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Esquisse D’un Programme
"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory, which is currently being extended based on his original proposal. Brief history Submitted in 1984, the ''Esquisse d'un Programme'' was a proposal submitted by Alexander Grothendieck for a position at the Centre National de la Recherche Scientifique. The proposal was not successful, but Grothendieck obtained a special position where, while keeping his affiliation at the University of Montpellier, he was paid by the CNRS and released of his t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoids
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial function replacing the binary operation; *''Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g:A \rightarrow B, h:B \rightarrow C, say. Composition is then a total function: \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C , so that h \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pro-finite Group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Theory
Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences. The notion is very close to the notion of algebraic structure, which, arguably, may be just a synonym. Saying that a theory is algebraic is a stronger condition than saying it is elementary. Informal interpretation An algebraic theory consists of a collection of ''n''-ary functional terms with additional rules (axioms). For example, the theory of groups is an algebraic theory because it has three functional terms: a binary operation ''a'' × ''b'', a nullary operation 1 (neutral element), and a unary operation ''x'' ↦ ''x''−1 with the rules of associativity, neutrality and inverses respectively. Other examples include: * the theory of semigroups * the theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuing Stacks
''Pursuing Stacks'' (french: À la Poursuite des Champs) is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the work is a generalized homotopy theory using higher category theory. The word "stacks" in the title refers to what are nowadays usually called " ∞-groupoids", one possible definition of which Grothendieck sketches in his manuscript. (The stacks of algebraic geometry, which also go back to Grothendieck, are not the focus of this manuscript.) Among the concepts introduced in the work are derivators and test categories. Some parts of the manuscript were later developed in: * * Overview of manuscript I. The letter to Daniel Quillen Pursuing stacks started out as a letter from Grothendieck to Daniel Quillen. In this letter he discusses Quillen's progress on the foundations for homotopy theory and remarked on the lack of progress since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moderated Theories
Moderation is the process of eliminating or lessening extremes. It is used to ensure normality throughout the medium on which it is being conducted. Common uses of moderation include: *Ensuring consistency and accuracy in the marking of student assessments. *A moderator may remove unsuitable contributions from the website, forum or chat room they represent in accordance with their moderation system. *A more proactive nuance is found in the Methodist church's use of the term for the heads of its conferences. *A neutron moderator is used to slow down neutrons in a nuclear reactor. *A way of life emphasizing perfect amounts of everything, not indulging in too much of one thing, hence moderation. History Ancient Greece Moderation is also a principle of life. In ancient Greece, the temple of Apollo at Delphi bore the inscription ''Meden Agan'' () - 'Nothing in excess'. Doing something "in moderation" means not doing it excessively. For instance, someone who moderates their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
