Alexander Grothendieck

views updated

Alexander Grothendieck


French Mathematician

Alexander Grothendieck is regarded by many as one of the preeminent mathematicians of the twentieth century. He is credited with establishing a new school of algebraic geometry, and his work garnered a Fields Medal in 1966 for advancement of K-theory (Grothendieck groups and rings).

Born in Berlin, Grothendieck emigrated to France 1941. He earned his doctorate at the University of Nancy in 1953, and thereafter served in several academic posts around the world, including Harvard, before returning to France. Although he concentrated his early efforts on advances in functional analysis, during his international travels he shifted the emphasis of his work and subsequently made substantial contributions to topology and algebraic geometry.

In 1959 Grothendieck accepted an appointment at the French Institute des Hautes Etudes Scientifiques (Institute for Advanced Scientific Studies). Concerned, however, over military funding of the Institute, Grothendieck eventually resigned his post in 1969. An ardent pacifist and environmentalist, Grothendieck returned to his undergraduate institution, the University of Montpellier, where he actively promoted military disarmament and farming without the use of pesticides. Although he remained a diligent teacher, by the 1980s Grothendieck had so withdrawn himself from the international mathematics community that he made few public appearances. In 1988 he rejected the Swedish Academy's Crafoord Prize (along with its monetary award) because of what Grothendieck publicly characterized as a growing dishonesty and politicization of science and mathematics.

Nearing retirement in the late 1980s, Grothendieck's published works branched into the philosophy of science and mathematics. Though his memoir, titled Récoltes et semailles (Harvest and Sowing), deals with a great many nonmathematical topics, Grothendieck wrote that "mathematical activity involves essentially three things: studying numbers, studying shapes and measuring distances." Grothendieck contended that all mathematical reasoning and divisions of study (for example, number theory, calculus, probability, topology, or algebraic geometry) branched from one or a combination of these methodologies.

Grothendieck's abstract and highly scholarly work built largely upon the work of French mathematicians André Weil (1906-1998), Jean-Pierre Serre (1926- ), and Russian-born mathematician Oscar Zariski (1899-1986). Together, these mathematicians laid the foundation for modern algebraic geometry. Because algebraic geometry borrows from both algebra and geometry, it has found practical application in both areas. The geometry of sets (elliptic curves, for example) can be studied with algebraic equations; it also enabled English mathematician Andrew John Wiles (1953- ) to formulate a proof of Fermat's Last Theorem. Other applications include solutions for conics and curves, commutative ring theory, and number theory (especially for the Diophantine type problems, including Fermat's Last Theorem).

The depth of Grothendieck's work remains largely inaccessible to all but the most learned and nimble mathematical minds. More accessible is his theory of schemes (that provided a base upon which certain Weil conjectures regarding number theory were solved) and his work in mathematical logic. Grothendieck placed a special emphasis on defining geometric objects in accordance with their underlying functions. In addition, he is credited with providing the algebraic definitions relevant to grouping of curves. Grothendieck's work ranged over a veritable landscape of modern and post-modern mathematics, borrowing from and making substantial contributions to various topics, including topological tensor products and nuclear spaces, sheaf cohomology as derived functions, number theory, and complex analysis. Grothendieck also won the attention of the mathematical community with his highly regarded major work on homological algebra, now commonly referred to as the Tohoku Paper.

Grothendieck's publications also include his celebrated 1960 work Eléments de géométrie algébrique (Elementary Algebraic Geometry).


About this article

Alexander Grothendieck

Updated About content Print Article