Henri Poincaré (1854-1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. He created new mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to asymptotic expansions. He revolutionized celestial mechanics, discovering deterministic chaos. In physics, he is one of the fathers of special relativity, and his work in the philosophy of sciences is illuminating.

For this book, about twenty world experts were asked to present one part of Poincaré's extraordinary work. Each chapter treats one theme, presenting Poincaré's approach, and achievements, along with examples of recent applications and some current prospects. Their contributions emphasize the power and modernity of the work of Poincaré, an inexhaustible source of inspiration for researchers, as illustrated by the Fields Medal awarded in 2006 to Grigori Perelman for his proof of the Poincaré conjecture stated a century before.

This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, or physics, or more generally by anyone who likes mathematical and physical ideas. Rather than presenting detailed proofs, the main ideas are explained, and a bibliography is provided for those who wish to understand the technical details.

Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.

Readership

Undergraduate students, graduate students, and research mathematicians interested in Poincaré's life and work.

Table of Contents

• E. Ghys -- Poincaré and his disk
• N. Bergeron -- Differential equations with algebraic coefficients over arithmetic manifolds
• E. Kowalski -- Poincaré and analytic number theory
• J.-P. Françoise -- The theory of limit cycles
• D. Cerveau -- Singular points of differential equations: On a theorem of Poincaré
• M. Nauenberg -- Periodic orbits of the three body problem: Early history, contributions of Hill and Poincaré, and some recent developments
• N. Anantharaman -- On the existence of closed geodesics
• F. Béguin -- Poincaré's memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections
• E. Ghys -- Variations on Poincaré's recurrence theorem
• G. Boffetta, G. Lacorata, and A. Vulpiani -- Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids
• A. Yger -- The concept of "residue" after Poincaré: Cutting across all of mathematics
• L. Bessières, G. Besson, and M. Boileau -- The proof of the Poincaré conjecture, according to Perelman
• J. Mawhin -- Henri Poincaré and the partial differential equations of mathematical physics
• P. Cartier -- Poincaré's calculus of probabilities
• M. M. France -- Poincaré and geometric probability
• P.-P. Grivel -- Poincaré and Lie's third theorem
• M. Le Bellac -- The Poincaré group
• Y. Pomeau -- Henri Poincaré as an applied mathematician
• G. Heinzmann -- Henri Poincaré and his thoughts on the philosophy of science