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Grothendieck's and .
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.
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.
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.
Grothendieck's early mathematical work was in functional analysis.
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.
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.
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last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.
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.
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.
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.
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.
* was able to prove the hard Lefschetz theorem ( part of Grothendieck's standard conjectures ) using his second proof of the Weil conjectures.
; 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.

political and views
In the 1930s, Carrel and Charles Lindbergh became close friends not only because of the years they worked together but also because they shared personal, political, and social views.
Reading about famous oratory aroused Johnson's interest in political dialogue and private debates with customers having opposing views on issues of the day.
Supporters of this view believe that “ to a hypothetical outside reader, presents Christianity as enlightened, harmless, even beneficent .” Some believe that through this work, Luke intended to show the Roman Empire that the root of Christianity is within Judaism so that the Christians “ may receive the same freedom to practice their faith that the Roman Empire afforded the Jews .” Those who support the view of Luke ’ s work as political apology generally draw evidence from the facts that Christians are found innocent of committing any political crime ( Acts 25: 25 ; 19: 37 ; 19: 40 ) and that Roman officials ’ views towards Christians are generally positive.
Although Franklin apparently reconsidered shortly thereafter, and the phrases were omitted from all later printings of the pamphlet, his views may have played a role in his political defeat in 1764.
His anti-fascist political views caused him a great deal of trouble with the establishment in Hungary.
During recent times, mainly during the apartheid reform and post-1994 eras, many more white Afrikaans-speaking people, mainly with " conservative " political views and of Trekboer and Voortrekker descent, have preferred to be called " Boers " or Boere-Afrikaners, rather than " Afrikaners ".
During the 1960s the political atmosphere began to gradually change, and attention was once again directed to Chaplin's films instead of his political views.
The honour had already been proposed in 1931 and 1956, but was vetoed after a Foreign Office report raised concerns over Chaplin's political views and private life ; it was felt that honouring him would damage both the reputation of the British honours system and relations with the United States.
Although the project started in 1958, Chaplin only received his star in 1970 because of his political views.
Michael Kelly, a Washington Post journalist and critic of anti-war movements on both the left and right, coined the term " fusion paranoia " to refer to a political convergence of left-wing and right-wing activists around anti-war issues and civil liberties, which he said were motivated by a shared belief in conspiracism or anti-government views.
Hearst's critics often assumed that all the strips in his papers were fronts for his own political and social views.
Other utilitarian-type views include the claims that the end of action is survival and growth, as in evolutionary ethics ( the 19th-century English philosopher Herbert Spencer ); the experience of power, as in despotism ( the 16th-century Italian political philosopher Niccolò Machiavelli and the 19th-century German Friedrich Nietzsche ); satisfaction and adjustment, as in pragmatism ( 20th-century American philosophers Ralph Barton Perry and John Dewey ); and freedom, as in existentialism ( the 20th-century French philosopher Jean-Paul Sartre ).
Prior to this, his political views had been conservative.
Mohism though popular at first due to its emphasis on brotherly love versus harsh Qin Legalism, fell out of favour during the Han Dynasty due to the efforts of Confucians in establishing their views as political orthodoxy.
Psychologist Felicia Pratto and her colleagues have found evidence to support the idea that a high Social Dominance Orientation ( SDO ) is strongly correlated with conservative political views, and opposition to social engineering to promote equality, though Pratto's findings have been highly controversial.
Modern history has rejected these views, instead characterising Domitian as a ruthless but efficient autocrat, whose cultural, economic and political program provided the foundation of the peaceful 2nd century.
As more and more of its demands were met during the 1990s – such as the direct popular election of Taiwan's president and all representatives in its Legislative Yuan, and open discussion of Taiwan's repressive past as represented in the 2-28 Incident and its long martial law aftermath – a greater variety of views could be advocated in the more liberal political atmosphere.
In 1980 Susan Wirth, a 35 year old teacher living in San Francisco, was abducted by her parents to be deprogramed from her leftist political views and activities.
The official opposition and political pressure groups, like the Muslim Brotherhood, were active in Egypt and made their views public.
In addition to the formal structure of the ELDR Party, there are convened at least two Political Leaders ' Meetings a year in order to exchange views on the items on the agenda of the European Council and more general views on the European political situation.
Environmental NGOs vary widely in political views and in the amount they seek to influence the environmental policy of the United States and other governments.

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