Henri Poincaré

How to use this course

Henri Poincaré published around five hundred papers and some thirty books in a career of thirty-three years. He changed complex analysis, differential equations, celestial mechanics, topology, mathematical physics and the philosophy of science, and he wrote best-selling essays for the general public. No single reader masters all of it. This course takes one lesson per major field and tries to show, in each, the idea that made the field his.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–05 cover his first great work (1880–1884, completed in 1907). Lessons 06–07 follow the road from differential equations to the three-body problem (1881–1899). Lessons 08–09 cover topology (1895–1904), and Lesson 10 relativity (1898–1906). His philosophy appears in Lesson 11.

Timeline

Nancy, 1854–1873

Jules Henri Poincaré was born on 29 April 1854 in the Cité Ducale quarter of Nancy, the capital of Lorraine. His father Léon was a physician and professor in the Nancy faculty of medicine; his mother, Eugénie Launois, gave both her children their early lessons. His younger sister Aline later married the philosopher Émile Boutroux. A cousin, Raymond Poincaré, became prime minister and then President of the Republic during the First World War; another, Lucien, became a senior administrator of French universities.

A bout of diphtheria in childhood left him weak for a time. He was clumsy, short-sighted and ambidextrous, and he had a remarkable memory: he could recall the page and line of something he had read. He entered the lycée in Nancy in 1862 and stayed there eleven years. He did well in every subject and won first prizes in the concours général, the national competition among lycée pupils.

The Franco-Prussian War of 1870–71 brought German occupation to Nancy, and the annexation of neighbouring Alsace and Moselle afterwards. Poincaré learned German during the occupation, which later let him read the German literature directly. He took his baccalauréats in letters and in sciences in 1871 and spent the following year in the mathematics class preparing for the grandes écoles. A story told by his teachers has him solving problems in his head and writing almost nothing down, a habit he kept for life.

Polytechnique, the Mines, the thesis, 1873–1879

In 1873 Poincaré entered the École Polytechnique in Paris in first place. His poor drawing and his clumsiness in physical exercises cost him marks, and he graduated second in 1875. He chose the Corps des Mines, the elite state engineering corps, and studied at the École des Mines from November 1875 to June 1878. He received his engineer’s degree in March 1879 and kept his rank in the Corps for the rest of his life, rising to chief engineer in 1893 and inspector general in 1910.

While a student at the Mines he wrote a doctoral thesis under Charles Hermite, Sur les propriétés des fonctions définies par les équations aux différences partielles, and defended it at the University of Paris in 1879. The examiners found the first results excellent and the rest a little confused, and they recommended the degree. The thesis already contains a theme of his later work: studying solutions of differential equations near their singular points without solving them.

In 1879 he served as an inspector of mines at Vesoul in eastern France. That August he investigated the Magny pit after a firedamp explosion that killed eighteen miners and wrote the official report on its causes. A few months later he left engineering for teaching.

Caen and the Fuchsian functions, 1879–1881

In December 1879 Poincaré became a lecturer in analysis at the University of Caen. In 1880 the Académie des sciences set as the subject of its Grand Prix the improvement of the theory of linear differential equations in one variable. Lazarus Fuchs had recently studied equations whose solutions have special properties under analytic continuation. Poincaré entered the competition, and in the essay and its supplements he began to study the functions obtained by inverting the ratio of two solutions.

In Science et méthode (1908) he described how the key idea arrived. During a geological excursion from Caen he boarded an omnibus at Coutances, and at the moment his foot touched the step he saw that the transformations he had used to define his functions were the same as those of non-Euclidean geometry. He did not check it then, and confirmed it later at leisure. The passage became a classic text on mathematical creativity.

