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.

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.

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.

According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup.

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

In 1905, Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group,

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.

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.

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.

By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at large times the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a graph manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.

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 ).

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

* 1913 – Raymond Poincaré is elected President of France.

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

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.

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 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 is the only solved Millennium problem.

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.

which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10 < sup >− 6 </ sup > Planck masses.

Rotation of 1 / 10 gives the Poincaré homology sphere, and rotation by 5 / 10 gives 3-dimensional real projective space.

In fact, whether one can smooth certain higher dimensional spheres was, until recently, an open problem — known as the smooth Poincaré conjecture.

Earlier in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher dimensional Poincaré conjecture.

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.

A two dimensional Poincaré section of the forced Duffing equation

In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to the flow of the system.

On higher dimensional complex manifolds, the Poincaré residue is used to describe the distinctive behavior of logarithmic forms along poles.

Poincaré duality constrains the topology of the cell decomposition, and thus the algebra of the Weyl group ; for instance, the top dimensional cell is unique ( it represents the fundamental class ), and corresponds to the longest element of a Coxeter group.

In particular, it has been useful in attempts to classify 3-manifolds and also in proving the higher dimensional Poincaré conjecture.

2m dimensional manifold M. In a neighborhood of each point p of M, by the Poincaré lemma, there is a 1-form θ with dθ = ω.

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

Lie groups are the symmetry groups used in the Standard Model of particle physics ; Point groups are used to help understand symmetry phenomena in molecular chemistry ; and Poincaré groups can express the physical symmetry underlying special relativity.

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.

The Poincaré algebra is the Lie algebra of the Poincaré group.

These are generators of a Lie group called the Poincaré group which is a semi-direct product of the group of translations and the Lorentz group.

The Poincaré polynomials of the compact simple Lie groups are:

In the theory of Lie algebras, the Poincaré – Birkhoff – Witt theorem ( Poincaré ( 1900 ), G. D. Birkhoff ( 1937 ), Witt ( 1937 ); frequently contracted to PBW theorem ) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra.

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group.

Thus a super-Poincaré algebra is a Z < sub > 2 </ sub > graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part.

The even part of the star Lie superalgebra is the direct sum of the Poincaré algebra and a reductive Lie algebra B ( such that its self-adjoint part is the tangent space of a real compact Lie group ).

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.