Henri Poincare : biography
The object of topology was distinctly determined by Felix Klein in his “Erlangen program” of 1872: it said that the object was geometry of invariants of unconditioned continuous transformation; in some relation it could be called qualitative geometry. As for the term “topology” (which was used instead of Analysis situs), it had been used before that by Johann Benedict Listing. Some of the most important concepts were brought in by Enrico Betti and Bernhard Riemann. However, the fundament of the sphere of science was created by Poincare. In addition to that he developed to be detailed enough for the space of any number of measuring. Poincare’s first article on the topic appeared in 1894.
The research in the sphere of geometry brought Poincare to the abstract topological determination of gomotopy and gomology. He was also the first one who brought in the principal concepts and invariants of combinative topology, such numbers as Betti’s number, fundamental group. Later he proved the formulae, which connected the number of edges; apex and ribbing of n-dimensional polyhedron (formulae by Eiler-Poincare).The scientists developed the first exact formulation of intuitive conception of dimension.
Astronomy and celestial mechanics
Poincare published his first classic monographs, which were called “New methods of celestial mechanics” (1892 – 1899) and “Lectures about celestial mechanics” (1905 – 1910). In those articles he successfully applied the results of his research to the problem of about movement of three bodies, having studied its behavior of solutions in details (periodicity, stability, asymptomacity and others). The scientist also brought in methods of minor parameter, immovable points, integral invariants, equations in its variations. Poincare also studied convergence of asymptotic decomposition. Having generalized the theorem of Bruins in 1887, Poincare proved that that the problem of three bodies could not be integrated on principal. In other words, the general solution of the problem of three bodies could not be expressed through algebraic or single-digit transcendent functions of coordinates and speed of bodies. His researching works of that sphere were achievements to be the most important achievements in the sphere of celestial mechanics since the time of Newton.
Those works by Poincare contended ideas, which later become the basic ones for mathematical theory of “chaos”, particularly Poincare’s theorem of recurrence, and general theory of dynamic systems.
In was Poincare who created the most important works about figures of equilibrium of gravitating revolving liquid. The scientist also brought in the important concept of points of bifurcation and proved the existence of figures of equilibrium, which were different from ellipsoid, including ring-shaped and pear-shaped figures. In addition to that the scientist studied its stability. For that discovery Poincare was awarded with the golden medal of London Royal Astronomy society in 1900.
Physics and other spheres of work
Being a member of the bureau of longitude, Poincare took part in measuring works of that organizations and published a few informative works about the problems of geodesy, gravimetry and the theory of flushes.
Since the end of 1880s and till the end of his life, Poincare devoted much time to electromagnetic theory by Maxwell and its advanced variant of Lorentz. He led active correspondence with Enrich Hertz and Lorentz, occasionally giving them good ideas. Particularly, Lorentz’s transformations were issued by Poincare and a more modern type of that, while Lorentz had offered a close variant before. However, Poincare often called them Lorentz’s transformations. Poincare’s contribution to developing the relative theory was also very important.
It was thanks to Poincare that young Antoine Henri Becquerel started to study the connection between phosphorescence and radiography rays in 1896. As the result of that research, the scientists came to the discovery of radioactivity of uranous compounds. Poincare was also the first researcher who output the law of radio waves fading.