complex manifolds

The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — m ...

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• Areas of mathematics - Foundations / general
• Areas of mathematics - Algebra
• Areas of mathematics - Analysis
• Areas of mathematics - Geometry
• Areas of mathematics - Applied mathematics
• Areas of mathematics - Probability and statistics
• Areas of mathematics - Computational sciences
• Areas of mathematics - Physical sciences
• Areas of mathematics - Non-physical sciences

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Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that r ...

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• Analytic geometry - Important themes of analytical geometry
• Analytic geometry - Example
• Analytic geometry - Other uses

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A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ...

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• Manifold - Introduction
• Manifold - Motivational example: the circle
• Manifold - Charts atlases and transition maps
• Manifold - Construction
• Manifold - Charts
• Manifold - Patchwork
• Manifold - Zeros of a function
• Manifold - Identifying points of a manifold
• Manifold - Cartesian products
• Manifold - Gluing along boundaries
• Manifold - Topological manifolds
• Manifold - Differentiable manifolds
• Manifold - Orientability
• Manifold - Möbius strip
• Manifold - Klein bottle
• Manifold - Real projective plane
• Manifold - Other types and generalizations of manifolds
• Manifold - History

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In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. Suppose given a Riemann surface S and on it a differential 1-form ω that is everywhere on S holomorphic, and fixing a point P on S from which to integrate. We can regard as a multi-valued function f(Q), or (better) an honest function of the chosen path C drawn on S from P to Q. Since S ...

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In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. Here OX is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented mo ...

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• Coherent sheaf - Coherent cohomology

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List of publications in mathematics - Euclid's Elements. Euclid Publication data: c. 300 BC Online version: Interactive Java version Description: This is probably not only the most important work in geometry but the most important work in mathematics. It contains many important results in geometry, number theory and the first algorithm as well. The Elements is still a valuable resource and a good introduction to algorithm. More tha ...

