Alexandre Grothendieck (1928-) Germany, France

Alexandre  Grothendieck (1928-) Germany, France

Grothendieck has done brilliant work in several areas of mathematics including number theory, geometry, topology, and functional analysis, but especially in the fields of algebraic geometry and category theory, both of which he revolutionized.

 He is most famous for his methods to unify different branches of mathematics, for example using algebraic geometry in number theory. Grothendieck is considered a master of abstraction, rigor and presentation. He has produced many important and deep results in homological algebra, most notably his etale cohomology. With these new methods, Grothendieck and his famous student Pierre Deligne were able to prove the Weil Conjectures. Grothendieck also developed the theory of sheafs, invented the theory of schemes, generalized the Riemann-Roch Theorem to revolutionize K-theory, developed Grothendieck categories, crystalline cohomology, infinity-stacks and more. The guiding principle behind much of Grothendieck's work has been Topos Theory, which he invented to harness the methods of topology. These methods and results have redirected several diverse branches of modern mathematics including number theory, algebraic topology, and representation theory.

Grothendieck's radical religious and political philosophies led him to retire from public life while still in his prime, but he is widely considered the greatest mathematician of the 20th century, and is sometimes called one of the greatest mathematical geniuses ever.