From February 1881 he published a rapid series of notes in the Comptes rendus, naming the new functions fuchsiennes after Fuchs. Felix Klein, who was working on the same circle of ideas in Leipzig, wrote to him in June 1881. Their correspondence over the following year is one of the great mathematical exchanges: courteous, competitive, and decisive for both. Klein objected to the honour given to Fuchs; Poincaré replied by naming another class of groups kleinéens. In 1882 the long memoir Théorie des groupes fuchsiens opened the first volume of Gösta Mittag-Leffler’s new journal Acta Mathematica.

In April 1881 he married Louise Poulain d’Andecy in Caen; they had three daughters and a son. That autumn he moved to the Faculty of Sciences in Paris, where he remained for the rest of his life.

Fuchsian groups and automorphic functions

For the general reader. The sine function repeats when you add \(2\pi\); elliptic functions repeat in two directions, like a wallpaper pattern on the plane. Poincaré found functions that repeat under the symmetries of a tiling of the hyperbolic plane, a world where triangles have angle sums less than \(\pi\). These functions turned out to parametrize every algebraic curve.

The hyperbolic plane

Let \(\mathbb H=\{z\in\mathbb C:\operatorname{Im}z>0\}\) with the metric \(ds=|dz|/\operatorname{Im}z\). The group \(\mathrm{PSL}_2(\mathbb R)\) acts by \(z\mapsto\gamma z=\frac{az+b}{cz+d}\) with \(ad-bc=1\), and these maps are exactly the orientation-preserving isometries. Geodesics are vertical lines and semicircles centred on the real axis. Equivalently one can use the unit disk \(\mathbb D\) with \(ds=2|dz|/(1-|z|^2)\). Both models carry Poincaré’s name.

A Fuchsian group is a discrete subgroup \(\Gamma\subset\mathrm{PSL}_2(\mathbb R)\). Discreteness is equivalent to \(\Gamma\) acting properly discontinuously on \(\mathbb H\). A fundamental domain is a closed region \(F\) whose translates \(\gamma F\) cover \(\mathbb H\) and overlap only along boundaries.

The theorem turns geometry into groups: draw a polygon, pair its sides, check angles, and a discrete group follows. Poincaré stated it in 1882 with versions allowing vertices on the boundary (cusps). The proof constructs the tiling by gluing copies of \(P\) and shows that the result is a covering of \(\mathbb H\), hence all of \(\mathbb H\) because \(\mathbb H\) is simply connected.

Theta series

A meromorphic function \(f\) on \(\mathbb D\) with \(f(\gamma z)=f(z)\) for all \(\gamma\in\Gamma\) is an automorphic function. To build them, Poincaré first built forms that transform with a factor.

Proof sketch

The transformation law is the chain rule: \(\Theta(\delta z)\delta'(z)^m=\sum_\gamma H(\gamma\delta z)\,(\gamma\delta)'(z)^m\), and \(\gamma\mapsto\gamma\delta\) permutes \(\Gamma\). For convergence, fix \(z_0\) and a small Euclidean disk \(U\) around it whose \(\Gamma\)-translates are pairwise disjoint (discreteness). The Euclidean area of \(\gamma U\) is \(\int_U|\gamma'|^2\,dA\), and since \(|\gamma'|^2\) is subharmonic this is at least \(|\gamma'(z_0)|^2\operatorname{area}(U)\). The images are disjoint subsets of \(\mathbb D\), so \(\sum_\gamma|\gamma'(z_0)|^2\le\pi/\operatorname{area}(U)\). The same bound holds uniformly for \(z_0\) in a compact set. Since \(|\gamma'(z)|\le 1/(1-|z|^2)\) by the Schwarz–Pick lemma, the terms with \(m\ge2\) are dominated by the \(m=2\) terms. Poincaré’s argument is essentially this area comparison.

When \(\mathbb D/\Gamma\) is compact of genus \(g\), Poincaré showed that the automorphic functions form a field of algebraic functions of one variable: any two are related by a polynomial equation, and the quotient is the Riemann surface of an algebraic curve. He also showed how the inverse of the ratio of two solutions of a suitable linear differential equation \(y''+p(x)y'+q(x)y=0\) is automorphic, which answered the question he had set out from.

