Information about Algebraic Geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.
The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.
Descartes's idea of coordinates is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real numbers, but first complex numbers, and then elements of an arbitrary field became acceptable. Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century occurred within abstract algebraic framework, with increasing emphasis being placed on 'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology and complex geometry.
One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by Grothendieck and Serre, is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a regular function and a regular map and extensively draws on sheaf theory. Another important difference lies in the scope of the subject. Grothendieck's idea of scheme provides the language and the tools for geometric treatment of arbitrary commutative rings and, in particular, bridges algebraic geometry with algebraic number theory. Andrew Wiles's celebrated proof of Fermat's last theorem is a vivid testament to the power of this approach. André Weil, Grothendieck, and Deligne also demonstrated that the fundamental ideas of topology of manifolds have deep analogues in algebraic geometry over finite fields.
In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere in three-dimensional Euclidean space R3 could be defined as the set of all points (x,y,z) with
A "slanted" circle in R3 can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations
A function f : An → A1 is said to be regular if it can be written as a polynomial, that is, if there is a polynomial p in k[x1,...,xn] such that f(t1,...,tn) = p(t1,...,tn) for every point (t1,...,tn) of An.
Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will write the regular functions on An as k[An].
We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[An]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in An where every polynomial in S vanishes. In other words,
A subset of An which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below).
Given a subset U of An, can one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanishes on U, so I(U) is always an ideal of k[An].
Two natural questions to ask are:
For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k[An] are always finitely generated.
An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a prime ideal of the polynomial ring.
It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.
Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V]. This ring is called the coordinate ring of V.
Since regular functions on V come from regular functions on An, there should be a relationship between their coordinate rings. Specifically, to get a function in k[V] we took a function in k[An], and we said that it was the same as another function if they gave the same values when evaluated on V. This is the same as saying that their difference is zero on V. From this we can see that k[V] is the quotient k[An]/I(V).
If V' is a variety contained in Am, we say that f is a regular function from V to V' if the range of f is contained in V'.
This makes the collection of all affine varieties into a category, where the objects are affine varieties and the morphisms are regular maps. The following theorem characterizes the category of affine varieties:
Consider the variety V(y - x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (x, x2) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller.
Compare this to the variety V(y - x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (x, x3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of V(y-x3) is different from the behavior "at infinity" of V(y - x2). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space.
The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y - x3) has a singularity at one of those extra points, but V(y - x2) is smooth.
While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry.
Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolò Fontana Tartaglia on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry.[3] The French mathematicians Franciscus Vieta and later René Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).
During the same period, Blaise Pascal and Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Isaac Newton and Gottfried Wilhelm Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the calculus of infinitesimals of Joseph Louis Lagrange and Leonard Euler.
See e.g. Dieudonné, Jean: "The historical development of algebraic geometry", Amer. Math. Monthly 79 (1972), 827--866. (MR46#7232) or his more complete "History of algebraic geometry. An outline of the history and development of algebraic geometry", Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif., 1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004)
The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the development of Riemann surfaces.
In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. Later, from about 1960, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilised in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli.
An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography.
While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for effective computation with concretely-given polynomials have also been developed. The most important is the technique of Gröbner bases which is employed in all computer algebra systems.
The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.
Descartes's idea of coordinates is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real numbers, but first complex numbers, and then elements of an arbitrary field became acceptable. Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century occurred within abstract algebraic framework, with increasing emphasis being placed on 'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology and complex geometry.
One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by Grothendieck and Serre, is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a regular function and a regular map and extensively draws on sheaf theory. Another important difference lies in the scope of the subject. Grothendieck's idea of scheme provides the language and the tools for geometric treatment of arbitrary commutative rings and, in particular, bridges algebraic geometry with algebraic number theory. Andrew Wiles's celebrated proof of Fermat's last theorem is a vivid testament to the power of this approach. André Weil, Grothendieck, and Deligne also demonstrated that the fundamental ideas of topology of manifolds have deep analogues in algebraic geometry over finite fields.
Zeros of simultaneous polynomials
)
Sphere and slanted circle
A "slanted" circle in R3 can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations
Affine varieties
First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We define An(k) (or more simply An, when k is clear from the context), called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, An is, for the moment, just a collection of points.A function f : An → A1 is said to be regular if it can be written as a polynomial, that is, if there is a polynomial p in k[x1,...,xn] such that f(t1,...,tn) = p(t1,...,tn) for every point (t1,...,tn) of An.
Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will write the regular functions on An as k[An].
