## Wednesday, October 31, 2012

### So far yet so close. Unmeasurably happy I found this!

The legacy of Jordan's canonical form on Poincar\'e's algebraic practices. This paper proposes a transversal overview on Henri Poincar\'e's early works (1878-1885). Our investigations start with a case study of a short note published by Poincar\'e on 1884 : "Sur les nombres complexes". In the perspective of today's mathematical disciplines - especially linear algebra -, this note seems completely isolated in Poincar\'e's works. This short paper actually exemplifies that the categories used today for describing some collective organizations of knowledge fail to grasp both the collective dimensions and individual specificity of Poincar\'es work. It also highlights the crucial and transversal role played in Poincar\'e's works by a specific algebraic practice of classification of linear groups by reducing the analytical representation of linear substitution to their Jordan's canonical forms. We then analyze in detail this algebraic practice as well as the roles it plays in Poincar\'e's works. We first provide a micro-historical analysis of Poincar\'e's appropriation of Jordan's approach to linear groups through the prism of the legacy of Hermite's works on algebraic forms between 1879 and 1881. This mixed legacy illuminates the interrelations between all the papers published by Poincar\'e between 1878 and 1885 ; especially between some researches on algebraic forms and the development of the theory of Fuchsian functions. Moreover, our investigation sheds new light on how the notion of group came to play a key role in Poincar\'e's approach. The present paper also offers a historical account of the statement by Jordan of his canonical form theorem. Further, we analyze how Poincar\'e transformed this theorem by appealing to Hermite's.

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.