Uniformization

For the general reader. A Riemann surface is any space that looks locally like the complex plane. There are infinitely many of them, of every shape. Poincaré and Klein conjectured, and Poincaré and Paul Koebe proved in 1907, that each one is a quotient of only three model surfaces: the sphere, the plane, and the disk. Geometry on any surface is then spherical, flat or hyperbolic.

Applied to the universal cover, the theorem says that every Riemann surface \(X\) is a quotient \(\tilde X/\Gamma\), with \(\tilde X\) one of the three models and \(\Gamma\cong\pi_1(X)\) a group of automorphisms acting freely and properly discontinuously. The sphere covers only itself. The plane covers \(\mathbb C\), \(\mathbb C^\ast\) and the tori \(\mathbb C/\Lambda\). The disk covers everything else, including every compact surface of genus \(g\ge2\), which therefore has the form \(\mathbb D/\Gamma\) for a Fuchsian group \(\Gamma\).

History

For algebraic curves, Klein and Poincaré announced the result in 1882–83, with arguments based on continuity in the space of all curves. In 1883 Poincaré went further and claimed that any multivalued analytic function could be made single-valued by automorphic functions. The arguments were incomplete. David Hilbert made rigorous uniformization the 22nd of his 1900 problems. Poincaré’s Sur l’uniformisation des fonctions analytiques and Koebe’s papers, both of 1907, settled it.

Proof outline (harmonic functions)

1. Let \(X\) be simply connected and non-compact. Exhaust \(X\) by compact regions \(K_1\subset K_2\subset\cdots\) and fix \(p\in X\). On each \(K_n\) solve the Dirichlet problem for the Green’s function \(G_n\), harmonic on \(K_n\setminus\{p\}\), zero on \(\partial K_n\), with a singularity \(-\log|z|\) at \(p\).
2. If the \(G_n\) converge to a Green’s function \(G\), simple connectivity gives a harmonic conjugate \(G^\ast\), and \(f=\exp(-(G+iG^\ast))\) is a single-valued holomorphic map \(X\to\mathbb D\). One shows that \(f\) is injective and onto.
3. If the \(G_n\) diverge, a similar construction with a dipole singularity produces a biholomorphism onto \(\mathbb C\). In the compact case one obtains a meromorphic function with a single simple pole, which is a biholomorphism onto \(\hat{\mathbb C}\).

Poincaré’s own 1907 proof also used an exhaustion and a limiting process (his méthode de balayage, sweeping out, belongs to the same circle of ideas for the Dirichlet problem).

Curves defined by a differential equation

For the general reader. Most differential equations cannot be solved by formulas. Poincaré proposed a different question: what do all the solutions look like together? Where do they spiral, where do they cross, which ones close up into cycles? This shift from formulas to pictures founded the modern theory of dynamical systems.

The Mémoire sur les courbes définies par une équation différentielle appeared in four parts in the Journal de mathématiques pures et appliquées from 1881 to 1886. Poincaré studied the planar system

with \(P,Q\) polynomials, and projected the plane onto a sphere so that the behaviour at infinity could be studied like any other point.

Singular points

At a zero of \((P,Q)\) let \(A\) be the Jacobian matrix, with trace \(\tau\) and determinant \(\Delta\). Poincaré’s names are still used:

• \(\Delta<0\): a col (saddle);
• \(\Delta>0\), \(\tau^2\ge4\Delta\), \(\tau\ne0\): a nœud (node);
• \(\Delta>0\), \(0<\tau^2<4\Delta\): a foyer (focus);
• \(\Delta>0\), \(\tau=0\): a centre for the linearization. Whether the nonlinear system has a true centre is the subtle centre problem he also studied.

The index of an isolated zero is the number of turns of the vector \((P,Q)\) as one goes once round a small circle about it: \(-1\) for a saddle and \(+1\) for nodes, foci and centres.

