Born to Jewish parents, he was a displaced person during much of his childhood due to the upheavals of the war. His father died in Auschwitz. After the war, young Grothendieck studied mathematics in France, writing his dissertation under Laurent Schwartz in functional analysis. He was at this time a leading expert in topological vector spaces. However he set this subject aside by 1957 in order to work in algebraic geometry and homological algebra.
Homological methods and sheaf theory had already been introduced in algebraic geometry by Leray and by Serre, but Grothendieck took them to a higher level. Among his insights, he shifted attention from the study of varieties to pairs of varieties related by a morphism, allowing a great generalization of many classical theorems such as the Riemann Roch theorem (already recently generalized to higher dimension by Hirzebruch); the introduction of non-closed points, which led to the theory of schemes; and the systematic use of nilpotents. His work is at a higher level of abstraction than prior versions of algebraic geometries, but due to its great power his theory of schemes has become established as the best foundations for this important field.
Perhaps Grothendieck's deepest accomplishment is the invention of the etale and l-adic cohomology theories, which explain an observation of Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers. Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck. This program culminated in the proofs of the Weil conjectures by Deligne in the early 1970's after Grothendieck had withdrawn from mathematics.
Grothendieck's radical left-wing and pacifist politics were doubtless born of his wartime experiences. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam war. He declined a scientific prize, stating that the money would be better spent on young mathematicians. He retired from scientific life around 1970. He is said to live in the Pyrenees, a Buddhist, and to entertain no visitors. Though he has been inactive for many years, he remains one of the greatest and most influential mathematicians of modern times.
- MacTutor biography of Grothendieck