We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[An]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in An where every polynomial in S vanishes. In other words,
A subset of An which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below).
Given a subset U of An, can one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanishes on U, so I(U) is always an ideal of k[An].
Two natural questions to ask are:
- Given a subset U of An, when is U = V(I(U))?
- Given a set S of polynomials, when is S = I(V(S))?
For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k[An] are always finitely generated.
An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a prime ideal of the polynomial ring.
Regular functions
Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular function on an algebraic set V contained in An is defined to be the restriction of a regular function on An, in the sense we defined above.It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.
Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V]. This ring is called the coordinate ring of V.
Since regular functions on V come from regular functions on An, there should be a relationship between their coordinate rings. Specifically, to get a function in k[V] we took a function in k[An], and we said that it was the same as another function if they gave the same values when evaluated on V. This is the same as saying that their difference is zero on V. From this we can see that k[V] is the quotient k[An]/I(V).
The category of affine varieties
Using regular functions from an affine variety to A1, we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in An. Choose m regular functions on V, and call them f1, ..., fm. We define a regular function f from V to Am by letting f(t1, ..., tn) = (f1, ..., fm). In other words, each fi determines one coordinate of the range of f.If V' is a variety contained in Am, we say that f is a regular function from V to V' if the range of f is contained in V'.
This makes the collection of all affine varieties into a category, where the objects are affine varieties and the morphisms are regular maps. The following theorem characterizes the category of affine varieties:
- The category of affine varieties is the opposite category to the category of finitely generated integral k-algebras and their homomorphisms.
Projective space
)
parabola (y=x2, blue) and cubic (y=x3, red) in projective space
Compare this to the variety V(y - x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (x, x3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of V(y-x3) is different from the behavior "at infinity" of V(y - x2). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space.
The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y - x3) has a singularity at one of those extra points, but V(y - x2) is smooth.
While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry.
The modern viewpoint
The modern approach to algebraic geometry redefines the basic objects. Varieties are subsumed in Alexander Grothendieck's concept of a scheme. Schemes start with the observation that if finitely generated reduced k-algebras are geometrical objects, then perhaps arbitrary commutative rings should also be geometrical objects. As such, schemes become both a more general algebro-geometric object, and a convenient language to describe those objects. This language of schemes has proved to be a valuable way of dealing with geometric concepts and has become a cornerstone of modern algebraic geometry.History
Prehistory: Before the 19th century
Some of the roots of algebraic geometry date back to the later work of the Hellenistic Greeks such as Archimedes and Apollonius on conic sections.[1] The geometrical methods of the Greeks were first applied in a recognizably algebraic setting by 10th century scholar, Abu’l Wafa al-Buznaji, an astronomer and mathematician, who wrote a book of applied geometry called Kitab al handasa or the Book of Geometry. In it he provided solutions of geometrical problems with one opening of the compass; constructions of a square equivalent to other squares; regular polyhedra; approximate construction of regular heptagon (taking for its side half the side of the equilateral triangle inscribed in the same circle); constructions of parabola by points; geometrical solution of x4 = a and x4 + ax3 = b. Then Muhammed Al Battani, a 10th century mathematician and astronomer from Baghdad , computed sine, tangent and cotangent tables from 0° to 90° with great accuracy. Last and not least Persian mathematician Omar Khayyám (born 1048 A.D.) discovered the general method of solving cubic equations by intersecting a parabola with a circle.[2] Each of these primordial developments in algebraic geometry dealt with questions of finding and describing the intersections of algebraic curves.Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolò Fontana Tartaglia on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry.[3] The French mathematicians Franciscus Vieta and later René Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).
During the same period, Blaise Pascal and Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Isaac Newton and Gottfried Wilhelm Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the calculus of infinitesimals of Joseph Louis Lagrange and Leonard Euler.
See e.g. Dieudonné, Jean: "The historical development of algebraic geometry", Amer. Math. Monthly 79 (1972), 827--866. (MR46#7232) or his more complete "History of algebraic geometry. An outline of the history and development of algebraic geometry", Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif., 1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004)
Nineteenth and early 20th century
It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of homogeneous polynomial forms, and more specifically quadratic forms, on projective space. Subsequently, Felix Klein studied projective geometry (along with other sorts of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of transformations on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher degree birational transformations. This weaker notion of congruence would later lead members of the 20th century Italian school of algebraic geometry to classify algebraic surfaces up to birational isomorphism.The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the development of Riemann surfaces.
