algebraic geometry

André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. He is known for his foundational work in number theory and algebraic geometry. He was a founding member, and de facto the early leader, of the influential Bourbaki group. The philosopher Simone Weil was his sister. André Weil - Life. Born in Paris to Alsatian parents who fled the annexation of Alsace-Lorraine to Germany, he studied in Paris, Rome and Göttingen and received his doctorate in 1928. He s ...

Including:

• André Weil - Life
• André Weil - Work
• André Weil - Books
• André Weil - External link
• André Weil - Bibliography

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Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki< ...

Including:

• Nicolas Bourbaki - Books by Bourbaki
• Nicolas Bourbaki - Influence on mathematics in general
• Nicolas Bourbaki - The group
• Nicolas Bourbaki - The Bourbaki perspective and its limitations
• Nicolas Bourbaki - Dieudonné as speaker for Bourbaki
• Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. Spectrum of a ring - Zariski topology. Spec(R) can be turned into a topological space as follows: a subset V of Spec(R) is closed if and only if there exists a subset I of R< ...

Including:

• Spectrum of a ring - Zariski topology
• Spectrum of a ring - Sheaves and schemes
• Spectrum of a ring - Functoriality
• Spectrum of a ring - Motivation from algebraic geometry
• Spectrum of a ring - External link

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

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In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fun ...

Including:

• Algebraic variety - Formal definitions
• Algebraic variety - Affine varieties
• Algebraic variety - Projective varieties
• Algebraic variety - Basic results
• Algebraic variety - Discussion and generalizations

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

Including:

• Category theory - Background
• Category theory - Historical notes
• Category theory - Categories objects and morphisms
• Category theory - Some properties of morphisms
• Category theory - Functors
• Category theory - Natural transformations and isomorphisms
• Category theory - Universal constructions limits and colimits
• Category theory - Equivalent categories
• Category theory - Further concepts and results
• Category theory - Higher-dimensional categories

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Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of its most extreme scientific personalities, with achievements over a short span of years that are still scarcely credible in their broad scope and sheer bulk, and an approach that antagonised even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and co-awarded the Crafoord Prize with Pierre Delig ...

Including:

• Alexander Grothendieck - Mathematical achievements
• Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles
• Alexander Grothendieck - Life
• Alexander Grothendieck - Childhood and studies
• Alexander Grothendieck - Politics and retreat from scientific community
• Alexander Grothendieck - Manuscripts written in the 1980s
• Alexander Grothendieck - Disappearance

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

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In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a complex torus that can be embedded into projective space. Also it is used for the generalization of this concept studied in algebraic geometry over fields more general than the complex numbers. One-dimensional abelian varieties are elliptic curves. Abelian variety - History and motivation. The success in the early nineteenth century of the theory of elliptic functions in giving a basis for the the ...

Including:

• Abelian variety - History and motivation
• Abelian variety - Analytic theory
• Abelian variety - Definition
• Abelian variety - Riemann conditions
• Abelian variety - The Jacobian of an algebraic curve
• Abelian variety - Abelian functions
• Abelian variety - Algebraic definition
• Abelian variety - Structure of the group of points
• Abelian variety - Polarization and dual abelian variety
• Abelian variety - Dual abelian variety
• Abelian variety - Polarizations
• Abelian variety - Polarizations over the complex numbers
• Abelian variety - Abelian scheme

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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 ...

Including:

• 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, 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 ...

Including:

• Coherent sheaf - Coherent cohomology

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Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. His collaboration with Paul Gordan led to the introduction of Clebsch-Gordan coefficients for spherical harmonics, which are now widely used in quantum mechanics. Together with Carl Neumann he founded the mathematical research journal Mathematische Annalen. Alfred Clebsch - External link. Biography at the MacTutor archive ...

Including:

• Alfred Clebsch - External link

Read more here: » Alfred Clebsch: Encyclopedia - Alfred Clebsch

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathemat ...

Including:

• Geometry - The earliest geometry
• Geometry - The Greek period c. 600 B.C. – 600 A.D.
• Geometry - Thales and Pythagoras
• Geometry - Plato
• Geometry - Euclid
• Geometry - Archimedes
• Geometry - After Archimedes
• Geometry - The Middle Ages Renaissance and Reformation
• Geometry - The 17th and early 18th centuries
• Geometry - The late 18th and 19th centuries
• Geometry - Non-Euclidean geometry
• Geometry - Introduction of mathematical rigor
• Geometry - Analysis situs or topology
• Geometry - The 20th century

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The Abel Prize is awarded annually by the King of Norway to outstanding mathematicians. In 2001 the government of Norway announced that the bicentennial of Norwegian mathematician Niels Henrik Abel's birth (which was 1802) would mark the commencement of a new prize for mathematicians, named after Abel. The Norwegian Academy of Science and Letters annually declares the winner of the Abel Prize after selection by a committee of five international mathematicians. The amount of money that comes with the prize is usually close to on ...

Including:

• Abel Prize - Laureates
• Abel Prize - External link

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In mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. It is named after its originator, Oscar Zariski. The Zariski topology is defined by defining the closed sets to be the sets consisting of the mutual zeroes of a set of polynomials. (See affine varieties) This is a topology, because two basic axioms can be checked. The intersection of any number of closed sets is the set of mutual zeros of the union of all the defining polynomials. The union of two closed sets, defined ...

Including:

• Zariski topology - Remarks

Read more here: » Zariski topology: Encyclopedia - Zariski topology

In mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. The main burden was that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modelled on the Riemann zeta func ...

Including:

• Weil conjectures - Background and history
• Weil conjectures - Statement of the Weil conjectures
• Weil conjectures - Examples
• Weil conjectures - The projective line
• Weil conjectures - Projective space
• Weil conjectures - Elliptic curves
• Weil conjectures - Weil cohomology

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Variety. Variety (linguistics) is a concept that includes for instance dialects, standard language and jargon. Variety (biology) is a rank in botany below that of species. Variety (plant) is a legal term. Variety (philately) is a term in stamp collecting. Variety (radio) is a format of radio programming. Variety (mineralogy) is a mineral-subform Variety is also the name of an entertainm

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In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve). Curve - Definitions. In ma ...

Including:

• Curve - Definitions
• Curve - Conventions and terminology
• Curve - Lengths of curves
• Curve - Differential geometry
• Curve - Algebraic curve
• Curve - History

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In algebraic geometry, the De Rham-Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question. Let be a sheaf on a topological space X and a resolution of by acyclic sheaves. Then where denotes the q-th sheaf cohomology group of X with coefficients in This article incorporates material from De Rham-Weil t ...

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