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List of publications in mathematics, List of publications in mathematics - Geometry, List of publications in mathematics - Euclid's Elements, List of publications in mathematics - La Géométrie, List of publications in mathematics - Logic, List of publications in mathematics - Begriffsschrift, List of publications in mathematics - Formulario mathematico, List of publications in mathematics - Principia Mathematica, List of publications in mathematics - Gödel's incompleteness theorem, List of publications in mathematics - Information theory, List of publications in mathematics - Number theory, List of publications in mathematics - Disquisitiones Arithmeticae, List of publications in mathematics - On the Number of Primes Less Than a Given Magnitude, List of publications in mathematics - Vorlesungen über Zahlentheorie, List of publications in mathematics - Number Theory An approach through history from Hammurapi to Legendre, List of publications in mathematics - Calculus, List of publications in mathematics - Philosophiae Naturalis Principia Mathematica, List of publications in mathematics - Newton's Principia for the Common Reader, List of publications in mathematics - Calculus and Calculus on Manifolds, List of publications in mathematics - Numerical analysis, List of publications in mathematics - Method of Fluxions, List of publications in mathematics - Game theory, List of publications in mathematics - Evolution and the Theory of Games, List of publications in mathematics - Theory of Games and Economic Behavior, List of publications in mathematics - On Numbers and Games, List of publications in mathematics - Winning Ways for your Mathematical Plays, List of publications in mathematics - Fractals, List of publications in mathematics - How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, List of publications in mathematics - Early manuscripts, List of publications in mathematics - Rhind Mathematical Papyrus, List of publications in mathematics - The Nine Chapters on the Mathematical Art, List of publications in mathematics - Archimedes Palimpsest, List of publications in mathematics - The Sand Reckoner, List of publications in mathematics - Textbooks, List of publications in mathematics - Course of Pure Mathematics, List of publications in mathematics - Art of Problem Solving, List of publications in mathematics - Metalogic: an Introduction to the Metatheory of Standard First Order Logic, List of publications in mathematics - Popular writing, List of publications in mathematics - Gödel Escher Bach, List of publications in mathematics - The World of Mathematics, List of publications in mathematics - Arithmetic, List of publications in mathematics - Arithmetick: or The Grounde of Arts, List of publications in mathematics - The Schoolmaster's Assistant Being a Compendium of Arithmetic both Practical and Theoretical, List of publications in mathematics - Abstract algebra, List of publications in mathematics - Moderne Algebra, List of publications in mathematics - Linear algebra, List of publications in mathematics - Algebraic geometry, List of publications in mathematics - Faisceaux Algébriques Cohérents, List of publications in mathematics - Géométrie Algébrique et Géométrie Analytique, List of publications in mathematics - Éléments de géométrie algébrique, List of publications in mathematics - Séminaire de géométrie algébrique, List of publications in mathematics - Algebraic Geometry, List of publications in mathematics - Universal algebra, List of publications in mathematics - Universal algebra, List of publications in mathematics - Group theory, List of publications in mathematics - Monoid, List of publications in mathematics - Topology, List of publications in mathematics - Topologie, List of publications in mathematics - Topology, List of publications in mathematics - General Topology, List of publications in mathematics - Graph theory, List of publications in mathematics - Category theory, List of publications in mathematics - Categories for the Working Mathematician, List of publications in mathematics - Category Theory for Computing Science, List of publications in mathematics - Order theory, List of publications in mathematics - Trigonometry, List of publications in mathematics - Differential geometry, List of publications in mathematics - Differential topology, List of publications in mathematics - Topology from the Differentiable Viewpoint, List of publications in mathematics - Algebraic topology, List of publications in mathematics - Algebraic Topology, List of publications in mathematics - Fractal geometry, List of publications in mathematics - Discrete mathematics, List of publications in mathematics - Combinatorics, List of publications in mathematics - Set theory, List of publications in mathematics - Grundzüge der Mengenlehre, List of publications in mathematics - Naive Set Theory, List of publications in mathematics - Cardinal and Ordinal Numbers, List of publications in mathematics - The Consistency of the Continuum Hypothesis, List of publications in mathematics - Set Theory and the Continuum Hypothesis, List of publications in mathematics - Set Theory: An Introduction to Independence Proofs, List of publications in mathematics - Optimization, List of publications in mathematics - The New Variational Method, List of publications in mathematics - Decomposition Principle for Linear Programs., List of publications in mathematics - Network Flows and General Matchings, List of publications in mathematics - Paths trees and Flowers, List of publications in mathematics - The complexity of theorem proving procedures, List of publications in mathematics - Reducibility among combinatorial problems, List of publications in mathematics - How good is the simplex algorithm?, List of publications in mathematics - Linear Programming and Polynomial time algorithms, List of publications in mathematics - New polynomial-time algorithm for linear programming, List of publications in mathematics - Interior Point Polynomial Algorithms in Convex Programming

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A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

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Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

See also:

Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

Read more here: » Manifold: Encyclopedia II - Manifold - Construction

The first version of sheaf cohomology to be defined was that based on Čech cohomology, in which the relatively small change was made of attributing to an open set U of a topological space X an element in F(U), an abelian group that 'varies' with U, rather than an abelian group A that is fixed ahead of time. This means that cochains are easy to write down rather concretely; in fact the model applications, such as the Cousin problems on meromorphic functions, stay within fairly familiar mathematical t ...

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Sheaf cohomology, Sheaf cohomology - Definitions, Sheaf cohomology - Applications, Sheaf cohomology - Euler characteristics

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A scheme S is a locally ringed space, so a fortiori a topological space, but the meaning of point of S are threefold: a point P of the underlying topological space; a T-valued point of S is a morphism from T to S, for any scheme T; a geometric point, where S is defined over (is equipped with a morphism to) Spec(K), where K is a field, is a morphism from Spec(K*) to S where < ...

