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Fine Moduli Scheme
In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept (algebraic spaces, algebraic stacks of Michael Artin). History Work of Grothendieck and David Mumford (see geometric invariant theory) opened up this area in the early 1960s. The more algebraic and abstract approach to moduli problems is to set them up as a representable functor question, then apply a criterion that singles out the representable functors for schemes. When this programmatic approach works, the result is a ''fine moduli scheme''. Under the influence of more geometric ideas, it suffices to find a scheme that gives the correct geometric points. This is more like the classical idea that the moduli problem is to express the algebraic structure naturally coming w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Invariant Theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an Group action (mathematics), action of a group on an algebraic variety (or scheme (mathematics), scheme) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopole (mathematics), monopoles. Background Invariant theory is concerned with a Group action (mathematics), group actio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instanton Bundle
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. Instantons are important in quantum field theory because: * they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and * they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory. Relevant to dynamics, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
