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The Scientific Legacy of Poincaré

Edited by: Éric Charpentier Université Bordeaux 1, Talence, France
Étienne Ghys École Normale Supérieure de Lyon, Lyon, France
Annick Lesne Université Pierre et Marie Curie, Paris, France
Translated by Joshua Bowman
A co-publication of the AMS and London Mathematical Society
Available Formats:
Hardcover ISBN: 978-0-8218-4718-3
Product Code: HMATH/36
List Price: $100.00 MAA Member Price:$90.00
AMS Member Price: $80.00 Electronic ISBN: 978-1-4704-1807-6 Product Code: HMATH/36.E List Price:$94.00
MAA Member Price: $84.60 AMS Member Price:$75.20
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List Price: $150.00 MAA Member Price:$135.00
AMS Member Price: $120.00 Click above image for expanded view The Scientific Legacy of Poincaré Edited by: Éric Charpentier Université Bordeaux 1, Talence, France Étienne Ghys École Normale Supérieure de Lyon, Lyon, France Annick Lesne Université Pierre et Marie Curie, Paris, France Translated by Joshua Bowman A co-publication of the AMS and London Mathematical Society Available Formats:  Hardcover ISBN: 978-0-8218-4718-3 Product Code: HMATH/36  List Price:$100.00 MAA Member Price: $90.00 AMS Member Price:$80.00
 Electronic ISBN: 978-1-4704-1807-6 Product Code: HMATH/36.E
 List Price: $94.00 MAA Member Price:$84.60 AMS Member Price: $75.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$150.00 MAA Member Price: $135.00 AMS Member Price:$120.00
• Book Details

History of Mathematics
Volume: 362010; 391 pp
MSC: Primary 30; 34; 37; 53; 32; 57; 16; 83; 00; 35; 01;

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.

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

• Chapters
• Introduction
• Poincaré and his disk
• Differential equations with algebraic coefficients over arithmetic manifolds
• Poincaré and analytic number theory
• The theory of limit cycles
• Singular points of differential equations: On a theorem of Poincaré
• Periodic orbits of the three body problem: Early history, contributions of Hill and Poincaré, and some recent developments
• On the existence of closed geodesics
• Poincaré’s memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections
• Variations on Poincaré’s recurrence theorem
• Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids
• The concept of “residue" after Poincaré: Cutting across all of mathematics
• The proof of the Poincaré conjecture, according to Perelman
• Henri Poincaré and the partial differential equations of mathematical physics
• Poincaré’s calculus of probabilities
• Poincaré and geometric probability
• Poincaré and Lie’s third theorem
• The Poincaré group
• Henri Poincaré as an applied mathematician
• Henri Poincaré and his thoughts on the philosophy of science

• Reviews

• 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
• Request Review Copy
Volume: 362010; 391 pp
MSC: Primary 30; 34; 37; 53; 32; 57; 16; 83; 00; 35; 01;

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.

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

• Chapters
• Introduction
• Poincaré and his disk
• Differential equations with algebraic coefficients over arithmetic manifolds
• Poincaré and analytic number theory
• The theory of limit cycles
• Singular points of differential equations: On a theorem of Poincaré
• Periodic orbits of the three body problem: Early history, contributions of Hill and Poincaré, and some recent developments
• On the existence of closed geodesics
• Poincaré’s memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections
• Variations on Poincaré’s recurrence theorem
• Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids
• The concept of “residue" after Poincaré: Cutting across all of mathematics
• The proof of the Poincaré conjecture, according to Perelman
• Henri Poincaré and the partial differential equations of mathematical physics
• Poincaré’s calculus of probabilities
• Poincaré and geometric probability
• Poincaré and Lie’s third theorem
• The Poincaré group
• Henri Poincaré as an applied mathematician
• Henri Poincaré and his thoughts on the philosophy of science
• 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
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