Jean Écalle (born 1947) is a French mathematician, specializing in dynamic systems, perturbation theory, and analysis. Écalle received, in 1974 from the

University of Paris-Saclay Paris-Saclay University (, ) is a combined Research institute, technological research institute and Public university, public research university in Orsay, France. Paris-Saclay was established in 2019 after the merger of four technical ''grandes ...

Orsay Orsay () is a Communes of France, commune in the Essonne Departments of France, department in Île-de-France in northern France. It is located in the southwestern suburbs of Paris, France, from the Kilometre Zero, centre of Paris. A fortifie ...

, a doctorate under the supervision of Hubert Delange with Thèse d'État entitled ''La théorie des invariants holomorphes''. He is a ''directeur de recherche'' (senior researcher) of the

Centre national de la recherche scientifique The French National Centre for Scientific Research (, , CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 staff, including 11,137 tenured researchers, 13,415 eng ...

(CNRS) and is a professor at the University of Paris-Saclay. He developed a theory of so-called " resurgent functions", analytic functions with isolated singularities, which have a special algebra of derivatives (''Alien calculus'', ''Calcul différentiel étranger''). "Resurgent functions" are divergent

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

whose Borel transforms converge in a neighborhood of the origin and give rise, by means of analytic continuation, to (usually) multi-valued functions, but these multi-valued functions have merely isolated singularities without singularities that form cuts with dimension one or greater. Écalle's theory has important applications to solutions of generalizations of Abel's integral equation; the method of resurgent functions provides for such solutions a (Borel) resummation method for dealing with divergent series arising from semiclassical asymptotic developments in quantum theory. He applied his theory to dynamic systems and to the interplay between diophantine small denominators and resonance involved in problems of

germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...

s of

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

s. Independently of Yulij Ilyashenko he proved that the number of

limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...

s of polynomial vector fields in the plane is finite, which

Henri Dulac Henri Claudius Rosarius Dulac (3 October 1870, Fayence – 2 September 1955, Fayence) was a French mathematician. Life Born in Fayence, France, Dulac graduated from École Polytechnique (Paris, class of 1892) and obtained a Doctorate in Math ...

had already tried to prove in 1923. This result is related to

Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topology ...

. In 1988 Écalle was the inaugural recipient of the of the

Académie des Sciences The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...

. He was in 1990 an Invited Speaker at

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

Kyoto Kyoto ( or ; Japanese language, Japanese: , ''Kyōto'' ), officially , is the capital city of Kyoto Prefecture in the Kansai region of Japan's largest and most populous island of Honshu. , the city had a population of 1.46 million, making it t ...

Selected publications

* '' Les Fonctions Résurgentes '', 3 volumes, pub. Math. Orsay, 1985 (downloadable pdf's availabl
here
* '' Cinq applications des fonctions résurgentes '', pub. Math. Orsay 1984
'' Singularities non abordables par la géométrie ''
Annales Inst. Fourier, 42, 1992, 73-164 * "Six Lectures on Transseries, Analytical Functions and the Constructive Proof of Dulac's Conjecture", in D. Schlomiuk's ''Bifurcations and Periodic Orbits of Vector Fields'', Kluwer 1993, 75-184 * with B. Vallet: ''Correction and linearization of resonant vector fields or diffeomorphisms'', Mathematische Zeitschrift 229, 1998, pp. 249–318 * "A Tale of Three Structures: The Arithmetic of Multizetas, the Analysis of Singularities, the Lie Algebra ARI", in BLJ Braaksma, GK Immink, Marius van der Put, J. Top (eds.
''Differential Equations and the Stokes Phenomenon''
World Scientific 2002, pp. 89–146 * Recent Advances in the Analysis of Divergence and Singularities, in C. Rousseau, Yu. Ilyashenko (Editor) ''Proceedings of the July 2002 Montreal Seminar on Bifurcation, Normal Forms and Finite Problems in Differential Equations'', Kluwer 2004, pp. 87–18

* '' Théorie des invariants holomorphes '', Pub. Math. Orsay 1974 * '' Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac '', Paris: Hermann 1992 * with Olivier Bouillot
"Invariants of identity-tangent diffeomorphisms: explicit formulae and effective computation."
arXiv preprint arXiv:1404.1042 (2014).

References

External links

* * {{DEFAULTSORT:Ecalle, Jean 1950 births Living people University of Paris alumni Academic staff of Paris-Saclay University Dynamical systems theorists 20th-century French mathematicians 21st-century French mathematicians