In 1933 Juliusz Schauder arrived in Paris on a Rockefeller scholarship to work with Hadamard. This led to a collaboration between Leray and Schauder and their joint work led to a paper Topologie et équations fonctionelles published in the Annales scientifiques de l'École normale Supérieure. This 1934 paper on topology and partial differential equations is of major importance:-
In this paper what is now known as Leray-Schauder degree (a homotopy invariant) is defined. This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.After his 1934 paper with Schauder, Leray published a paper on algebraic topology in the following year on the topology of Banach spaces. He then returned to work on analysis, in particular studying differential equations arising from hydrodynamics. He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations. He examined not only the existence and uniqueness of solutions but he showed that the solutions remained smooth for only a finite time after which turbulent solutions arise. In producing this theory Leray introduced many ideas of functional analysis which have today become standard tools.
In 1936 Leray was appointed Professor at the Faculty of Science at Nancy. World War II began in 1939 and Leray served as an army officer. He was captured in 1940 and sent to a prisoner of war camp in Austria where he remained until the end of the war in 1945. While at the camp Leray and some of his fellow captives organised a "université en captivité" and Leray became its rector. Not wishing the Germans to know that he was an expert in hydrodynamics, since he feared that if they found out he would be forced to undertake war work for them, Leray claimed to be a topologist. He worked only on topological problems for the years he was held captive in the camp.
Although he had undertaken some topological work it was not easy for Leray to work on the topic without reading topological literature. He was able to obtain some papers through Hopf who was at this time in Zürich but much of Leray's work was done independently of the developments which had taken place in the subject. After his release in 1945 Leray published a three part work Algebraic topology taught in captivity.
Leray continued to work on topological questions after his return to Paris where he became professor at the Collège de France in 1947. For Leray :-
... algebraic topology should not only study the topology of a space, i.e. algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e. topological invariants of a similar nature for continuous maps.Following this line he published papers which introduced sheaves, and the spectral sequence of a continuous map.
In the 1950s Leray worked in a number of areas. He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem. In particular he published a paper on the Cauchy problem for equations with variable coefficients in 1956. In 1957 he explained the aims of his work in this area:-
We propose to study globally the linear Cauchy problem in the complex case, then in the real hyperbolic case, assuming that the given data is analytic.He was able to generalise results in the theory of ordinary linear analytic differential equations to obtain similar results for partial differential equations. Leray's work on the Cauchy problem led him to study residues theory. In 1959 he :-
... developed a general residue theory on complex manifolds and applied it to the investigation of concrete integrals depending on parameters arising from solving the Cauchy problem.In  Ekeland sums up Leray's achievements as follows:-
Leray was so far ahead of his time because of his tremendous technical capability and geometrical insight. In his hands, energy estimates for partial differential equations became combined with ideas from algebraic topology (such as fixed point theorems) in a highly original combination which cracked open the toughest problems. He was the first to adopt the modern point of view, whereby a function is not a complicated relation between two sets of variables, but a point in some infinite dimensional space ... Leray [can be said to have been] the first modern analyst.Leray received many honours. He was a member of the Academy of Sciences from 1953 and he was elected to the National Academy of Sciences in the United States in 1965. The following year he was elected to the USSR Academy of Sciences. He was also a member of the Royal Academy of Belgium, a fellow of the Royal Society of London, and a member of the academies of Milan, Boston, Göttingen, Turin, Palermo, Warsaw, and Lincei. In 1967 he was awarded an honorary doctorate from the University of Chicago. The citation stated:-
Mathematician of penetration and originality, whose inventions revolutionized partial differential equations and algebraic topology.He was awarded the Malaxa prize in 1938, the Feltrinelli prize in 1971, the Wolf prize in 1979 and the M V Lomonosov Gold Medal in 1988. He was also made Commandeur de la Légion d'Honneur.
We should end this biography with some comments on Leray's lecturing style :-
He was a mild mannered, dapper man with a grey moustache, who squinted at his audience and lost it rather quickly; but he continued to write on the blackboard amidst a respectful silence, confident that the mathematics were there for all to see and needed no further explanation.
Article by: J J O'Connor and E F Robertson