Alexander Grothendieck was a mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the 20th century.
John Torrence Tate Jr. was an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He was awarded the Abel Prize in 2010.
In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie (SGA) was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris. The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series.
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s.
In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by Alexander Grothendieck (1966a), who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a scheme.
In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product.
Michael Artin is an American mathematician of Armenian origin and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.
Yuri Ivanovich Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
Masaki Kashiwara is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module theory, Hodge theory, sheaf theory and representation theory.
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use.
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by Alexander Grothendieck and developed by Pierre Berthelot (1974).
"Esquisse d'un Programme" is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory, which is currently being extended based on his original proposal.
Jean-Marc Fontaine was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death.
In mathematics, a Dieudonné module introduced by Jean Dieudonné, is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.
Luc Illusie is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences.
In algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic p. They were introduced by Barsotti (1962) under the name equidimensional hyperdomain and by Tate (1967) under the name p-divisible groups, and named Barsotti–Tate groups by Grothendieck (1971).
In algebraic geometry, F-crystals are objects introduced by Mazur (1972) that capture some of the structure of crystalline cohomology groups. The letter F stands for Frobenius, indicating that F-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny".
The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as the Tôhoku paper, was published in 1957 in the Tôhoku Mathematical Journal. It has revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology. It removed the need to distinguish the cases of modules over a ring and sheaves of abelian groups over a topological space.
Frans Oort is a Dutch mathematician, specializing in algebraic geometry.
The original article was a Google-aided translation of the corresponding article in German Wikipedia.
