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.

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.

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.

He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961 ; in 1962 he generalized the ideas in a 107 page paper that established the h-cobordism theorem.

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.

: Poincaré duality then implies that

: Poincaré duality then implies that

The Weil conjectures for the cohomology below the middle dimension follow from this by applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality.

This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.

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 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 ).

One then has a class of models which deviate from Poincaré symmetry near the Planck scale but still flows towards an exact Poincaré group at very large length scales.

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.

The company was founded after World War I, when the then French Prime Minister Raymond Poincaré rejected the idea of forming a partnership with Royal Dutch Shell in favour of creating an entirely French oil company.

Having survived the Holocaust, after the liberation became between 1944 – 1946 chief of researches at the Institut Henri Poincaré at Paris University, then until 1948 worked at the University of London.

From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of complete induction is not provable by general logic.

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

He then studied with Picard, Poincaré, Painlevé, Jordan, Darboux, and Goursat at the Sorbonne in Paris from 1898 to 1900.

He then moved to the Henri Poincaré Institute in Paris to pursue postdoctoral studies on a scholarship granted by the French Government.

A consequence is that the Generalized Poincaré conjecture is true in PL for dimensions greater than four – the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere.

If let's say gravity is an emergent theory of a fundamentally flat theory over a flat Minkowski spacetime, then by Noether's theorem, we have a conserved stress-energy tensor which is Poincaré covariant.

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.

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.

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.

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.

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.

* John Morgan and Gang Tian posted a paper on the arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture ( which is somewhat easier than the full geometrization conjecture ) and expanded this to a book.

John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006, declaring that " in 2003, Perelman solved the Poincaré Conjecture.

In December 2006, the journal Science honored the proof of Poincaré conjecture as the Breakthrough of the Year and featured it on its cover.

Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold.

In November 2002, Russian mathematician Grigori Perelman posted the first of a series of eprints on arXiv outlining a solution of the Poincaré conjecture.

* The Geometry of 3-Manifolds ( video ) A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.

* The slides used by Yau in a popular talk on the Poincaré conjecture.

Conventions which used the Franc Poincaré included the Convention for the Unification of Certain Rules Relating to International Carriage by Air, the International Convention on Civil Liability for Oil Pollution Damage and the International Convention on the Establishment of an International Fund for Compensation for Oil Pollution Damage.

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 ).

In his autobiography, mathematician Gian-Carlo Rota tells of casually browsing the mathematical stacks of Sterling Library and stumbling on a handwritten mailing list, attached to some of Gibbs's course notes, which listed over two hundred notable scientists of his day, including Poincaré, Hilbert, Boltzmann, and Mach.

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.

Other famous problems on his list include the Poincaré conjecture, the P = NP problem, and the Navier-Stokes equations, all of which have been designated Millennium Prize Problems by the Clay Mathematics Institute.

where σ is a 3-vector composed of the Pauli matrices ( used here as generators for the Lie group SL ( 2, C )) and n and m are real 3-vectors on the Poincaré sphere corresponding to one of the propagation modes of the medium.

* Poincaré recurrence theorem, Henri Poincaré's theorem on dynamical systems

Henri Poincaré gave a keynote address on mathematical physics, including an outline for what would eventually became known as special relativity.

Following the armistice with Germany ending the First World War, the French army entered Metz in November 1918 and Philippe Pétain received his marshal's baton from French President Raymond Poincaré and Prime Minister Georges Clémenceau on the Esplanade garden.