Hardcover ISBN:  9780821847183 
Product Code:  HMATH/36 
List Price:  $100.00 
MAA Member Price:  $90.00 
AMS Member Price:  $80.00 
Electronic ISBN:  9781470418076 
Product Code:  HMATH/36.E 
List Price:  $94.00 
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Book DetailsHistory of MathematicsVolume: 36; 2010; 391 ppMSC: 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.ReadershipUndergraduate students, graduate students, and research mathematicians interested in Poincaré's life and work.

Table of Contents

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

Lowdimensional 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


Additional Material

Reviews

The articles are very well written, indeed, and are of course autonomous. But even nonspecialists 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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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

Lowdimensional 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 nonspecialists 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