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Stratification Theory
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a 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 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 locally finite and (c) (axiom of frontier) if two strata ''A'', ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Germ
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word ''local'' has some meaning. Name The name is derived from ''cereal germ'' in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain. Formal definition Basic definition Given a point ''x'' of a topological space ''X'', and two maps f, g: X \to Y (where ''Y'' is any set), then f and g define the same germ at ''x'' if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harder–Narasimhan Stratification
In algebraic geometry and complex geometry, the Harder–Narasimhan stratification is any of a stratification of the moduli stack of principal ''G''-bundles by locally closed substacks in terms of "loci of instabilities". In the original form due to Harder and Narasimhan, ''G'' was the general linear group; i.e., the moduli stack was the moduli stack of vector bundles, but, today, the term refers to any of generalizations. The scheme-theoretic version is due to Shatz and so the term "Shatz stratification" is also used synonymously. The general case is due to Behrend. References * Further reading * Nitin NitsureSchematic Harder-Narasimhan Stratification {{algebraic-geometry-stub Algebraic geometry Stratifications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stratified Morse Theory
In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky and Robert MacPherson. The main point of the theory is to consider functions f : M \to \mathbb R and consider how the stratified space f^(-\infty,c] changes as the real number c \in \mathbb R changes. Morse theory of stratified spaces has uses everywhere from pure mathematics topics such as braid groups and Lawrence–Krammer representation, representations to robot motion planning and potential theory. A popular application in pure mathematics is Morse theory on manifolds with boundary, and manifolds with corners. See also * Digital Morse theory * Discrete Morse theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology com ... * Level-set method ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perverse Sheaf
The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular. The concept was introduced in the work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne and Ofer Gabber (1982) as a consequence of the Riemann-Hilbert correspondence, which establishes a connection between the derived categories regular holonomic D-modules and constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more general complexes thereof); a perverse sheaf ''is'' in general represented by a complex of sheaves. The concept of perverse sheaves is already implicit in a 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules. A key observation was that the intersection homology of Mark Goresky and Robert MacPherson could be described using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equisingularity
In algebraic geometry, an equisingularity is, roughly, a family of singularities that are not non-equivalent and is an important notion in singularity theory. There is no universal definition of equisingularity but Oscar Zariski's equisingularity is the most famous one. Zariski's equisingualrity, introduced in 1971 under the name " algebro-geometric equisingularity", gives a stratification that is different from the usual Whitney stratification on a real or complex algebraic variety. See also *stratified space In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset ... References Further reading *https://mathoverflow.net/questions/299314/a-general-definition-of-an-equisingular-family-of-singular-varieties algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Closed
In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied: * E is the intersection of an open set and a closed set in X. * For each point x\in E, there is a neighborhood U of x such that E \cap U is closed in U. * E is open in its closure \overline. * The set \overline\setminus E is closed in X. * E is the difference of two closed sets in X. * E is the difference of two open sets in X. The second condition justifies the terminology ''locally closed'' and is Bourbaki's definition of locally closed. To see the second condition implies the third, use the facts that for subsets A \subseteq B, A is closed in B if and only if A = \overline \cap B and that for a subset E and an open subset U, \overline \cap U = \overline \cap U. Examples The interval (0, 1] = (0, 2) \cap , 1/math> is a locally closed subset of \Reals. For another example, consider the relative interior D of a closed d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stratification (mathematics)
Stratification has several usages in mathematics. In mathematical logic In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q_1 \wedge \dots \wedge Q_n \wedge \neg Q_ \wedge \dots \wedge \neg Q_ \rightarrow P is stratified if and only if there is a stratification assignment S that fulfills the following conditions: # If a predicate P is positively derived from a predicate Q (i.e., P is the head of a rule, and Q occurs positively in the body of the same rule), then the stratification number of P must be greater than or equal to the stratification number of Q, in short S(P) \geq S(Q). # If a predicate P is derived from a negated predicate Q (i.e., P is the head of a rule, and Q occurs negatively in the body of the same rule), then the stratification number of P must be greater than the stratification numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, and which excludes some pathological spaces that do not correspond to intuitive notions of spaces. Some authors consider an o-minimal structure to be a candidate for realizing tame topology in the real case. There are also some other suggestions. 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 ea ... References * External links *https://ncatlab.org/nlab/show/tame+topology Algebraic analysis Str ... [...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. Brief history Submitted in 1984, the ''Esquisse d'un Programme'' was a proposal 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 teaching obligations. Grothendieck held this position from 1984 till 1988. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
