Alexander Grothendieck

Alexander Grothendieck ( ; ; born 28 March 1928) is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations. This new perspective led to revolutionary advances across many areas of pure mathematics.

Within algebraic geometry itself, his theory of schemes has become the universally accepted language for all further technical work. His generalization of the classical Riemann-Roch theorem launched the study of algebraic and topological K-theory. His construction of new cohomology theories has left deep consequences for algebraic number theory, algebraic topology, and representation theory. His creation of topos theory has had an impact on set theory and logic.

One of his most celebrated achievements is the discovery of the first arithmetic Weil cohomology theory: the ℓ-adic étale cohomology. This key result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre Deligne. To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands program.

Grothendieck’s way of thinking has influenced generations of mathematicians long after his departure from mathematics. His emphasis on the role of universal properties brought category theory into the mainstream as an important organizing principle. His notion of abelian category is now the basic object of study in homological algebra. His conjectural theory of motives has been a driving force behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration.

Driven by deep personal and political convictions, Grothendieck left the Institut des Hautes Études Scientifiques, where he had been appointed professor and accomplished his greatest work, after a dispute over military funding in 1970. His mathematical activity essentially ceased after this, and he devoted his energies to political causes. He formally retired in 1988 and within a few years moved to the Pyrenees, where he currently lives in isolation from human society.