Poincaré also proved the analogue on a surface of genus \(p\), with \(2-2p\) on the right. Heinz Hopf extended it in 1926 to manifolds of any dimension: the sum of indices of a vector field equals the Euler characteristic.

Limit cycles and the return map

A limit cycle is a periodic orbit with no other periodic orbit nearby. To study one, Poincaré took a segment \(\Sigma\) crossing the orbit transversally (an arc sans contact) and followed each point of \(\Sigma\) until it first returned to \(\Sigma\). This defines the first-return map or Poincaré map \(\Pi:\Sigma\to\Sigma\). Periodic orbits correspond to fixed points of \(\Pi\); the cycle is attracting when \(|\Pi'(x^\ast)|<1\). For a planar field \(F=(P,Q)\) and a cycle of period \(T\),

Consequently, a compact annulus with no zeros that every orbit enters and none leaves must contain a periodic orbit. Poincaré proved the result for analytic systems; Ivar Bendixson gave the general proof.

The three-body problem and chaos

For the general reader. Is the solar system stable? To win a prize offered by the King of Sweden, Poincaré tried to show that three bodies attracting each other by gravity behave predictably. He found the opposite. Some orbits are so tangled that a tiny change in the starting point leads to a completely different future. This is the first discovery of what is now called chaos.

The competition

In 1885 Gösta Mittag-Leffler announced in Acta Mathematica a prize to mark the sixtieth birthday of King Oscar II of Sweden and Norway. The jury was Mittag-Leffler, Karl Weierstrass and Charles Hermite. The first of four questions asked for series solutions of the \(n\)-body problem converging uniformly for all time. Twelve entries arrived by June 1888. Poincaré’s memoir on the restricted three-body problem won, and the result was announced on the king’s birthday, 21 January 1889. The prize was a gold medal and 2,500 kronor.

While the memoir was being printed, the editor Edvard Phragmén asked questions about its convergence arguments. Working through them, Poincaré found in late November 1889 a far deeper error: he had assumed that certain asymptotic surfaces join smoothly, which implied stability. They need not. The printed copies were recalled and destroyed, and Poincaré paid for the reprinting, more than 3,500 kronor, well above the prize itself. The corrected memoir, Sur le problème des trois corps et les équations de la dynamique, appeared in Acta Mathematica 13 in 1890. He expanded it into the three volumes of Les méthodes nouvelles de la mécanique céleste (1892, 1893, 1899).

Recurrence

Proof. Let \(B=\{x\in A: T^nx\notin A\text{ for all }n\ge1\}\). The sets \(B,T^{-1}B,T^{-2}B,\dots\) are pairwise disjoint: if \(x\in T^{-j}B\cap T^{-k}B\) with \(j<k\), then \(y=T^jx\in B\) and \(T^{k-j}y\in B\subset A\), contradicting the definition of \(B\). They all have measure \(\mu(B)\), and their total measure is at most 1, so \(\mu(B)=0\). Applying this to \(T^k\) for each \(k\) shows that almost every point returns infinitely often. \(\square\)

By Liouville’s theorem the Hamiltonian flow preserves phase-space volume, and on a bounded energy surface the theorem applies. Ernst Zermelo later used it against Ludwig Boltzmann’s derivation of irreversibility.

Integrals and small divisors

The three-body problem has the Hamiltonian

on an 18-dimensional phase space, with ten classical integrals: energy, three components of momentum, three of the uniform motion of the centre of mass, and three of angular momentum. Heinrich Bruns proved in 1887 that there are no further algebraic integrals. Poincaré studied a near-integrable system in action-angle variables, \(H=H_0(I)+\mu H_1(I,\varphi)+O(\mu^2)\), with \(\varphi\) on a torus.

