The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

He is especially known for his foundational work in number theory and algebraic geometry.

He made substantial contributions in many areas, the most important being his discovery of profound connections between algebraic geometry and number theory.

Alexander Grothendieck (; ; born 28 March 1928 ) is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry.

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

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

His conjectural theory of motives has been a driving force behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration.

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.

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.

This letter and successive parts were distributed from Bangor ( see External Links below ): in an informal manner, as a kind of diary, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks.

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray.

In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

This periodicity modulo 8 is known in mathematics not only in the theory of Clifford algebras, but also in algebraic topology, in KO-theory ; see spin representation.

It is also commonly used in mathematics in algebraic solutions representing quantities such as angles.

Another example of an algebraically closed field is the field of ( complex ) algebraic numbers.

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers.

** The golden ratio is algebraic since it is a root of the polynomial.

* The set of algebraic numbers is countable ( enumerable ).

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

If its minimal polynomial has degree, then the algebraic number is said to be of degree.

An algebraic number of degree 1 is a rational number.

* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

This can be rephrased by saying that the field of algebraic numbers is algebraically closed.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

All numbers which can be obtained from the integers using a finite number of integer additions, subtractions, multiplications, divisions, and taking nth roots ( where n is a positive integer ) are algebraic.

The converse, however, is not true: there are algebraic numbers which cannot be obtained in this manner.

An algebraic integer is an algebraic number which is a root of a polynomial with integer coefficients with leading coefficient 1 ( a monic polynomial ).

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

The obvious way to define the tensor product of two Banach spaces A and B is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm.

C *- algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

* axiomatic, algebraic and constructive quantum field theory

It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to non-locality as well as to quantum teleportation and to superdense coding, whereas non-locality is defined according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.

In this formalism ( which is closely related to the C *- algebraic formalism ) the collapse of the wave function corresponds to a non-unitary quantum operation.

But they may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways ( through Groenewold's 1946 star product ), consistent with the uncertainty principle of quantum mechanics.

The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states of a variable or unknown number of particles from a single particle Hilbert space.

The algebraic properties of functional integrals are used to develop series used to calculate properties in quantum electrodynamics and the standard model.

Though the concept of observing a subject from different points in space and time simultaneously ( multiple or mobile perspective ) developed by Metzinger and Gleizes was not derived directly from Albert Einstein's theory of relativity, it was certainly influenced in a similar way, through the work of Jules Henri Poincaré ( particularly Science and Hypothesis ), the French mathematician, theoretical physicist and philosopher of science, who made many fundamental contributions to algebraic topology, celestial mechanics, quantum theory and made an important step in the formulation of the theory of special relativity.

Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.

The very simplest nontrivial example corresponds to two copies of R locally acting on each other and results in a quantum group ( given here in an algebraic form ) with generators p, K, K < sup >- 1 </ sup >, say, and coproduct

In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.

It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling.

In the C *- algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i. e. mappings from physical observables to their expected measurement outcome.

Quantum algorithms may also be grouped by the type of problem solved, for instance see the survey on quantum algorithms for algebraic problems.

In this view, the quantum logic approach resembles more closely the C *- algebraic approach to quantum mechanics ; in fact with some minor technical assumptions it can be subsumed by it.

** Fock space, an algebraic system used in quantum mechanics for a variable or unknown number of particles

Drinfeld introduced the notion of a quantum group ( independently discovered by Michio Jimbo at the same time ) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the Quantum inverse scattering method, and the Drinfeld – Sokolov reduction in the theory of solitons.

" is the title of two articles ( by Hilary Putnam and Michael Dummett ) that discuss the idea that the algebraic properties of logic may, or should, be empirically determined ; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic as a consistent logical rendering of reality.

They also fall in the related framework introduced by Haag and Kastler, called algebraic quantum field theory.

He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms