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.

Let ( a, L ) be an element of the Poincaré group ( the inhomogeneous Lorentz group ).

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:

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

Concerning the observables, and states | v ), we get a representation U ( a, L ) of Poincaré group, on integer spin subspaces, and U ( a, A ) of the inhomogeneous SL ( 2, C ) on half-odd-integer subspaces, which acts according to the following interpretation:

The fundamental Poincaré – Birkhoff – Witt theorem gives a precise description of U ( L ); the most important consequence is that L can be viewed as a linear subspace of U ( L ).

* 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

In 2007, he won the Levi L. Conant Prize for his expository paper, " The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations " ( Notices of the AMS 2004 ), and in 2008 he gave the first Levi Conant Lecture at Conant's former employer, the Worcester Polytechnic Institute.

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.

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.

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.

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.

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.

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.

But this issue later was discussed in a more detailed way by Poincaré, who showed that the thermodynamic problem within Le Sage models remained unresolved.

As Maxwell and Poincaré showed, inelastic collisions lead to a vaporization of matter within fractions of a second and the suggested solutions were not convincing.

One example is the planetary movement of three bodies: even if there is no simple solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos.

He showed that the transformations are a consequence of the Principle of Least Action and developed the properties of the Poincaré stresses.

Laue also showed that non-electrical forces are needed to ensure the proper Lorentz transformation properties, and for the stability of matter – he could show that the " Poincaré stresses " ( as mentioned above ) are a natural consequence of relativity theory so that the electron be a closed system.

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 ).

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 Poincaré group is the full symmetry group of any relativistic field theory.

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.

In the spring of 1914 an exceptionally radical chamber was elected, and for a while it seemed that they would be unable to agree upon any one for Premier, but finally, he was appointed Prime Minister on 13 July 1914, by President Poincaré.

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 ).

Poincaré's plans for the seizing of " productive pledges " and the occupation of the Ruhr valley met with the opposition of the other powers, while Poincaré on his side was not to be moved from his designs, which were of a military and political rather than economic character, by any offers of England for a mutual cancellation of debts.

Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring ; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.

This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theories provided one has a Thom Isomorphism for that homology theory.

Early results achieved in discussions with Anatole de Monzie were dismissed by the opposition rallied around Poincaré, and, after being revived by the short-lived cabinet of Édouard Herriot, talks ended without any result.

Recall that any vector space V over a field has a basis ; this is a set S such that any element of V is a unique ( finite ) linear combination of elements of S. In the formulation of Poincaré – Birkhoff – Witt theorem we consider bases of which the elements are totally ordered by some relation which we denote ≤.

More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.

The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to the n-sphere ; it was solved by Stephen Smale in dimensions five and higher, Michael Freedman in dimension 4, and for dimension 3 by Grigori Perelman in 2005.

This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish – Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link.

In his paper Electromagnetic phenomena in a system moving with any velocity smaller than that of light, Lorentz ( 1904 ) was following the suggestion of Poincaré and attempted to create a formulation of Electrodynamics, which explains the failure of all known aether drift experiments, i. e. the validity of the relativity principle.

Eventually, in 1953 Einstein described the advances of his theory ( although Poincaré already stated in 1905 that Lorentz invariance is a general condition for any physical theory ):

Only a few theoretical physicists like Lorentz, Poincaré, Abraham or Langevin, still believed in the existence of an aether in any form.

Ernst Zermelo in his 1908 A new proof of the possibility of a well-ordering presents an entire section " b. Objection concerning nonpredicative definition " where he argued against " Poincaré ( 1906, p. 307 ) states that a definition is ' predicative ' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it ".