The obstacle is the small divisors. Perturbation series for quasi-periodic motion with frequencies \(\omega=\partial H_0/\partial I\) contain terms \(\frac{h_k}{k\cdot\omega}\), \(k\in\mathbb Z^n\setminus\{0\}\), and \(k\cdot\omega\) can be arbitrarily small. Poincaré showed that the series of Lindstedt used by astronomers are generally divergent, while still useful as asymptotic series. Whether some of them converge was settled by Kolmogorov, Arnold and Moser (KAM, 1954–63): for most Diophantine frequencies, invariant tori survive.

The homoclinic tangle

Reduce the restricted problem to an area-preserving map of a surface section. A hyperbolic fixed point \(p\) has a stable curve \(W^s(p)\) and an unstable curve \(W^u(p)\). In his first memoir Poincaré assumed they coincide. In the correction he showed they can cross transversally at a homoclinic point \(q\ne p\). Then every iterate \(T^nq\) is also homoclinic, and the curves must fold back and forth across one another infinitely often. In the Méthodes nouvelles he wrote that the figure was so complicated he would not attempt to draw it. Stephen Smale’s horseshoe (1960s) made the structure explicit, and it implies sensitive dependence on initial conditions.

In 1912, weeks before his death, Poincaré published his “last geometric theorem”: an area-preserving homeomorphism of an annulus that rotates the two boundary circles in opposite directions has at least two fixed points. He could not prove it in general. George David Birkhoff proved it in 1913.

Analysis Situs

For the general reader. To understand the global behaviour of solutions in many dimensions, Poincaré needed to know the shape of the spaces they live in, and no adequate theory existed. So he created one. In a long memoir of 1895 and five complements he invented the basic tools of algebraic topology: counting holes of each dimension, and recording how loops can be deformed.

Analysis Situs appeared in the Journal de l’École Polytechnique in 1895. Poincaré defined manifolds both as subsets of \(\mathbb R^N\) cut out by equations and by gluing the faces of polyhedra. He called a \(k\)-dimensional submanifold homologous to zero when it bounds a \((k+1)\)-dimensional one, and defined the \(k\)-th Betti number as the maximal number of \(k\)-cycles independent under homology (his own convention counted one more). In modern terms \(b_k=\operatorname{rank}H_k(M;\mathbb Z)\).

The left side depends on the decomposition, the right side only on the space. For polyhedral surfaces this is Euler’s \(V-E+F\).

Poincaré’s first proof, by intersecting cycles transversally, was criticized by the Danish mathematician Poul Heegaard in his 1898 thesis. In the first complement (1899) Poincaré gave a new proof using a triangulation and its dual cell decomposition, and he introduced the torsion coefficients to handle the cycles that bound only when taken several times. The combinatorial dual-cell argument is still the standard intuition.

The fundamental group

In §12 of the 1895 memoir Poincaré defined the groupe fondamental: the group of closed paths from a base point up to continuous deformation, under concatenation. He observed that commuting its elements gives back the first homology. He used it to show that Betti numbers do not determine a 3-manifold, by exhibiting manifolds glued from a cube with the same Betti numbers and different fundamental groups.

The Poincaré conjecture

For the general reader. On the surface of a ball, any rubber band can be shrunk to a point; on a doughnut it cannot. Poincaré asked whether the three-dimensional version of this test is enough to recognise the three-dimensional sphere. The question took ninety-nine years and a new geometric method to answer.

From a mistake to a question

The second complement (1900) ended with a claim: a closed 3-manifold with the Betti numbers and torsion of the sphere is the sphere. In the fifth complement (1904, Rendiconti del Circolo Matematico di Palermo) he refuted his own claim. Using a genus-2 Heegaard diagram he constructed a 3-manifold with the homology of \(S^3\) whose fundamental group is non-trivial, of order 120. It is now called the Poincaré homology sphere. Herbert Seifert and Constantin Weber identified it in 1933 with the quotient of \(S^3\) by the binary icosahedral group, obtained by gluing opposite faces of a dodecahedron with a twist of \(\pi/5\).

Poincaré then asked whether a 3-manifold with trivial fundamental group could fail to be the sphere, and left the question open.

