Alexander Grothendieck (1928–2014) Mathematician Who Rebuilt Algebraic Geometry

Home , Alexander Grothendieck, Fields Medal

COMMENT OBITUARY Alexander Grothendieck (1928–2014) Mathematician who rebuilt algebraic geometry. S

lexander Grothendieck, who died his discovery of how all schemes have a É on 13 November, was considered by topology. Topology had been thought to many to be the greatest mathemati- belong exclusively to real objects, such as Acian of the twentieth century. His unique spheres and other surfaces in space. But skill was to burrow into an area so deeply that Grothendieck found not one but two ways its inner patterns on the most abstract level to endow all schemes, even the discrete ones, revealed themselves, and solutions to old with a topology, and especially with the REGEMORTER/IH H. VAN problems fell out in straightforward ways. fundamental invariant called cohomology. Grothendieck was born in Berlin in 1928 With a brilliant group of collaborators, he to a Russian Jewish father and a German gained deep insight into theories of coho- Protestant mother. After being separated mology, and established them as some of from his parents at the age of five, he was the most important tools in modern math- briefly reunited with them in France just ematics. Owing to the many connections before his father was interned and then trans- that schemes turned out to have to various ported to Auschwitz, where he died. Around mathematical disciplines, from algebraic 1942, Grothendieck arrived in the village of geometry to number theory to topology, Le Chambon-sur-Lignon, a centre of resist- there can be no doubt that Grothendieck’s ance against the Nazis, where thousands of work recast the foundations of large parts of refugees were hidden. It was probably here, twenty-first-century mathematics. at the secondary school Collège Cévonol, that Grothendieck left the IHÉS in 1970 for his fascination for mathematics began. reasons not entirely clear to anyone. He In 1945, Grothendieck enrolled at the turned from maths to the problems of envi- University of Montpellier. He completed ronmental protection, founding the activist his doctoral thesis on topological vector group Survivre. With a breathtakingly naive spaces at the University of Nancy in 1953, spirit (that had served him well in math- and spent a short time teaching in Brazil. enormously, Grothendieck made his first ematics), he believed that this movement His most revolutionary work happened hugely significant innovation. He proposed could change the world. When he saw that it between 1954 and 1970, mainly at the Insti- that a geometric object called a scheme was was not succeeding, in 1973, he returned to tute of Advanced Scientific Studies (IHÉS) in associated to any commutative ring — that is, maths, teaching at the University of Montpel- a suburb of Paris. His strength and dedica- a set in which addition and multiplication are lier. Despite writing thousand-page treatises tion were legendary: throughout his 15 years defined and multiplication is commutative, on yet-deeper structures connecting alge- in mainstream mathematics, he would work a × b = b × a. Before Grothendieck, mathema- bra and geometry (still unpublished), his long hours in the unheated attic of his house ticians considered only the case in which the research was only meagrely funded by the seven days a week. He was awarded the ring is the set of functions on the variety that CNRS, France’s main basic-research agency. Fields medal in 1966 for his work in alge- are expressible as polynomials in the coordi- Grothendieck could be very warm. Yet the braic geometry. nates. In any geometry, local parts are glued nightmares of his childhood had made him a Algebraic geometry is the field that studies together in some fashion to create global complex person. He remained on a Nansen the solutions of sets of polynomial equations objects, and this worked for schemes too. passport his whole life — a document issued by looking at their geometric properties. An example might help in illustrating for stateless people and refugees who could For instance, a circle is the set of solutions of how novel this idea was. A simple ring can not obtain travel documents from a national x2 + y2 = 1, and in general such a set of points be generated if we make a ring from expres- authority. For the last two decades of his life is called a variety. Traditionally, algebraic sions a + bε, in which a and b are ordinary real he broke off from the maths community, geometry was limited to polynomials with numbers but ε is a variable with only ‘very his wife, a later partner and even his chil- real or complex coefficients, but just before small’ values, so small that we decide to set dren. He sought total solitude in the village Grothendieck’s work, André Weil and Oscar ε2 = 0. The scheme corresponding to this ring of Lasserre in the foothills of the Pyrenees. Zariski had realized that it could be con- consists of only one point, and that point is Here he wrote remarkable self-analytical nected to number theory if you allowed the allowed to move the infinitesimal distance ε works on topics ranging from maths and polynomials to have coefficients in a finite but no further. The possibility of manipulat- philosophy to religion. ■ field. These are a type of number that are ing infinitesimals was one great success of added like the hours on a clock — 7 hours schemes. But Grothendieck’s ideas also had David Mumford is professor emeritus after 9 o’clock is not 16 o’clock, but 4 o’clock important implications in number theory. at Harvard University in Cambridge, — and it creates a new discrete type of variety, The ring of all integers, for example, defines Massachusetts, and at Brown University in one variant for each prime number p. a scheme that connects finite fields to real Providence, Rhode Island, USA. John Tate But the proper foundations of this enlarged numbers, a bridge between the discrete and is professor emeritus at Harvard University, view were unclear, and this is where, inspired classical worlds, having one point for each and at the University of Texas at Austin. by the ideas of the French mathematician prime number and one for the classical world. e-mails: [email protected]; Jean-Pierre Serre, but generalizing them Probably his best-known work was [email protected]