence applies virtually to all local analytic dynamical systems with resonant linear part and the combinatorics of moulds can help to manifest it. Mould calculus is also of interest in the opposite situation, when the linear part is non-resonant and a Diophantine condition is used to control the small divisor problem. A lot remains to be done to exploit all the ideas that Écalle's machinery suggests in various contexts: parabolic renormalization in one variable, holomorphic dynamics in several variables, Hamiltonian small divisor problems, but also classification of geometrical structures with singularities, holomorphic PDEs, etc.

Alien calculus also provides a tool to prove analytic independence results; another tantalising application is thus in the theory of o-minimal structures, where this could help to prove the quasianalyticity of certain algebras of functions and then the o-minimality of the corresponding structures (for instance algebras generated by normalisation diffeomorphisms of a saddle-node vector field).

The intrinsic operator setting of quantum mechanics suggests applications of mould calculus to the perturbation theory of quantum mechanics which should, as a by-product, lead to new results in semiclassical analysis; in particular, one can hope to find a quantum resummation algorithm that would be uniform with respect to the semi-classical limit.

Mould calculus can be presented as a rich combinatorial environment of Hopf-algebraic nature. As such, it has connections with the theory of noncommutative symmetric functions, as developed by the French school of algebraic combinatorics, for which the use of Hopf-algebraic techniques appears as one of the main successes in the field during the last twenty years, and also with the Connes-Kreimer Hopf algebras of trees and graphs and their applications to perturbative quantum field theory. Combining these ideas stemming from various domains and points of view should result in a unified system of concepts and methods. The hope is that insisting on combinatorial methods and effectivity will result in a renewal of resurgence theory, its applications and interactions (much beyond the domain of local dynamics), while the structures suggested by resurgence and mould calculus will enrich algebraic combinatorics (the domain of its applications as well as the realm of its objects). As for the computational side, the development of software in collaboration of experts in computer algebra is envisaged; the Sage-Combinat system seems to be particularly well-suited for this purpose.

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