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.

La Longue Marche à travers la théorie de Galois Long March Through Galois Theory is an approximately 1600-page handwritten manuscript produced by Grothendieck during the years 1980 – 1981, containing many of the ideas leading to the Esquisse d ' un programme ( see below, and also a more detailed entry ), and in particular studying the Teichmüller theory.

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

* Birkhoff – Grothendieck theorem

* Brieskorn – Grothendieck resolution

* Grothendieck – Katz p-curvature conjecture

* Grothendieck – Ogg – Shafarevich formula

* Grothendieck – Riemann – Roch theorem

* Tarski – Grothendieck set theory

To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone – Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

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

The MML is built on the axioms of the Tarski – Grothendieck set theory.

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

Some notable mathematicians include Archimedes of Syracuse, Leonhard Euler, Carl Gauss, Johann Bernoulli, Jacob Bernoulli, Aryabhata, Brahmagupta, Bhaskara II, Nilakantha Somayaji, Omar Khayyám, Muhammad ibn Mūsā al-Khwārizmī, Bernhard Riemann, Gottfried Leibniz, Andrey Kolmogorov, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, Alexander Grothendieck, David Hilbert, Alan Turing, von Neumann, Kurt Gödel, Joseph-Louis Lagrange, Georg Cantor, William Rowan Hamilton, Carl Jacobi, Évariste Galois, Nikolay Lobachevsky, Rene Descartes, Joseph Fourier, Pierre-Simon Laplace, Alonzo Church, Nikolay Bogolyubov and Pierre de Fermat.

The idea of a Grothendieck topology ( also known as a site ) has been characterised by John Tate as a bold pun on the two senses of Riemann surface.

In mathematics, specifically in algebraic geometry, the Grothendieck – Riemann – Roch theorem is a far-reaching result on coherent cohomology.

The Grothendieck – Hirzebruch – Riemann – Roch theorem sets both theorems in a relative situation of a morphism between two manifolds ( or more general schemes ) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves.

Conversely, complex analytic analogues of the Grothendieck – Hirzebruch – Riemann – Roch theorem can be proved using the families index theorem.

The Grothendieck – Riemann – Roch theorem relates the push forward maps

In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck Hirzebruch Riemann Roch formula to

The arithmetic Riemann – Roch theorem extends the Grothendieck – Riemann – Roch theorem to arithmetic schemes.

( Grothendieck Riemann Roch )" on MathOverflow.

The Grothendieck – Riemann – Roch theorem from about 1956 is usually cited as the key moment for the introduction of this circle of ideas.

) Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of ' integrable connections on algebraic varieties ', generalising the theory of linear differential equations on Riemann surfaces.

* Grothendieck existence theorem

Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.

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.

He reworked the tannakian category theory in his paper for the Grothendieck Festschrift, employing Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate Weil cohomology.

As a partial converse, a theorem of Grothendieck asserts that any torus of finite type is quasi-isotrivial, i. e., split by an etale surjection.

Alexander Grothendieck discovered K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann-Roch theorem.

The correspondence defines an exact functor from the category of sheaves over to the category of sheaves of. The following two statements are the heart of Serre's GAGA theorem ( as extended by Grothendieck, Neeman et al.

The semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of F.

Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.

However, the Garden of Eden theorem can be generalized beyond Euclidean spaces, to cellular automata defined on the elements of an amenable group or a sofic group ; the proof of this generalization uses the Ax – Grothendieck theorem, an analogous relation between injectivity and bijectivity in algebraic geometry.

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.

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.

Grothendieck was co-awarded ( but declined ) the Crafoord Prize with Pierre Deligne in 1988.

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.