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Coarse Moduli Space
In algebraic geometry, a moduli scheme is a moduli space that exists in the category of schemes developed by French mathematician 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 stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Level Structure (algebraic Geometry)
In algebraic geometry, a level structure on a space ''X'' is an extra structure attached to ''X'' that shrinks or eliminates the automorphism group of ''X'', by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of ''X''. In applications, a level structure is used in the construction of moduli spaces; a moduli space is often constructed as a quotient. The presence of automorphisms poses a difficulty to forming a quotient; thus introducing level structures helps overcome this difficulty. There is no single definition of a level structure; rather, depending on the space ''X'', one introduces the notion of a level structure. The classic one is that on an elliptic curve (see #Example: an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in . Level structures on elliptic curves Classically, level structures on elliptic curves E = \mathbb/\L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semistable Vector Bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles \mathbfGL_n is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of \mathbb^1 by \mathcal(1) there is an exact sequence0 \to \mathcal(-1) \to \mathcal\oplus \mathcal \to \mathcal(1) \to 0which represents a non-zero element v \in \text^1(\mathcal(1), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Vector Bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles \mathbfGL_n is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of \mathbb^1 by \mathcal(1) there is an exact sequence0 \to \mathcal(-1) \to \mathcal\oplus \mathcal \to \mathcal(1) \to 0which represents a non-zero element v \in \text^1(\mathcal(1), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Covering
In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme * Étale algebra Other * Étale (mountain) Étale is a mountain of Savoie and Haute-Savoie, France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Sa ... in Savoie and Haute-Savoie, France See also * Étalé space * Etail, or online commerce {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Algebraic Numbers
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental and include and . There are countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue measure) are transcendental. Examples * All rational numbers are algebraic. Any rational number, express ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belyi's Theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data. Quotients of the upper half-plane It follows that the Riemann surface in question can be taken to be the quotient :''H''/Γ (where ''H'' is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is ''not'' the conclusion that any such curve is a modular curve. Belyi functions A Belyi function is a holomorphic map from a compact Riemann surface ''S'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
János Kollár
János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry. Professional career Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 under the direction of Teruhisa Matsusaka with a thesis on canonical threefolds. He was Junior Fellow at Harvard University from 1984 to 1987 and professor at the University of Utah from 1987 until 1999. Currently, he is professor at Princeton University. Contributions Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic surfaces, for pioneering the notion of rational connectedness (''i.e.'' extending the theory of rationally connected varieties for varieties over the complex field to varieties over local fields), and finding counterexamples to a conjecture of John Nash. (In 1952 Nash conjectured a converse to a famous theorem he proved, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
