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.

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 ).

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 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 ).

The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold ( up to homeomorphism ) with these properties.

Predicativism was first studied in detail by Henri Poincaré and Hermann Weyl in Das Kontinuum, where they showed that much of elementary real analysis can be conducted in a predicative manner starting with only the natural numbers.

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.

In 1917, an orangutan escaped from a nearby ménagerie, entered the palace and was said to have tried to haul the wife of President Raymond Poincaré into a tree only to be foiled by Élysée guards.

In 1911, he began a series of important conferences in physics, known as the Solvay Conferences, whose participants included luminaries such as Max Planck, Ernest Rutherford, Marie Curie, Henri Poincaré, and ( then only 32 years old ) Albert Einstein.

and more generally linear recursive sequences are exactly the sequences generated by rational functions ; thus the Poincaré series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursive sequence.

The Poincaré lemma states that if X is a contractible open subset of R < sup > n </ sup >, any smooth closed p-form α defined on X is exact, for any integer p > 0 ( this has content only when p ≤ n ).

Some of the names remain familiar to the average reader: Kaiser Wilhelm II of Germany, French President Raymond Poincaré, Britain's First Lord of the Admiralty Winston Churchill, and a young soldier named Charles de Gaulle who fought for France ( given only honorable mention ), among others.

Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEV's.

For example, it contains a chapter named " The Relativity Theory of Poincaré and Lorentz ", where Whittaker credited Henri Poincaré and Lorentz for developing special relativity, and he attributed to Albert Einstein's relativity paper only little importance.

Lattice field theory differs from these in that it keeps manifest gauge invariance, but sacrifices manifest Poincaré invariance — recovering it only after renormalization.

William Browder ( 1962 ) proved that a simply-connected compact polyhedron with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.

The above examples only include the works of Poincaré, and yet Brouwer named other mathematicians as Pre-Intuitionists too ; Borel and Lebesgue.

It lies to the west of the walled plain Poincaré, another enormous formation only slightly larger than Planck.

The only conserved quantities in a " realistic " theory with a mass gap, apart from the generators of the Poincaré group, must be Lorentz scalars.

Every quantum field theory satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible.

For example, with a value of 4, there is only one point on the Poincaré map, because the function yields a periodic orbit of period one, or if the value is set to 12. 8, there would be six points corresponding to a period six orbit.

The French Premier Raymond Poincaré was deeply reluctant to order the Ruhr occupation and took this step only after the British had rejected his proposals for more moderate sanctions against Germany.

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.

where is a p-form in n-space and S is the p-dimensional boundary of the ( p + 1 )- dimensional region T. Goursat also used differential forms to state the Poincaré lemma and its converse, namely, that if is a p-form, then if and only if there is a ( p − 1 )- form with

However, Lorentz's local time was only an auxiliary mathematical tool to simplify the transformation from one system into another – it was Poincaré in 1900 who recognized that " local time " is actually indicated by moving clocks.

Poincaré and Weyl argued that impredicative definitions are problematic only when one or more underlying sets are infinite.