Let ( a, L ) and ( b, M ) be two Poincaré transformations, and let us denote their group product by ( a, L ).

Poincaré wrote in the sense of his conventionalist philosophy in 1889: " Whether the ether exists or not matters little-let us leave that to the metaphysicians ; what is essential for us is, that everything happens as if it existed, and that this hypothesis is found to be suitable for the explanation of phenomena.

Poincaré sphere with 6 common types of polarizations labeled

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.

However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology 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.

Poincaré sphere diagram

This space is closely related to the Poincaré sphere, which is the spherical surface occupied by completely polarized states in the space of the vector

Paths taken by vectors in the Poincaré sphere under birefringence.

In general the situation is more complicated and can be characterized as a rotation in the Poincaré sphere about the axis defined by the propagation modes ( this is a consequence of the isomorphism of SU ( 2 ) with SO ( 3 )).

In terms of the Stokes parameters, the total intensity is reduced while vectors in the Poincaré sphere are " dragged " towards the direction of the favored mode.

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.

Examples include the 3-sphere, the Poincaré homology sphere, Lens spaces.

It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen.

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.

* Henri Poincaré discovers the Poincaré sphere, leading him to formulate the Poincaré conjecture.

For regions in R < sup > 3 </ sup > more complicated than this, the latter statement might be false ( see Poincaré lemma ).

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 ).

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.

* Lettre de M Pierre Boutroux a M Mittag-Leffler, in The mathematical heritage of Henri Poincaré 2 ( Providence, R. I., 1983 ), 441-445.

Verdier duality is the appropriate generalization to ( possibly singular ) geometric objects, such as analytic spaces or schemes, while intersection homology was developed R. MacPherson and M. Goresky for stratified spaces, such as real or complex algebraic varieties, precisely so as to generalise Poincaré duality to such stratified spaces.

By the Poincaré lemma, the θ < sup > i </ sup > locally will have the form dx < sup > i </ sup > for some functions x < sup > i </ sup > on the manifold, and thus provide an isometry of an open subset of M with an open subset of R < sup > n </ sup >.

* R. Bott, L. Tu Differential Forms in Algebraic Topology: a classic reference for differential topology, treating the link to Poincaré duality and the Euler class of Sphere bundles

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.

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.

Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map.

These points are a Poincaré section S ( γ, x < sub > 0 </ sub >), of the orbit.

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 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.

Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition.

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.

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.

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.

Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called " self-inverse " fractals.

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.

Historically, the concept of fundamental group first emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Henri Poincaré and Felix Klein, where it describes the monodromy properties of complex functions, as well as providing a complete topological classification of closed surfaces.

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

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.

The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a " grave disease " infecting the discipline of mathematics, and Kronecker's public opposition and personal attacks included describing Cantor as a " scientific charlatan ", a " renegade " and a " corrupter of youth.

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