In fact, whether one can smooth certain higher dimensional spheres was, until recently, an open problem — known as the smooth Poincaré conjecture.

Newman made important contributions leading to an invitation to present his work at the 1962 International Congress of Mathematicians in Stockholm at the age of 65, and proved a Generalized Poincaré conjecture for topological manifolds in 1966.

The Poincaré conjecture asserts that the same is true for 3-dimensional spaces.

In mathematics, the Poincaré conjecture ( ; ) is a theorem about the characterization of the three-dimensional sphere ( 3-sphere ), which is the hypersphere that bounds the unit ball in four-dimensional space.

The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.

On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of Hamilton's ( who first suggested using the Ricci flow for the solution ).

The Poincaré conjecture is the only solved Millennium problem.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first time this had been bestowed in the area of mathematics.

Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture.

In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere .< ref > Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.

For dimensions greater than three, one can pose the Generalized Poincaré conjecture: is a homotopy n-sphere homeomorphic to the n-sphere?

In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem.

In 1982 Michael Freedman proved the Poincaré conjecture in dimension four.

This so-called smooth Poincaré conjecture, in dimension four, remains open and is thought to be very difficult.

Milnor's exotic spheres show that the smooth Poincaré conjecture is false in dimension seven, for example.

The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons.

Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture.

In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.

As explained before, this theory is not a violation of Poincaré symmetry as much as a deformation of it and there is an exact de Sitter symmetry.

This consequently solved in the affirmative the Poincaré conjecture, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology.

Raymond Poincaré, Aristide Briand and Tardieu before him had offered ministerial posts to Herriot's Radicals, but to no avail.

After a very brief crisis Raymond Poincaré presented himself before Parliament with a new cabinet containing several members of the previous one.

It was almost twenty years later, when quantum mechanics was formulated in terms of the Schrödinger equation, that the connection was made to atomic spectra ; a connection with the mathematical physics of vibration had been suspected before, as remarked by Henri Poincaré, but rejected for simple quantitative reasons, absent an explanation of the Balmer series.

Ton-That and Tran conclude that Poincaré had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff.

However, Max von Laue quickly rebutted those claims by saying that the inertia of electromagnetic energy was long known before Hasenöhrl, especially by the works of Henri Poincaré ( 1900 ) and Max Abraham ( 1902 ), while Hasenöhrl only used their results for his calculation on cavity radiation.

To better understand Einstein's step, a summary of the situation before 1905, as it was described above, shall be given ( it must be remarked that Einstein was familiar with the 1895 theory of Lorentz, and " Science and Hypothesis " by Poincaré, but not their papers of 1904-1905 ):

So according to Darrigol Poincaré understood local time as a physical effect just like length contraction-in contrast to Lorentz, who used the same interpretation not before 1906.

Since the space is then a pseudo-Euclidean space, the rotation is a representation of a hyperbolic rotation, although Poincaré did not give this interpretation, his purpose being only to explain the Lorentz transformation in terms of the familiar Euclidean rotation.

The suggestion of a secondary re-radiation mechanism for wave models attracted the interest of JJ Thomson, but was not taken very seriously by either Maxwell or Poincaré, because it entails a gross violation of the second law of thermodynamics ( huge amounts of energy spontaneously being converted from a colder to a hotter form ), which is one of the most solidly established of all physical laws.

Early results achieved in discussions with Anatole de Monzie were dismissed by the opposition rallied around Poincaré, and, after being revived by the short-lived cabinet of Édouard Herriot, talks ended without any result.

It was the first entirely state-run oil company in the world, the second being the French Compagnie française des pétroles ( CFP, French Company of Petroleum ), created in 1924 by the conservative Prime Minister Raymond Poincaré.

Some founders of the theory are considered to be Isaac Newton and Bernhard Riemann, with main contributors being Niels Henrik Abel, Henri Poincaré, Max Noether, among others.

In a September 1904 lecture in St. Louis named The Principles of Mathematical Physics, Poincaré draw some consequences from Lorentz's theory and defined ( in modification of Galileo's Relativity Principle and Lorentz's Theorem of Corresponding States ) the following principle: " The Principle of Relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion.

The same opinion was expressed in the Chamber of Deputies by the deputies Bos, Millerand, and Poincaré, the latter being one of the ministers of 1894 who took advantage of this opportunity to " unburden his conscience.

* The Poincaré theorem ( proven by Grigori Perelman )

The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf.

Chaos theory and the sensitive dependence on initial conditions was described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890.

It was in the work of Poincaré that these dynamical systems themes developed.

The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive — even shocking — that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections.

In 1905, Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group,

The work of Lorentz was mathematically perfected by Henri Poincaré who formulated on many occasions the Principle of Relativity and tried to harmonize it with electrodynamics.

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology.

Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere.

The Poincaré sphere was the first example of a homology sphere, a manifold that had the same homology as a sphere, of which many others have since been constructed.

To establish that the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new topological invariant, the fundamental group, and showed that the Poincaré sphere had a fundamental group of order 120, while the 3-sphere had a trivial fundamental group.

The Franc Poincaré is a unit of account that was used in the international regulation of liability.

Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students ( for example in the Banach – Schauder theorem ) and other mathematicians ( in particular Bouwer and Poincaré and Birkhoff ).

The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux and Eugene Parker.

On July 24 French prime minister Raymond Poincaré resigned for medical reasons ; he was succeeded by Aristide Briand.

One of the major innovations made by the Salon Cubists, independently of Picasso and Braque, was that of simultaneity, drawing to greater or lesser extent on theories of Henri Poincaré, Ernst Mach, Charles Henry, and Henri Bergson.

Earlier in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher dimensional Poincaré conjecture.