Within algebraic geometry itself, his theory of schemes has become the universally accepted language for all further technical work.

During this time he had officially as students Michel Demazure ( who worked on SGA3, on group schemes ), Luc Illusie ( cotangent complex ), Michel Raynaud, Jean-Louis Verdier ( cofounder of the derived category theory ) and Pierre Deligne.

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it.

He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory.

He adapted the use of non-closed generic points, which led to the theory of schemes.

It was a big breakthrough for the theory of quantum information, when quantum error correction codes and fault-tolerant quantum computation schemes were discovered.

Digital signature schemes share basic prerequisites that — regardless of cryptographic theory or legal provision — they need to have, meaning:

Along with Fabien Morel, Voevodsky introduced a homotopy theory for schemes.

There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.

In the theory of schemes of Grothendieck these points are all reconciled: but the general scheme is far from having the immediate geometric content of a variety.

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one.

In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC ( and even in much weaker theories ).

* Red Book of Varieties and Schemes a series of lecture notes by mathematician David Mumford on the theory of schemes

Random oracles have long been considered in computational complexity theory ( e. g. Bennett & Gill ), and many schemes have been proven secure in the random oracle model, for example OAEP and PSS.

Prouty presented " a quartet of the greatest propaganda schemes ever put forth by man " that included Darwin's theory of evolution and Heisenberg's uncertainty principle.

In theory, most Chinese characters as encoded by Han unification and similar schemes could be treated as precomposed characters, since they can be reduced ( decomposed ) to their constituent strokes and ideograph descriptions, though Unicode does not take this approach that would certainly be on the cutting edge of text storage and layout.

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.

The initial development of the theory of group schemes was due to Alexandre Grothendieck, Michel Raynaud, and Michel Demazure in the early 1960s.

However, one can take a projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or in the theory of the fundamental group scheme, and these are affine of infinite type.

In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i. e.,

For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme.

The Rusk belief in balanced defense, replacing the Dulles theory of massive retaliation, removes a grave danger that has existed.

Falling somewhere in a category between Einstein's theory and sand fleas -- difficult to see but undeniably there, nevertheless -- is the tropical green `` city '' of Islandia, a string of offshore islands that has almost no residents, limited access and an unlimited future.

Nevertheless, the theory that the determining influence of the hypothalamic balance has a profound influence on the clinical behavior of neuropsychiatric patients has not yet been tested on an adequate number of patients.

We should expect that general phonologic theory should be as adequate for tone as for consonants and vowels, but it has not been.

Social theory has no more right to expect results from meaningless questions, than physics has the right to expect a theological solution to the wave-particle controversy.

This theory has been put so clearly and precisely that it deserves criticism of the same kind, and this I will do my best to supply.

-- On the basis of a differentiability assumption in function space, it is possible to prove that, for materials having the property that the stress is given by a functional of the history of the deformation gradients, the classical theory of infinitesimal viscoelasticity is valid when the deformation has been infinitesimal for all times in the past.

-- The theory of elasticity of Gaussian networks has been developed on a more general basis and the equations of state relating variables of pressure, volume, temperature, stress and strain have been precisely formulated.

Modern physics has developed the theory that all matter consists of minute waves of energy.

Group selection theory has criticized by many other evolutionary scientists.

Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.

Tylor also theorized about the origins of religious beliefs in human beings, proposing a theory of animism as the earliest stage, and noting that " religion " has many components, of which he believed the most important to be belief in supernatural beings ( as opposed to moral systems, cosmology, etc .).

Later in the 1960s and 1970s, Edmund Leach and his students Mary Douglas and Nur Yalman, among others, introduced French structuralism in the style of Lévi-Strauss ; while British anthropology has continued to emphasize social organization and economics over purely symbolic or literary topics, differences among British, French, and American sociocultural anthropologies have diminished with increasing dialogue and borrowing of both theory and methods.

Anthropology has been used in Britain to provide an alternative explanation for the Financial crisis of 2007 – 2010 to the technical explanations rooted in economic and political theory.

Brønsted-Lowry acid-base theory has several advantages over Arrhenius theory.

*: The article on Whorf states " Drawing on Nietzsche's ideas of perspectivism Alfred Korzybski developed the theory of general semantics which has been compared to Whorf's notions of linguistic relativity.

A common theory about the building is that the rounded feature to the left of centre, terminating at the top in a turret and cross, represents the lance of Saint George ( patron saint of Catalonia, Gaudi's home ), which has been plunged into the back of the dragon.

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

His construction of new cohomology theories has left deep consequences for algebraic number theory, algebraic topology, and representation theory.

His creation of topos theory has had an impact on set theory and logic.