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Topic: Algebraic geometry and analytic geometry

	PlanetMath: algebraic geometry
		Algebraic geometry is the study of algebraic objects using geometrical tools.
		Algebraic groups are essentially matrix group schemes, and as such allow the tools of algebraic geometry to be applied to their study.
		This is version 11 of algebraic geometry, born on 2004-03-19, modified 2005-02-26.
	planetmath.org /encyclopedia/AlgebraicGeometry.html (2516 words)

	Geometry
		The next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are represented as ordered pairs or triples of numbers.
		The central notion in geometry is that of congruence.
		In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translations.
	www.ebroadcast.com.au /lookup/encyclopedia/ge/Geometry.html (266 words)

	Algebraic geometry and analytic geometry - Wikipedia, the free encyclopedia
		Where algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.
		Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces.
		It states that an analytic subspace of complex projective space that is closed in the strong topology is an algebraic subvariety (closed for the Zariski topology).
	en.wikipedia.org /wiki/Algebraic_geometry_and_analytic_geometry (780 words)

	analytic geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05
		branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic.
		In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x-axis.
		Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. Leibniz in the late 17th cent.
	www.bartleby.com /65/an/analytGeo.html (471 words)

	Analytic Geometry \| World of Mathematics
		Analytic geometry is a branch of mathematics which uses algebraic equations to describe the size and position of geometric figures on a coordinate system.
		The link between algebra and geometry was made possible by the development of a coordinate system which allowed geometric ideas, such as point and line, to be described in algebraic terms like real numbers and equations.
		Using the ideas of analytic geometry, it is possible to calculate the distance between the two points A and B, represented by the line segment AB which connects the points.
	www.bookrags.com /research/analytic-geometry-wom (2360 words)

	Analytic geometry - Wikipedia, the free encyclopedia
		Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra.
		The fact that Euclidean geometry is interpretable in the language of analytic geometry (that is, every theorem of one is a theorem of the other) is a key step of Alfred Tarski's proof that Euclidean geometry is consistent and decidable.
		Analytic geometry, for algebraic geometers, is also the name for the theory of (real or) complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables (or sometimes real ones).
	en.wikipedia.org /wiki/Analytic_geometry (484 words)

	Analytic Geometry
		Analytic geometry, otherwise known as coordinate geometry or cartesian geometry, is the brainchild of Pierre de Fermat and Rene Descartes.
		The cartesian plane, the basis of analytic geometry, allows algebraic equations to be graphically represented, in a process called graphing.
		Analytic geometry is the study of points, curves and lines defined by algebraic expressions.
	library.thinkquest.org /C0110248/geometry/analytic.htm (169 words)

	Analytic Geometry
		He considered the relation between geometric loci and algebraic equations in two or more variables, as well as the framework for this, a system of axes where lengths can be measured against.
		It is believed that he formulated the idea whilst watching a fly crawl along the ceiling of his room near a corner--he began expressing the path of the fly in terms of distance from the walls.
		The algebraic notation that we use today was introduced by Descartes, where unknown quantities were represented by the last few letters of the alphabet and the constants by the first few.
	library.thinkquest.org /C0110248/geometry/history5.htm (449 words)

	Algebraic Geometry (Site not responding. Last check: 2007-10-09)
		As the name suggests, algebraic geometry is a marriage of algebra and geometry.
		This should not be confused with analytic geometry, which is also a marriage of algebra and geometry.
		In contrast, algebraic geometry, an advanced topic in graduate mathematics, begins with an equation and builds the corresponding shape in n space.
	www.mathreference.com /ag,intro.html (158 words)

	14: Algebraic geometry
		Algebraic geometry combines the algebraic with the geometric for the benefit of both.
		Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery.
		Note that many computations in algebraic geometry are really computations in polynomials rings, hence computational commutative algebra applies.
	www.math.niu.edu /~rusin/known-math/index/14-XX.html (523 words)

	Amazon.ca: Algebraic Geometry: Books: Robin Hartshorne (Site not responding. Last check: 2007-10-09)
		Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris.
		That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic.
		It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables.
	www.amazon.ca /Algebraic-Geometry-Robin-Hartshorne/dp/0387902449 (2077 words)