The generalized form asks whether every closed \(n\)-manifold homotopy equivalent to \(S^n\) is homeomorphic to it. Stephen Smale proved this for \(n\ge5\) in 1961 using the h-cobordism method, and Michael Freedman for \(n=4\) in 1982; both received the Fields Medal. Grigori Perelman proved the original case in three arXiv preprints of 2002–03, completing Richard Hamilton’s programme of Ricci flow with surgery, and in fact proved William Thurston’s geometrization conjecture. He declined the Fields Medal in 2006 and the Clay Millennium Prize in 2010.

The dynamics of the electron

For the general reader. Around 1900 physicists could not detect the Earth’s motion through the supposed ether, and Hendrik Lorentz had invented transformations to explain why. Poincaré argued that simultaneity at a distance is a convention, stated a general principle of relativity, and in 1905 showed that Lorentz’s transformations form a group leaving the equations of electromagnetism unchanged. Albert Einstein reached the theory independently the same year, from different premises.

The path

• 1898. La mesure du temps (Revue de métaphysique et de morale): we have no direct intuition of the simultaneity of distant events; it is defined by conventions chosen for convenience.
• 1900. In a paper for the Lorentz jubilee volume of the Archives néerlandaises, he interpreted Lorentz’s “local time” as the time read by moving clocks synchronized by light signals, and discussed the momentum carried by electromagnetic radiation, which behaves like a fluid of mass density \(\varepsilon/c^2\) for energy density \(\varepsilon\).
• 1904. At the Congress of Arts and Science in St. Louis he stated the principle of relativity: the laws of physics must be the same for a stationary observer and for one in uniform translation.
• 1905. On 5 June the note Sur la dynamique de l’électron was presented to the Académie (Comptes rendus 140); the full memoir appeared in the Rendiconti del Circolo Matematico di Palermo 21 (1906).

The Lorentz group

In units with \(c=1\), Poincaré wrote the transformations as

He observed that with \(t\sqrt{-1}\) as a fourth coordinate the transformations are rotations in a four-dimensional space, two years before Hermann Minkowski developed the space-time geometry. He also found the transformation law for velocities, added non-electromagnetic stresses (the Poincaré stresses) to keep Lorentz’s deformable electron in equilibrium, and suggested that gravitation should propagate with the speed of light as ondes gravifiques, gravitational waves.

Poincaré and Einstein

Einstein’s paper of 1905, received on 30 June, derived the same transformations from two postulates and treated the new times and lengths as the physically real ones. Poincaré kept the ether as an undetectable but convenient reference and distinguished true from apparent time. He never cited Einstein’s relativity papers, and Lorentz, who credited Poincaré’s mathematical completion of his theory, also wrote that the theory of relativity was Einstein’s work. Historians still debate the balance; the mathematics of the Lorentz group is unambiguously Poincaré’s. The inhomogeneous group including translations is now called the Poincaré group.

Paris, the Academies and philosophy of science

At the Paris Faculty of Sciences Poincaré held three chairs in turn: physical and experimental mechanics (1885), mathematical physics and probability (1886, succeeding Gabriel Lippmann) and mathematical astronomy and celestial mechanics (1896, succeeding Félix Tisserand). He also taught analysis at the École Polytechnique (1883–97). He changed the topic of his course almost every year, from optics and electricity to elasticity, thermodynamics, capillarity, potential theory and probability, and students published the lectures as a long series of volumes. He joined the Bureau des Longitudes in 1893 and worked on geodesy and the international standardization of time.

He kept a regular routine, working on mathematics from ten to noon and from five to seven, and did not plan his papers in advance. He founded no school, and his formal doctoral students were few; they included Louis Bachelier, whose 1900 thesis on speculation introduced Brownian motion into finance, and Dimitrie Pompeiu (1905).