Twentieth century
van der Waerden, Oscar Zariski, André Weil and others attempted to develop a rigorous foundation for algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals.In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. Later, from about 1960, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilised in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli.
An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography.
While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for effective computation with concretely-given polynomials have also been developed. The most important is the technique of Gröbner bases which is employed in all computer algebra systems.
See also
- Important publications in algebraic geometry
- List of algebraic surfaces
- Root-finding algorithm
Notes
1. ^ Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press. pp. 108, 90.
2. ^ Kline ibid, pp. 193-195.
3. ^ Kline ibid, p. 279.
2. ^ Kline ibid, pp. 193-195.
3. ^ Kline ibid, p. 279.
References
A classical textbook, predating schemes:- W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 1. Cambridge University Press. ISBN 0-521-46900-7.
- Hodge, W. V. D.; Pedoe, Daniel (1994). Methods of Algebraic Geometry: Volume 2. Cambridge University Press. ISBN 0-521-46901-5.
- Hodge, W. V. D.; Pedoe, Daniel (1994). Methods of Algebraic Geometry: Volume 3. Cambridge University Press. ISBN 0-521-46775-6.
- David Cox; John Little, Donal O'Shea (1997). Ideals, Varieties, and Algorithms, second edition, Springer-Verlag. ISBN 0-387-94680-2.
- Phillip Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.
- Joe Harris (1995). Algebraic Geometry: A First Course. Springer-Verlag. ISBN 0-387-97716-3.
- David Mumford (1995). Algebraic Geometry I: Complex Projective Varieties, 2nd ed., Springer-Verlag. ISBN 3-540-58657-1.
- Miles Reid (1988). Undergraduate Algebraic Geometry. Cambridge University Press. ISBN 0-521-35662-8.
- Igor Shafarevich (1995). Basic Algebraic Geometry I: Varieties in Projective Space, 2nd ed., Springer-Verlag. ISBN 0-387-54812-2.
- David Eisenbud; Joe Harris (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5.
- Alexander Grothendieck (1960). Éléments de géométrie algébrique. Publications mathématiques de l'IHÉS.
- Alexander Grothendieck (1971). Éléments de géométrie algébrique, 2nd ed., Springer-Verlag. ISBN 3-540-05113-9.
- Robin Hartshorne (1997). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9.
- David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians, 2nd ed., Springer-Verlag. ISBN 3-540-63293-X.
- Igor Shafarevich (1995). Basic Algebraic Geometry II: Schemes and Complex Manifolds. Springer-Verlag. ISBN 0-387-54812-2.
- Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System
- Algebraic geometry entry on PlanetMath
- Algebraic Equations and Systems of Algebraic Equations at EqWorld: The World of Mathematical Equations
Major fields of mathematics |
|---|
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.
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commutative algebra studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the
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Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics.
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Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
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Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.
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In mathematics, equation solving is the problem of finding what values (numbers, functions, sets, etc.) fulfill a condition stated as an equality (an equation). Usually, this condition involves expressions with variables (or unknowns
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algebraic variety is essentially a (finite or infinite) set of points where a polynomial (in one or more variables) attains, or a set of such polynomials all attain, a value of zero.
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In mathematics, a solution set is a set of possible values that a variable can take on in order to satisfy a given set of conditions (which may include equations and inequalities).
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In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. is a polynomial.
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In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.
Circles are simple closed curves which divide the plane into an interior and exterior.
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Circles are simple closed curves which divide the plane into an interior and exterior.
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parabola (from the Greek: παραβολή) (IPA pronunciation: /pəˈrab(ə)lə/
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In algebraic geometry, the word lemniscate refers to any of several figure-eight or ∞ shaped curves, of which the best known is the Lemniscate of Bernoulli. It is also sometimes used to refer to the ∞ symbol used in mathematics as a symbol for infinity.
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Cassini oval is a set (or locus) of points in the plane such that each point p on the oval bears a special relation to two other, fixed points q1 and q2: the product of the distance from p to q1
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René Descartes (French IPA: [ʁə'ne de'kaʁt]) (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius
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coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. Also they make calculations possible in projective space just as Cartesian coordinates do in Euclidean space.
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Projective geometry is a non-metrical form of geometry. First developed by Desargues in the 17th century, it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others.
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Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
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In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables.
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Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to algebraic geometry, homological algebra, and functional analysis.
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Jean-Pierre Serre
Born September 15 1926
Bages, Pyrénées-Orientales, France
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Born September 15 1926
Bages, Pyrénées-Orientales, France
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regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined.
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regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined.
..... Read more.
..... Read more.