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last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.

Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of ' objects ' explicitly depending on parameters, as the basic field of study, rather than a single such object.

Alexander Grothendieck's work during the ` Golden Age ' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis.

The bulk of Grothendieck's published work is collected in the monumental, and yet incomplete, Éléments de géométrie algébrique ( EGA ) and Séminaire de géométrie algébrique ( SGA ).

However, Grothendieck's standard conjectures remain open ( except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures ), and the analogue of the Riemann hypothesis was proved by, using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.

It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work ( the Tohoku paper ).

Earlier in his career Mackey did significant work in the duality theory of locally convex spaces, which provided tools for subsequent work in this area, including Alexander Grothendieck's work on topological tensor products.

He also indicated the role of the Brauer group, via Grothendieck's theory of global Azumaya algebras, in accounting for obstructions to the Hasse principle, setting off a generation of further work.

There Grothendieck's period conjecture for an abelian variety A states that the transcendence degree of its period matrix is the same as the dimension of the associated Mumford – Tate group, and what is known by work of Pierre Deligne is that the dimension is an upper bound for the transcendence degree.

At the time of his birth Grothendieck's mother was married to Johannes Raddatz, a German journalist, and his birthname was initially recorded as Alexander Raddatz.

While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from IHÉS, those who knew him say that the causes of the rupture ran deeper.

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 first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

Thus, V ( S ) is " the same as " the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals ; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.

* was able to prove the hard Lefschetz theorem ( part of Grothendieck's standard conjectures ) using his second proof of the Weil conjectures.

Grothendieck's key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any category, not just the category of open subsets of a space.

During 1964-1967 at the Forschungsinstitut für Mathematik at the ETH in Zurich he worked on the Category of Categories and was especially influenced by Pierre Gabriel's seminars at Oberwolfach on Grothendieck's foundation of algebraic geometry.

Alexander Grothendieck's version of the Riemann – Roch theorem was originally conveyed in a letter to Jean-Pierre Serre around 1956 – 7.

Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism ; Serre duality was recovered as the case of the morphism of a non-singular projective variety ( or complete variety ) to a point.

Grothendieck's political views were radical and pacifist.

Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's that there is a deep connection between the topological characteristics of a variety and its diophantine ( number theoretic ) properties.

There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology.

However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article .< ref > A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebra with a view toward algebraic geometry / David Eisenbud ( 1995 ), which uses " h < sub > A </ sub >" to mean the covariant hom-functor.

This is done by studying the zeta functions of the even powers E < sup > k </ sup > of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups.

pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.

; Grothendieck's Galois theory: A very abstract approach from algebraic geometry, introduced to study the analogue of the fundamental group.

A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme ( over a base category ), abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.

The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's Éléments de géométrie algébrique and Mumford's.

In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry.

Notes on Grothendieck's Galois Theory http :// arxiv. org / abs / math / 0009145v1

Grothendieck's use of these universes ( whose existence cannot be proved in ZFC ) led to some uninformed speculation that étale cohomology and its applications ( such as the proof of Fermat's last theorem ) needed axioms beyond ZFC.

As a traditionally trained biologist with little mathematical experience, Mayr was often highly critical of early mathematical approaches to evolution such as those of J. B. S.

Logical empiricism ( aka logical positivism or neopositivism ) was an early 20th century attempt to synthesize the essential ideas of British empiricism ( e. g. a strong emphasis on sensory experience as the basis for knowledge ) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein.

One of the early ( and portable ) languages that had 4GL properties was Ramis developed by Gerald C. Cohen at Mathematica, a mathematical software company.

It is not known how the name " knapsack problem " originated, though the problem was referred to as such in the early works of mathematician Tobias Dantzig ( 1884 – 1956 ), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined.

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.

Although the existence of molecules has been accepted by many chemists since the early 19th century as a result of Dalton's laws of Definite and Multiple Proportions ( 1803 – 1808 ) and Avogadro's law ( 1811 ), there was some resistance among positivists and physicists such as Mach, Boltzmann, Maxwell, and Gibbs, who saw molecules merely as convenient mathematical constructs.

Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces.

Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period.

See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the very concept of mathematical probability.

David Diringer noted that " the first mention of Egyptian documents written on leather goes back to the Fourth Dynasty ( c. 2550-2450 BCE ), but the earliest of such documents extant are: a fragmentary roll of leather of the Sixth Dynasty ( c. twenty-fourth century BCE ), unrolled by Dr. H. Ibscher, and preserved in the Cairo Museum ; a roll of the Twelfth Dynasty ( c. 1990-1777 BCE ) now in Berlin ; the mathematical text now in the British Museum ( MS. 10250 ); and a document of the reign of Ramses II ( early thirtheenth century BCE ).".

Greek contributions to science — including works of geometry and mathematical astronomy, early accounts of biological processes and catalogs of plants and animals, and theories of knowledge and learning — were produced by philosophers and physicians, as well as practitioners of various trades.

Mien taught Banach French and most likely encouraged him in his early mathematical pursuits.

During the early 20th century, chemists such as Gilbert N. Lewis, Merle Randall, and E. A. Guggenheim applied the mathematical methods of Gibbs to the analysis of chemical processes.

Further examples include some of the early versions of the pixel shader languages embedded in Direct3D and OpenGL extensions, or a series of mathematical formulae in a spreadsheet with no cycles.

Segner worked with Euler on some of the early mathematical theories of turbine design.

Some of the mathematical description work on curves was developed in the early 1940s by Robert Issac Newton from Pawtucket, Rhode Island.

In the 3rd century, Liu Hui would provide commentary on this important early Chinese mathematical treatise.

In mathematical astronomy, his fame is due to the introduction of the astronomical globe, and his early contributions to understanding the movement of the planets.

Though his education and early work were mathematical, especially geometrical, Frege's thought soon turned to logic.

Its history dates back to December 1954, when Saab got a license to build its own copy of BESK, an early Swedish computer design using vacuum tubes, from Matematikmaskinnämnden ( the Swedish governmental board for mathematical machinery ).

The work is notable for its early use of proof by mathematical induction, and pioneering work in combinatorics.

Professor Diaconis received a MacArthur Fellowship in 1982, and in 1992 published ( with Dave Bayer ) a paper entitled " Trailing the Dovetail Shuffle to Its Lair " ( a term coined by magician Charles Jordan in the early 1900s ) which established rigorous results on how many times a deck of playing cards must be riffle shuffled before it can be considered random according to the mathematical measure total variation distance.