**History of Mathematics**

Volume: 36;
2010;
391 pp;
Hardcover

MSC: Primary 30; 34; 37; 53; 32; 57; 16; 83; 00; 35; 01;

**Print ISBN: 978-0-8218-4718-3
Product Code: HMATH/36**

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

**Electronic ISBN: 978-1-4704-1807-6
Product Code: HMATH/36.E**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

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#### Supplemental Materials

# The Scientific Legacy of Poincaré

Share this page *Edited by *
*Éric Charpentier; Étienne Ghys; Annick Lesne*

Translated by Joshua Bowman

A co-publication of the AMS and the London Mathematical Society

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.

#### Readership

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

#### Reviews & Endorsements

The articles are very well written, indeed, and are of course autonomous. But even non-specialists will want to sample these wares. The mathematics is presented clearly and very accessible, and the numerous historical accounts and asides make add an additional welcome cultural element to whole experience.

[This book] is bound to be a hit across the mathematical spectrum: it has something for every one interested in any aspect of Poincaré's work, which is to say, something for every one.

-- MAA Reviews

#### Table of Contents

# Table of Contents

## The Scientific Legacy of Poincare

- Cover Cover11
- Title page iii4
- Contents v6
- List of authors xi12
- Translator’s note xiii14
- Introduction 116
- Poincaré and his disk 1732
- Differential equations with algebraic coefficients over arithmetic manifolds 4762
- Poincaré and analytic number theory 7388
- The theory of limit cycles 87102
- Singular points of differential equations: On a theorem of Poincaré 99114
- Periodic orbits of the three body problem: Early history, contributions of Hill and Poincaré, and some recent developments 113128
- On the existence of closed geodesics 143158
- Poincaré’s memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections 161176
- Variations on Poincaré’s recurrence theorem 193208
- Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids 207222
- The concept of “residue" after Poincaré: Cutting across all of mathematics 225240
- The proof of the Poincaré conjecture, according to Perelman 243258
- Henri Poincaré and the partial differential equations of mathematical physics 257272
- Poincaré’s calculus of probabilities 279294
- Poincaré and geometric probability 293308
- Poincaré and Lie’s third theorem 307322
- The Poincaré group 329344
- Henri Poincaré as an applied mathematician 351366
- Henri Poincaré and his thoughts on the philosophy of science 373388
- Back Cover Back Cover1410