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.

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.

The Poincaré conjecture, before being proven, was one of the most important open questions in topology.

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.

This behaviour under CPT is the same as the statement that the particle and its antiparticle lie in the same irreducible representation of the Poincaré group.

The flow now defines a map, the Poincaré map F: S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x < sub > 0 </ sub >.

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.

In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere.

On July 31, 1914, coincidentally the same day that the noted antimilitarist Jean Jaurès was assassinated, Raymond Poincaré signed a decree making Zaharoff a commander of the Legion of Honour.

Jules Henri Poincaré ( 1854 – 1912 ), also a distinguished physicist and mathematician, belonged to another branch of the same family.

They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a " reduction mod p " of a complex variety tend to have the same rank as the singular cohomology groups.

The CPT theorem implies that in a unitary, Poincaré invariant theory, which is a theory in which the S-matrix is a unitary matrix and the same Poincaré generators act on the asymptotic in-states as on the asymptotic out-states, the supersymmetry algebra must contain an equal number of left-handed and right-handed Weyl spinors.

In topology, Poincaré duality also reverses dimensions ; it corresponds to the fact that, if a topological manifold is respresented as a cell complex, then the dual of the complex ( a higher dimensional generalization of the planar graph dual ) represents the same manifold.

After Einstein derived special relativity formally from the ( at first sight counter-intuitive ) assumption that the speed of light is the same to all observers, Hermann Minkowski built on mathematical approaches used in non-euclidean geometry and on the mathematical work of Lorentz and Poincaré.

In the same paper Henri Poincaré ( 1900b ) found another way of combining the concepts of mass and energy.

So Henri Poincaré ( 1898 ) in his paper The Measure of Time drew some important consequences of this process and explained that astronomers, in determining the speed of light, simply assume that light has a constant speed, and that this speed is the same in all directions.

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.

Lorentz and Poincaré had also adopted these same principles, as necessary to achieve their final results, but didn't recognize that they were also sufficient, and hence that they obviated all the other assumptions ( especially the stationary aether ) underlying Lorentz's initial derivations.

On that occasion, he noted that the formal mathematical content of Poincaré paper on the center of mass ( 1900b ) and his own paper were mainly the same, although the physical interpretation was different in light of relativity.

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.

Already in his philosophical writing on time measurements ( 1898 ) Poincaré wrote that astronomers like Ole Rømer, in determining the speed of light, simply assume that light has a constant speed, and that this speed is the same in all directions.

Poincaré went on to note that Rømer also had to assume that Jupiter's moons obey Newton's laws, including the law of gravitation, whereas it would be possible to reconcile a different speed of light with the same observations if we assumed some different ( probably more complicated ) laws of motion.

Lorentz and Poincaré had also adopted these same principles, as necessary to achieve their final results, but didn't recognize that they were also sufficient, and hence that they obviated all the other assumptions underlying Lorentz's initial derivations ( many of which later turned out to be incorrect ).

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 intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0.

Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

After important contributions of Hendrik Lorentz and Henri Poincaré, in 1905, Albert Einstein solved the problem with the introduction of special relativity, which replaces classical kinematics with a new theory of kinematics that is compatible with classical electromagnetism.

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand drawn images of a similar function, which is now called the Koch curve ( see Figure 2 ).

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

( The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations.

As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations ( isomorphisms for instance ) between them.

Like all subatomic particles, hadrons are assigned quantum numbers corresponding to the representations of the Poincaré group: J < sup > PC </ sup >( m ), where J is the spin quantum number, P the intrinsic parity ( or P-parity ), and C, the charge conjugation ( or C-parity ), and the particle's mass, m. Note that the mass of a hadron has very little to do with the mass of its valence quarks ; rather, due to mass – energy equivalence, most of the mass comes from the large amount of energy associated with the strong interaction.

* 1913 – Raymond Poincaré is elected President of France.

* The Poincaré group is a 10 dimensional Lie group of affine isometries of the Minkowski space.

The more general set of transformations that also includes translations is known as the Poincaré group.

Henri Poincaré is regarded as the last mathematician to excel in every field of the mathematics of his time.

From a fundamental physics perspective, mass is the number describing under which the representation of the little group of the Poincaré group a particle transforms.

In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary ( a closed 3-manifold ).