He entered public life when science was misused. In the Dreyfus affair he wrote to the court-martial at Rennes in 1899 criticizing the pseudo-probabilistic analysis of the bordereau, and in 1904, at the request of the Court of Cassation, he signed with Gaston Darboux and Paul Appell a report that demolished it. The report was used in the 1906 rehabilitation of Alfred Dreyfus.

Philosophy of science

Poincaré’s essays, collected in La science et l’hypothèse (1902), La valeur de la science (1905) and Science et méthode (1908), sold widely and are still read. Their central themes:

• Conventionalism in geometry. Experiment cannot decide whether physical space is Euclidean, because any observation can be accommodated by changing physics instead of geometry. The choice of geometry is a convention, guided by convenience, and Euclidean geometry is the most convenient. He wrote that one geometry cannot be more true than another, only more convenient.
• Structural realism. Theories change, but the relations they capture survive. Fresnel’s and Maxwell’s optics disagree about what light is and agree on its equations.
• Induction and intuition. Mathematical induction is a synthetic judgement a priori, which logic cannot reduce. Against the logicism of Bertrand Russell and Louis Couturat he insisted on the role of intuition, and he blamed the paradoxes of set theory on impredicative definitions, which define an object by reference to a totality that contains it.
• Invention. In a 1908 lecture he described discovery as unconscious work followed by sudden illumination and conscious verification, with the Coutances omnibus as his example.

Last years, death and legacy, 1906–1912

Honours accumulated. Poincaré was elected to the Académie des sciences in 1887 and served as its president in 1906. He became a Fellow of the Royal Society in 1894 and received its Sylvester Medal in 1901, the Gold Medal of the Royal Astronomical Society in 1900, the Bolyai Prize and the Matteucci Medal in 1905, and the Bruce Medal of the Astronomical Society of the Pacific in 1911. On 5 March 1908 he was elected to the Académie française, a rare honour for a scientist. He was nominated many times for the Nobel Prize in Physics without receiving it.

In 1908 he fell ill at the International Congress of Mathematicians in Rome, and Gaston Darboux read his address on the future of mathematics. The illness was a prostate condition. He kept working at full speed. In 1911 he attended the first Solvay Conference in Brussels, where the problem of quanta was debated. His 1912 paper Sur la théorie des quanta proved that Planck’s radiation law requires a genuine discontinuity in energy, an argument that helped convince physicists that the quantum hypothesis was unavoidable.

He underwent surgery in July 1912. He seemed to be recovering, then died of an embolism in Paris on 17 July 1912, aged 58. After a funeral at Saint-Jacques-du-Haut-Pas he was buried in the family vault at the Montparnasse cemetery.

Legacy

• Dynamical systems: the qualitative method, return maps, recurrence, homoclinic chaos and small divisors led through Birkhoff, Kolmogorov, Arnold, Moser and Smale to modern ergodic theory and chaos theory.
• Topology: homology, duality and the fundamental group are the foundation of algebraic topology. His conjecture drove 3-manifold topology for a century, through Thurston to Perelman.
• Complex analysis and number theory: Fuchsian groups, Poincaré series and uniformization underlie the theory of modular and automorphic forms and Teichmüller theory.
• Physics: the Lorentz group and the Poincaré group are the symmetry groups of relativistic field theory; his quantum argument of 1912 is a landmark of early quantum theory.
• Memorials: the Institut Henri Poincaré in Paris, founded in 1928; a lunar crater; the Henri Poincaré Prize of the International Association of Mathematical Physics; the former Université Henri-Poincaré in his native Nancy.

