Influential Mathematicians: Jacques Hadamard
Last week we covered David Hilbert, the mathematician for whom Hilbert spaces were named after. Today we’ll cover Jacques Hadamard, the mathematician for whom as you can guess the Hadamard gate is named after!
Who was Jacques Hadamard?
The School of Mathematics and Statistics University of St Andrew has a pretty comprehensive bio on Jacques Hadamard that I've summarised for you below.
Jacques Hadamard began his schooling at the Lycée Charlemagne where his father taught. In his first few years at school he was good at all subjects except mathematics. He excelled in particular in Greek and Latin.
In 1884 Hadamard took the entrance examinations for École Polytechnique and École Normale Supérieure; he was placed first in both examinations. He chose the École Normale Supérieure, where he soon made friends with his fellow students including Duhem and Painlevé. Among his teachers were Jules Tannery, Hermite, Darboux, Appell, Goursat and Émile Picard. Already at this stage he began to undertake research, investigating the problem of finding an estimate for the determinant generated by coefficients of a power series. He graduated from the École Normale Supérieure on 30 October 1888.
While undertaking research for his doctorate he worked as a school teacher. At first he was attached to the Lycée de Caen but without teaching duties. From June 1889 he taught at the Lycée Saint-Louis and then from September 1890 at the Lycée Buffon where he taught for three years. Although his research went extremely well, his teaching was less appreciated, probably because he demanded more of his pupils than their abilities allowed. His one great success was teaching Fréchet, and the two corresponded over a period of about nine years.
Hadamard obtained his doctorate in 1892 for a thesis on functions defined by Taylor series. This work on functions of a complex variable was one of the first to examine the general theory of analytic functions, in particular his thesis contained the first general work on singularities. In the same year Hadamard received the Grand Prix des Sciences Mathématiques for his paper ‘Determination of the number of primes less than a given number’. The topic proposed for the prize, concerning filling gaps in Riemann's work on zeta functions, had been put forward by Hermite with his friend Stieltjes in mind. Stieltjes had claimed in 1885 to have proved the Riemann hypothesis but had never published his "proof" and, after the prize topic was announced in 1890, Stieltjes discovered a gap in his "proof" which he was unable to fill. He never submitted an entry for the prize but Hadamard, between the time his thesis was submitted and his oral examination, realised that his results could be applied to zeta functions. His paper on entire functions and zeta functions was awarded first prize.
Hadamard went on to receive the Prix Poncelet in 1898 for his research achievements over the preceding ten years. His research turned more towards mathematical physics from the time he took up the posts in Paris, yet he always argued strongly that he was a mathematician, not a physicist. He believed that it was the rigour of his work which made it mathematics. In particular he worked on the partial differential equations of mathematical physics producing results of outstanding importance. His famous 1898 work on geodesics on surfaces of negative curvature laid the foundations of symbolic dynamics. Among the topics he considered were elasticity, geometrical optics, hydrodynamics and boundary value problems. He introduced the concept of a well-posed initial value and boundary value problem.
Hadamard continued to receive prizes for his research and he was further honoured in 1906 with election as President of the French Mathematical Society. In 1909 he was appointed to the chair of mechanics at the Collège de France. In the following year he published Leçons sur le calcul des variations which helped lay the foundations of functional analysis (he introduced the word functional). Then in 1912 he was appointed as professor of analysis at the École Polytechnique where he succeed Jordan.
Quantum Formalism Influence
Although Hadamard did not directly contribute to the quantum formalism, his seminal work ‘Leçons sur le calcul des variations’ laid the foundations for the likes of Maurice Fréchet (his student!) to start the early development of a branch of pure mathematics now known as functional analysis. This subject is at the heart of the mathematical formalism of quantum mechanics thanks to John von Neumann.
This it for today, I hope you enjoyed this brief biography of one of the greatest mathematicians of 20th century. I’ll end with a recommendation of some influential books written by Hadamard:
‘Lectures on Cauchy's Problem in Linear Partial Differential Equations’
‘The Mathematician's Mind: The Psychology of Invention in the Mathematical Field’
Author: Bambordé, co-founder at Zaiku Group. Please take the survey if you haven’t already done so - we’ll keep it up until the end of this month!
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