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, 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.

The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set.

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.

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.

These symmetries are grouped into the Poincaré group and internal symmetries and the Coleman – Mandula theorem showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a compact internal symmetry group or if there is no mass gap, the conformal group with a compact internal symmetry group.

However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in.

Then the theory is invariant under the Poincaré group if for every ray Ψ of the Hilbert space and every group element ( a, L ) is given a transformed ray Ψ ( a, L ) and the transition probability is unchanged by the transformation:

The fields are covariant under the action of Poincaré group, and they transform according to some representation S of the Lorentz group, or SL ( 2, C ) if the spin is not integer:

It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice-versa ; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere.

This is because, among other reasons, according to the principle of mass-energy equivalence, if the Earth was absorbing the energy of the ultramundane flux at the rate necessary to produce the observed force of gravity ( i. e. by using the values calculated by Poincaré ), its mass would be doubling in each fraction of a second.

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.

We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum ( corresponding to spacetime translations ) lies on and in the positive light cone.

Specifically, if two manifolds, X and Y, intersect transversely in a manifold M, the homology class of the intersection is the Poincaré dual of the cup product of the Poincaré duals of X and Y.

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 ).

The modern statement of the Poincaré duality theorem is in terms of homology and cohomology: if M is a closed oriented n-manifold, and k is an integer, then there is a canonically defined isomorphism from the k-th cohomology group H < sup > k </ sup >( M ) to the ( n − k ) th homology group H < sub > n − k </ sub >( M ).

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.

is called Poincaré map for orbit γ on the Poincaré section S through point p if

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.

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.

They also engaged the work of contemporary philosophers and scientists, such as Karl Pearson, Ernst Mach, Henri Poincaré, William James and John Dewey in an attempt to move, in the words of Boas ' student Robert Lowie, from " a naively metaphysical to an epistemological stage " as a basis for revising the methods and theories of anthropology.

The intense political and judicial scandal that ensued divided French society between those who supported Dreyfus ( the Dreyfusards ), such as Anatole France, Henri Poincaré and Georges Clemenceau, and those who condemned him ( the anti-Dreyfusards ), such as Hubert-Joseph Henry and Edouard Drumont, the director and publisher of the anti-semitic newspaper La Libre Parole.

The affair saw the emergence of the " intellectuals " – academics and others with high intellectual achievements who took positions on grounds of higher principle – such as Émile Zola, novelists Octave Mirbeau and Anatole France, mathematicians Henri Poincaré and Jacques Hadamard, and Lucien Herr, librarian of the École Normale Supérieure.

A positively curved universe is described by spherical geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold ( such as the Poincaré dodecahedral space ), all of which are quotients of the 3-sphere.

In the late 19th and early 20th centuries, leading mathematicians and scientists such as Hermann von Helmholtz ( 1896 ) and Henri Poincaré ( 1908 ) began to reflect on and publicly discuss their creative processes.

The insights of Poincaré and von Helmholtz were built on in early accounts of the creative process by pioneering theorists such as Graham Wallas and Max Wertheimer.

These are important in that many finite sequences can usefully be interpreted as generating functions, such as the Poincaré polynomial, and others.

It was proposed by, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

" On 22 December 2006, the journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first such recognition in the area of mathematics.

The set of all such translations, rotations, inversions and boosts ( called Poincaré transformations ) forms the Poincaré group.

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.

The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed point theorem in this context.

Edmond took little active part in banking but pursued artistic and philanthropic interests, helping to found scientific research institutions such as the Institut Henri Poincaré, the Institut de Biologie physico-chimique, the pre-Centre National de la Recherche Scientifique, the Casa Velázquez in Madrid, and the French Institute in London.

From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential A < sub > α </ sub > such that

For this reason, in physical applications one considers an action of the supersymmetry algebra on the four fermionic directions of R < sup > 4 | 4 </ sup > such that they transform as a spinor under the Poincaré subalgebra.