University of David
He spent the year 1968–1969 as a Sloan Fellow at the IHES near Paris, where Grothendieck was based. The following year, spent at the Institute for Advanced Study in Princeton, was the most fertile of his life, and he produced a torrent of new work. Perhaps the most exciting development at the time was a proof of the Adams conjecture, which identifies—in terms of K-theory and its Adams operations—the direct summand in the stable homotopy groups of spheres which comes from the orthogonal groups. Quillen had already given an outline proof of this three years earlier, showing how it follows from the expected properties of Grothendieck’s étale homotopy theory for algebraic varieties in characteristic p.2 Meanwhile, however, he had been carefully studying the work of the algebraic topologists centered in Chicago, who had used ideas of infinite loop space theory to calculate the homology of many important classifying spaces. He now realized that the crucial idea of his first proof amounted to saying that the classifying spaces of the discrete group GLn(¯Fp) and of the Lie group GLn(C) have the same homology away from the prime p, and that this could be proved directly. (Here ¯Fp denotes the algebraic closure of the field with p elements.) This led straight to his development of algebraic K-theory, which is the achievement he is nowmost remembered for; but before coming to that I shall mention a few other things.