Connections

People linked to Henri Poincaré across this collection. A name without a link has no page here yet. See Henri Poincaré in the network of all 337 lives →

Teachers and advisors

• Charles Hermite 1879

Students

• Dimitrie Pompeiu 1905
• Louis Bachelier 1900

Influenced by

• Bernhard Riemann
• Eugenio Beltrami
• János Bolyai
• Nikolai Lobachevsky
• Pierre-Simon Laplace
• Lazarus Fuchs 1880-1881

Influenced

• Albert Einstein
• Heinz Hopf
• Henri Becquerel 1896
• Hermann Minkowski
• Ivar Fredholm 1899
• Jacques Hadamard
• James Jeans
• L. E. J. Brouwer
• Louis de Broglie
• Paul Ehrenfest
• Solomon Lefschetz
• William Thurston
• Élie Cartan
• Émile Borel
• George David Birkhoff
• Ivar Bendixson 1901

Colleagues and collaborators

• Albert Einstein 1905-1912
• David Hilbert
• Gaston Darboux 1904
• Gösta Mittag-Leffler 1882-1912
• Hendrik Lorentz 1895-1912
• Karl Weierstrass 1885-1889
• Ludwig Boltzmann
• Paul Langevin
• Paul Lévy
• Sofia Kovalevskaya
• Thomas Joannes Stieltjes 1886-1894
• Edvard Phragmén 1889
• Paul Appell 1904
• Poul Heegaard 1898-1899

Rivals

• Aleksandr Lyapunov
• Bertrand Russell
• Felix Klein 1881-1882
• Paul Koebe 1907

Family

• Pierre Boutroux
• Émile Boutroux

Also mentioned in this life

• Isaac Barrow
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Joseph Fourier
• Augustin-Jean Fresnel
• Carl Friedrich Gauss
• William Rowan Hamilton
• Andrey Kolmogorov
• Joseph Liouville
• James Clerk Maxwell
• John Milnor
• August Ferdinand Möbius
• Max Planck
• Gregorio Ricci-Curbastro
• Oswald Teichmüller

Fields

• complex analysis 13 lives
• differential equations 13 lives
• celestial mechanics 16 lives
• dynamical systems 7 lives
• algebraic topology 6 lives
• mathematical physics 45 lives
• philosophy of science 7 lives

On CoursesHub

• Hilbert & Poincaré · History of Mathematics
• Complex Analysis · Mathematics for Physics
• ODEs / PDEs / Green Functions · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Henri Poincaré, MacTutor History of Mathematics.
2. Henri Poincaré, Wikipedia; Henri Poincaré, Wikipédia (French).
3. J. Gray, Henri Poincaré: A Scientific Biography, Princeton University Press, 2013.
4. F. Verhulst, Henri Poincaré: Impatient Genius, Springer, 2012.
5. Institut Mittag-Leffler, The prize competition of King Oscar II.
6. J. Barrow-Green, Poincaré and the Three Body Problem, AMS/LMS History of Mathematics 11, 1997.
7. Archives Henri Poincaré, Nancy: the Poincaré correspondence project.

Mathematics and physics

1. H. Poincaré, Théorie des groupes fuchsiens, Acta Math. 1 (1882) 1–62.
2. H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, J. Math. Pures Appl. (1881–1886).
3. H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math. 13 (1890) 1–270.
4. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, 3 vols., Gauthier-Villars, 1892–1899.
5. H. Poincaré, Analysis Situs, J. École Polytechnique (2) 1 (1895); English translation in Papers on Topology, trans. J. Stillwell, AMS, 2010.
6. H. Poincaré, Cinquième complément à l’Analysis Situs, Rend. Circ. Mat. Palermo 18 (1904).
7. H. Poincaré, Sur la dynamique de l’électron, C. R. Acad. Sci. Paris 140 (1905); Rend. Circ. Mat. Palermo 21 (1906). Text on Wikisource.
8. H. Poincaré, Sur l’uniformisation des fonctions analytiques, Acta Math. 31 (1907).
9. H. de Saint-Gervais, Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem, EMS, 2016.
10. Poincaré conjecture, Wikipedia; J. Milnor, The Poincaré conjecture, Clay Mathematics Institute problem description.
11. Relativity priority dispute, Wikipedia; S. Walter, Breaking in the 4-vectors (2007), and O. Darrigol, The genesis of the theory of relativity, Séminaire Poincaré 1 (2005).
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