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Concept# Henri Poincaré

Summary

Jules Henri Poincaré (UKˈpwæ̃kɑreɪ, ; ɑ̃ʁi pwɛ̃kaʁe; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime.
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.
Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the

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Related lectures (37)

The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

This research work analyses Theo van Doesburg's Counter-constructions presented in Paris in 1923 at the "L'Effort Moderne" exhibition. These Counter-constructions stand as icons of the Modern Movement due to their role as precursors of a new "boundless" spatiality, a role first intuitively perceived by Le Corbusier or Mies van der Rohe and then theorized by Sigfried Giedion. We have attempted in this research a theoretical study of the Counter-constructions' position within the architectural field. That is to say bringing together their aspect of "spatial manifest" and the Dadaist persona of Theo van Doesburg; this alleged contradiction between the progressive and negative dimension of the Counter-constructions being the subject of this research. It is based on the following problematic: understanding what is at stake in the disjointing of the polychrome plane (as a textile surface) and the suppression of the architectural boundary under the guise of spatial continuity ("the breaking of the enclosure" mentioned by Theo van Doesburg and his interest in the notion of mathematical continuum through the theories of the fourth dimension: Henri Poincaré's Analysis situs). That amounts to studying the modalities connecting the notions of dressing and of continuum to a destructive drive, "throwing out" the inside of the house. The methodology turns towards psychoanalysis through the concepts of Verneinung [negation] (Freud) and Moi-peau [Skin-ego] (Anzieu).

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar, equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar, equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, mu CH and mu DP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.