Alexander Grothendieck was born in Berlin to anarchist parents: a Ukrainian father from an ultimately Hassidic family, Alexander " Sascha " Shapiro aka Tanaroff, and a mother from a German Protestant family, Johanna " Hanka " Grothendieck ; both of his parents had broken away from their early backgrounds in their teens.

The marriage was dissolved in 1929 and Shapiro / Tanaroff acknowledged his paternity, but never married Hanka Grothendieck.

In 1939 Grothendieck went to France and lived in various camps for displaced persons with his mother, first at the Camp de Rieucros, and subsequently lived for the remainder of the war in the village of Le Chambon-sur-Lignon, where he was sheltered and hidden in local boarding-houses or pensions.

Many of the refugee children being hidden in Chambon attended Cévenol and it was at this school that Grothendieck apparently first became fascinated with mathematics.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam War ( The Life and Work of Alexander Grothendieck, American Mathematical Monthly, vol.

The Grothendieck Festschrift was a three-volume collection of research papers to mark his sixtieth birthday ( falling in 1988 ), and published in 1990.

While Grothendieck was at the IHÉS, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a mandarin of the scientific world.

The Grothendieck – Riemann – Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.

Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck.

This program culminated in the proofs of the Weil conjectures, the last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.

This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme.

The first of the Weil conjectures was proved by Bernard Dwork, and a completely different proof of the first two conjectures via l-adic cohomology was given by Alexander Grothendieck.

Littlewood's inequality on bilinear forms was a forerunner of the later Grothendieck tensor norm theory.

In classical algebraic geometry ( that is, the subject prior to the Grothendieck revolution of the late 1950s and 1960s ) the Zariski topology was defined in the following way.

This took two decades ( it was a central aim of the work and school of Alexander Grothendieck ) building up on initial suggestions from Serre.

Grothendieck proved an analogue of the Lefschetz fixed point formula for l-adic cohomology theory, and by applying it to the Frobenius automorphism F was able to prove the following formula for the zeta function.

The question of points was close to resolution by 1950 ; Alexander Grothendieck took a sweeping step ( appealing to the Yoneda lemma ) that disposed of it — naturally at a cost, that every variety or more general scheme should become a functor.

In the light of later work ( c. 1970 ), ' descent ' is part of the theory of comonads ; here we can see one way in which the Grothendieck school bifurcates in its approach from the ' pure ' category theorists, a theme that is important for the understanding of how the topos concept was later treated.

There was perhaps a more direct route available: the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules.

* Fields Prize in Mathematics: Michael Atiyah, Paul Joseph Cohen, Alexander Grothendieck and Stephen Smale

Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

Grothendieck lived with his parents until 1933 in Berlin.

# Yoga of the Grothendieck – Riemann – Roch theorem ( K-theory, relation with intersection theory ).

A category together with a choice of Grothendieck topology is called a site.

To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead.

* Grothendieck – Serre Correspondence ( 2003 ), bilingual edition, edited with Pierre Colmez

* Grothendieck expressed the zeta function in terms of the trace of Frobenius on l-adic cohomology groups, so the Weil conjectures for a d-dimensional variety V over a finite field with q elements depend on showing that the eigenvalues α of Frobenius acting on the ith l-adic cohomology group H < sup > i </ sup >( V ) of V have absolute values | α |= q < sup > i / 2 </ sup > ( for an embedding of the algebraic elements of Q < sub > l </ sub > into the complex numbers ).

The subject can be said to begin with Alexander Grothendieck ( 1957 ), who used it to formulate his Grothendieck – Riemann – Roch theorem.

Grothendieck needed to work with coherent sheaves on an algebraic variety X.

The characterisation was by means of categories ' with enough colimits ', and applied to what is now called a Grothendieck topos.

The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.

Alternatively, Alexander Grothendieck replaced these with a slightly smaller set of axioms:

In particular, one may define the Chern classes of E in the sense of Grothendieck, denoted by expanding this way the class, with the relation:

He was both an influential researcher and teacher, with students such as Bernard Malgrange, Jacques-Louis Lions, François Bruhat and Alexander Grothendieck.

The urgency ( to put it that way ) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence ( see FGA ) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.

The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted, which is the inverse limit of the cyclic additive groups Z / nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index.

After a dispute with Springer, Grothendieck refused the permission for reprints of the series.

After completing a doctorate under the supervision of Alexander Grothendieck, he worked with him at the Institut des Hautes Études Scientifiques ( IHÉS ) near Paris, initially on the generalization within scheme theory of Zariski's main theorem.

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.

The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as Artin and SGA 4.