|
Mixed Hodge Module
In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair of a filtered D-module (M, F^\bullet) together with a perverse sheaf \mathcal such that the functor from the Riemann–Hilbert correspondence sends (M, F^\bullet) to \mathcal. This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered. This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves. Abstract structure Before going into the nitty gri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hodge Theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings—Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, . Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Mac Lane says Alexander Grothendieck defined the abelian category, but there is a reference that says Eilenberg's disciple, Buchsbaum, proposed the concept in his PhD thesis, and Groth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Generalized Manifolds
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
1605
Events January–March * January 1 – William Shakespeare's play ''A Midsummer Night's Dream'', copyrighted 1600, is given its earliest recorded performance, and witnessed by the Viscount Dorchester. * January 7 – Shakespeare's play '' Henry V'', copyrighted 1600, is given its earliest recorded performance, presented by the Lord Chamberlain's Men for King James I of England. * January 15 – Shakespeare's play ''Love's Labour's Lost'', copyrighted 1598, is given its second recorded performance, probably presented at the home of the Earl of Southampton for Queen Anne, wife of King James I of England. * January 16 – The first part of Miguel de Cervantes' satire on the theme of chivalry, ''Don Quixote'' (''El ingenioso hidalgo don Quixote de la Mancha'', "The Ingenious Hidalgo Don Quixote of La Mancha"), is published in Madrid. One of the first significant novels in the western literary tradition, it becomes a global bestseller almost at once. * February 3 – The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Deligne Cohomology
In mathematics, Deligne cohomology sometimes called Deligne-Beilinson cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians. For introductory accounts of Deligne cohomology see , , and . Definition The analytic Deligne complex Z(''p'')D, an on a complex analytic manifold ''X'' is0\rightarrow \mathbf Z(p)\rightarrow \Omega^0_X\rightarrow \Omega^1_X\rightarrow\cdots\rightarrow \Omega_X^ \rightarrow 0 \rightarrow \dotswhere Z(''p'') = (2π i)''p''Z. Depending on the context, \Omega^*_X is either the complex of smooth (i.e., ''C''∞) differential forms or of holomorphic forms, respectively. The Deligne cohomology is the ''q''-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit of the diagram\begin & & \mathbb \\ & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mixed Motives (math) , a person who is of multiracial descent
*
*
{{disambiguation ...
Mixed is the past tense of ''mix''. Mixed may refer to: * Mixed (United Kingdom ethnicity category), an ethnicity category that has been used by the United Kingdom's Office for National Statistics since the 2001 Census Music * ''Mixed'' (album), a compilation album of two avant-garde jazz sessions featuring performances by the Cecil Taylor Unit and the Roswell Rudd Sextet See also * Mix (other) * Mixed breed, an animal whose family are from different breeds or species * Mixed ethnicity The term multiracial people refers to people who are mixed with two or more races (human categorization), races and the term multi-ethnic people refers to people who are of more than one ethnicity, ethnicities. A variety of terms have been used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Verdier Duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism. Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves. Verdier duality Verdier duality states that (subject to suitable finiteness conditions disc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Faithful Functor
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor. Formal definitions Explicitly, let ''C'' and ''D'' be ( locally small) categories and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function :F_\colon\mathrm_(X,Y)\rightarrow\mathrm_(F(X),F(Y)) for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be *faithful if ''F''''X'',''Y'' is injectiveJacobson (2009), p. 22 *full if ''F''''X'',''Y'' is surjectiveMac Lane (1971), p. 14 *fully faithful (= full and faithful) if ''F''''X'',''Y'' is bijective for each ''X'' and ''Y'' in ''C''. Properties A faithful functor need not be injective on objects or morphisms. That is, two objects ''X'' and ''X''′ may map to the same object in ''D'' (which is why the range of a full and faithful functor is not necessarily iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Morihiko Saito
Morihiko Saitō (, ''Saitō Morihiko'', born 1961) is a Japanese mathematician, specializing in algebraic analysis and algebraic geometry. Education and career After graduating from Aiko High School in Matsuyama, Saito completed undergraduate study in mathematics at the University of Tokyo and in 1979 completed the master's program there. In 1986 he received his D.Sc. from Kyoto University. After working as a research assistant at Kyoto University's Research Institute for Mathematical Sciences, he was appointed there an associate professor. In 1988/1990 he introduced the theory of mixed Hodge modules, based on the theory of D-modules in algebraic analysis, the theory of perverse sheaves, and the theory of variation of Hodge structures and mixed Hodge structures (introduced by Pierre Deligne) in algebraic geometry. This led, among other things, to a generalization of the fundamental decomposition theorems of Alexander Beilinson, Joseph Bernstein, Deligne, and Ofer Gabber a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mixed Hodge Structure
In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. In mixed Hodge theory, where the decomposition of a cohomology group H^k(X) may have subspaces of different weights, i.e. as a direct sum of Hodge structures :H^k(X) = \bigoplus_i (H_i, F_i^\bullet) where each of the Hodge structures have weight k_i. One of the early hints that such structures should exist comes from the long exact sequence \dots \to H^(Y) \to H^i_c(U) \to H^i(X) \to \dotsassociated to a pair of smooth projective varieties Y \subset X . This sequence suggests that the cohomology groups H^i_c(U) (for U = X - Y ) should have differing weights coming from both H^(Y) and H^i(X) . Motivation Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hodge Structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989). Hodge structures Definition of Hodge structures A pure Hodge structure of integer weight ''n'' consists of an abelian group H_ and a decomposition of its complexification H into a direct sum of complex subspaces H^, where p+q=n, with the property that the complex conjugate of H^ is H^: :H := H_\otimes_ \Complex = \bigoplus\n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