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Glossary of scheme theory, Glossary of scheme theory - Points, Glossary of scheme theory - Local properties, Glossary of scheme theory - Properties of the underlying space, Glossary of scheme theory - Properties of scheme morphisms, Glossary of scheme theory - Affine morphism, Glossary of scheme theory - Closed immersion, Glossary of scheme theory - Étale morphism, Glossary of scheme theory - Finite morphism, Glossary of scheme theory - Flat morphism, Glossary of scheme theory - Open immersion, Glossary of scheme theory - Projective morphism, Glossary of scheme theory - Proper morphism, Glossary of scheme theory - Separated morphism, Glossary of scheme theory - Unramified morphism

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There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order. Algebraic geometry and analytic geometry - Riemann's existence theorem. Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on rami ...

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Algebraic geometry and analytic geometry, Algebraic geometry and analytic geometry - Introduction, Algebraic geometry and analytic geometry - Important results, Algebraic geometry and analytic geometry - Riemann's existence theorem, Algebraic geometry and analytic geometry - The Lefschetz principle, Algebraic geometry and analytic geometry - Chow's theorem, Algebraic geometry and analytic geometry - Serre's GAGA, Algebraic geometry and analytic geometry - Moishezon manifolds, Algebraic geometry and analytic geometry - Reference

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In a covering map the Euler-Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z → zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler-Poincaré characteristic 0), but with the whole disk the Euler-Poincaré characteristic is 1, n-1 being the 'lost' points ...

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Ramification, Ramification - In complex analysis, Ramification - In algebraic topology, Ramification - In algebraic number theory, Ramification - In local fields

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To rephrase this definition in a way that will work in any category C that has sufficient structure, we note that we can write the objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing": Here the first map is the product of the restriction maps resU,Ui,:F(U)→F(Ui) and each pair of arrows represents the two restrictions resUi,Ui∩Uj:USee also:

Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

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Let Ωp,q be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections Since so this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space ...

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Dolbeault cohomology, Dolbeault cohomology - Construction of the cohomology groups, Dolbeault cohomology - Dolbeault's theorem, Dolbeault cohomology - Proof

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Here is an example of a problem from the USAMTS that can be solved via analytic geometry: Problem: In a convex pentagon ABCDE, the sides have lengths 1, 2, 3, 4, and 5, though not necessarily in that order. Let F, G, HSee also:

Analytic geometry, Analytic geometry - Important themes of analytical geometry, Analytic geometry - Example, Analytic geometry - Other uses

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The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois ...

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Areas of mathematics, Areas of mathematics - Foundations / general, Areas of mathematics - Algebra, Areas of mathematics - Analysis, Areas of mathematics - Geometry, Areas of mathematics - Applied mathematics, Areas of mathematics - Probability and statistics, Areas of mathematics - Computational sciences, Areas of mathematics - Physical sciences, Areas of mathematics - Non-physical sciences

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The name is in honour of Charles-Emile Picard's theories, in particular of divisors on algebraic surfaces. The Picard group of the spectrum of a Dedekind domain is its ideal class group. The invertible sheaves on projective space , for k a field, are the twisting sheaves , so the Picard group of is isomorphic to . The Picard group of the affine line with two origins over See also:

Picard group, Picard group - Examples, Picard group - Picard scheme

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Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomo ...

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Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts, atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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Subsequently there were further technical extensions (for example in Godement's book), and areas of application. For example, sheaves were applied to transformation groups; as an inspiration to homology theory in the form of Borel-Moore homology for locally compact spaces; to representation theory in the Borel-Bott-Weil theorem; as well as becoming standard in algebraic geometry and complex manifolds. The particular needs of étale cohomology were more about reinterpreting sheaf in sheaf cohomology, than cohomologySee also:

Sheaf cohomology, Sheaf cohomology - Definitions, Sheaf cohomology - Applications, Sheaf cohomology - Euler characteristics

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Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to itself acquire a sense of ordering, which we can think of as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, we can choose charts so that overlapping regions agree on their "handedness"; these are orientable manifo ...

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Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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