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Tame Topology
In mathematics, a tame topology is a hypothetical topology proposed by Alexander Grothendieck in his research program ''Esquisse d’un programme'' under the French name ''topologie modérée'' (moderate topology). It is a topology in which the theory of dévissage can be applied to stratified structures such as semialgebraic or semianalytic sets. Some authors consider an o-minimal structure to be a candidate for realizing tame topology in the real case. Some other authors have claimed that their theory of conically smooth stratified spaces can achieve tame topology. See also *Thom's first isotopy lemma In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f : M \to N between smooth manifolds and S \subset M a closed Whitney stratified subset, if f, _S is proper and f, _A is a submersion for eac ... References * External links *https://ncatlab.org/nlab/show/tame+topology Algebraic analysis Geometry education Mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Dévissage
In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent. Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme. They did this by defining an object called a relative dévissage which is well-suited to certain kinds of inductive arguments. They used this technique to give a new criterion for a module to be flat. As a consequence, they were able to simplify and generalize the results of EGA IV 11 on descent of flatness. The word ''dévissage'' is French for ''unscrewing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stratified Space
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a Stratification (mathematics)#In topology, stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space. On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant sheaf, locally constant on each stratum. Among the several ideals, Grothendieck's ''Esquisse d’un programme'' considers (or proposes) a stratified space with what he calls the tame topology. A stratified space in the sense of Mather Mather gives the following definition of a stratified space. A ''prestratification'' on a topological space ''X'' is a partition of ''X'' into subsets (called strata) such that (a) each stratum is locally closed, (b) it is loca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semialgebraic Set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and inequalities (of the form Q(x_1,...,x_n) > 0), or any finite union of such sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Properties Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another such (as case o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semianalytic Set
In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real power series to be positive there). Subanalytic sets still have a reasonable local description in terms of submanifolds. Formal definitions A subset ''V'' of a given Euclidean space ''E'' is semianalytic if each point has a neighbourhood ''U'' in ''E'' such that the intersection of ''V'' and ''U'' lies in the Boolean algebra of sets generated by subsets defined by inequalities ''f'' > 0, where f is a real analytic function. There is no Tarski–Seidenberg theorem for semianalytic sets, and projections of semianalytic sets are in general not semianalytic. A subset ''V'' of ''E'' is a subanalytic set if for each point there exists a relatively compact semianalytic set ''X'' in a Euclidean space ''F'' of dimension at least a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
O-minimal Structure
In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every definable subset ''X'' ⊂ ''M'' (with parameters taken from ''M'') is a finite of intervals and points. O-minimality can be regarded as a weak form of . A structure ''M'' is o-minimal if and only if every formula with one free variable and parameters in ''M'' is equivalent to a quantifier-free formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Thom's First Isotopy Lemma
In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f : M \to N between smooth manifolds and S \subset M a closed Whitney stratified subset, if f, _S is proper and f, _A is a submersion for each stratum A of S, then f, _S is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when N = \mathbb. In that case, the lemma constructs an isotopy from the fiber f^(a) to f^(b); whence the name "isotopy lemma". The local trivializations that the lemma provide preserve the strata. However, they are generally not smooth (not even C^1). On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic. The lemma is also valid for a more general stratified space such as a stratified space in the sense of Mather but still with the Whitney conditions (or some other conditions). The lemma is also valid for the stratification that satisfies Bekk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunctions. Semantically, it is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician Mikio Sato in 1959. This can be seen as an algebraic geometrization of analysis. It derives its meaning from the fact that the differential operator is right-invertible in several function spaces. It helps in the simplification of the proofs due to an algebraic description of the problem considered. Microfunction Let ''M'' be a real-analytic manifold of dimension ''n'', and let ''X'' be its complexification. The sheaf of microlocal functions on ''M'' is given as :\mathcal^n(\mu_M(\mathcal_X) \otimes \mathcal_) where * \mu_M denotes the microlocalization functor, * \mathcal_ is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
