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Localized Chern Class
In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's ''intersection theory'', as an algebraic counterpart of the similar construction in algebraic topology. The notion is used in particular in the Riemann–Roch-type theorem. S. Bloch later generalized the notion in the context of arithmetic schemes (schemes over a Dedekind domain) for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a degeneration (algebraic geometry), degenerating family of algebraic varieties (in the mixed characteristic case). Definitions Let ''Y'' be a pure-dimensional regular scheme of finite type over a field or discrete valuation ring and ''X'' a closed subscheme. Let E_ denote a complex of vector bundles on ''Y'' :0 = E_ \to E_n \to \dots \to E_m \to E_ = 0 that is exact on Y - X. The local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Roch-type Theorem
In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al. Formulation due to Baum, Fulton and MacPherson Let G_* and A_* be functors on the category ''C'' of schemes separated and locally of finite type over the base field ''k'' with proper morphisms such that *G_*(X) is the Grothendieck group of coherent sheaves on ''X'', *A_*(X) is the rational Chow group of ''X'', *for each proper morphism ''f'', G_*(f), A_*(f) are the direct images (or push-forwards) along ''f''. Also, if f: X \to Y is a (global) local complete intersection morphism; i.e., it factors as a closed regular embedding X \hookrightarrow P into a smooth scheme ''P'' followed by a smooth morphism P \to Y, then let :T_f = _X- _/math> be the class in the Grothendieck group of vector bundles on ''X''; it is independent of the factorization and is ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Scheme
In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. Background The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals \mathfrak \in \text(\mathbb) and finite places v_p : \mathbb^* \to \mathbb, but there also exists a place at infinity v_\infty, given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying \text(\mathbb) into a complete space \overline which has a prime lying at infinity. Arakelov's original construction studies one such theory, where a definition of divisors is constructor for a scheme \mathfrak of relative dimension 1 over \text(\mathcal_K) such that it extends to a Riemann surface X_\infty = \mathfrak(\mathbb) for every valuation at infinity. In addition, he equips these Riemann surfaces with Hermitian metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneration (algebraic Geometry)
In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism :\pi: \mathcal \to C, of a variety (or a scheme) to a curve ''C'' with origin 0 (e.g., affine or projective line), the fibers :\pi^(t) form a family of varieties over ''C''. Then the fiber \pi^(0) may be thought of as the limit of \pi^(t) as t \to 0. One then says the family \pi^(t), t \ne 0 ''degenerates'' to the ''special'' fiber \pi^(0). The limiting process behaves nicely when \pi is a flat morphism and, in that case, the degeneration is called a flat degeneration. Many authors assume degenerations to be flat. When the family \pi^(t) is trivial away from a special fiber; i.e., \pi^(t) is independent of t \ne 0 up to (coherent) isomorphisms, \pi^(t), t \ne 0 is called a general fiber. Degenerations of curves In the study of moduli of curves, the important point is to understand the boundaries of the moduli, which amounts to understand deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivariant Chow Group
In mathematics, a bivariant theory was introduced by Fulton and MacPherson , in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring. On technical levels, a bivariant theory is a mix of a homology theory and a cohomology theory. In general, a homology theory is a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the category of (nice) spaces to the category of rings. A bivariant theory is a functor both covariant and contravariant; hence, the name “bivariant”. Definition Unlike a homology theory or a cohomology theory, a bivariant class is defined for a map not a space. Let f : X \to Y be a map. For such a map, we can consider the fiber square : \begin X' & \to & Y' \\ \downarrow & & \downarrow \\ X & \to & Y \end (for example, a blow-up.) Intuitively, the consideration of all the fiber squares like the above can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Bundle
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics. Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Construction
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discrete mathematics *Graph of a function *Graph of a relation *Graph paper *Chart, a means of representing data (also called a graph) Computing *Graph (abstract data type), an abstract data type representing relations or connections *graph (Unix), Unix command-line utility *Conceptual graph, a model for knowledge representation and reasoning Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also *Complex network *Graf *Graff (other) *Graph database *Grapheme, in linguistics *Graphemics *Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") *List of information graphics software *Statistical graphics Statistical graphics, also known as statistical graphical techniques, are graphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definitions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, that at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