	Algebraic Geometry (Site not responding. Last check: 2007-10-09)
		Algebraic Geometry is a subject with historical roots in analytic geometry.
		It subsumes most of commutative algebra and much of algebraic number theory, and overlaps with differential geometry, modern "analytic geometry" (complex manifolds), Lie groups, representation theory, theoretical physics, and to a lesser extent the theory of partial differential equations.
		In addition to being one of the central disciplines of pure mathematics, algebraic geometry has developed an applied side which is linked to problems in computational complexity and the theory of algorithms, symbolic computation, robotics, control theory, computational geometry, geometric modeling, image recognition, computer vision, and scientific visualization.
	www.math.tamu.edu /~Peter.Stiller/agpage.html (224 words)

	IRMAR: Real algebraic geometry, symbolic computation and complexity
		Some are related with "classical" algebraic geometry and algebra (for instance quadratic forms and sums of squares, the importance of which is well known since Hilbert's 17th problem).
		Analytic geometry and model theory are used for the study of "tame topology" (or o-minimal structures), which provide a framework where nice properties (finiteness properties, for instance) of algebraic objects still hold.
		Part of our activity in symbolic computation is related to algorithms in real algebraic geometry and addresses this decision problem and generalisations of it, or also effective versions of results such as the "real Nullstellensatz".
	www.math.univ-rennes1.fr /geomreel/index.html.en.html (573 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Analytic geometry studies solution sets of polynomial equations f(x, y) = 0 or g(x, y, z) = 0, where f and g are polynomials of degree 1 or 2.
		Algebraic geometry begins where analytic geometry leaves off: it studies solution sets to polynomial equations of degree d in n variables, where d and n are arbitrary positive integers (and more generally, to systems of such polynomial equations).
		A negative side effect of this recent wave of creative activity in algebraic geometry is that the subject has become more technical and consequently, more difficult to enter.
	islab.oregonstate.edu /emails/islmail-01/0003 (233 words)

	analytic geometry — FactMonster.com
		In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope
		In analytic geometry, one of the four regions of the plane determined by two lines,...
		differential geometry - differential geometry, branch of geometry in which the concepts of the calculus are applied to...
	www.factmonster.com /ce6/sci/A0803868.html (418 words)

	A Unified Algebraic Framework for Classical Geometry
		Though fundamental ideas of classical geometry are permanently imbedded and broadly applied in mathematics and physics, the subject itself has practically disappeared from the modern mathematics curriculum.
		Because the three geometries are obtained by interpreting null vectors of the same Minkowski space differently, natural correspondences exist among geometric entities and constraints of these geometries.
		Geometric Algebra was applied to hyperbolic geometry by H. Li in [L97], stimulated by Iversen's book [I92] on the algebraic treatment of hyperbolic geometry and by the paper of Hestenes and Ziegler [HZ91] on projective geometry with Geometric Algebra.
	modelingnts.la.asu.edu /html/UAFCG.html (2056 words)

	PlanetMath: analytic algebraic function
		is said to be real-analytic algebraic or a Nash function.
		-analytic algebraic if each component function is analytic algebraic.
		This is version 4 of analytic algebraic function, born on 2005-12-05, modified 2005-12-09.
	planetmath.org /encyclopedia/AnalyticAlgebraicMapping.html (187 words)

	The Math Forum - Math Library - Geometry (Site not responding. Last check: 2007-10-09)
		A collection of handouts for a two-week summer workshop entitled 'Geometry and the Imagination', led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 17-28, 1991.
		A short article designed to provide an introduction to geometry, including classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory.
		The FoCM's primary aim is to further the understanding of the deep relationships between mathematical analysis, topology, geometry and algebra and the computational process as they are evolving together with the modern computer.
	mathforum.org /library/topics/geometry (2331 words)

	Ben Richert: Math 614, Fall 2001 (Site not responding. Last check: 2007-10-09)
		As well as providing the foundation for algebraic geometry, complex analytic geometry, and algebraic number theory, this field has developed into a beautiful and deep theory in its own right, with applications for nearly every algebraist.
		Algebraic geometers, number theorists, algebraic combinatorialists, lie theorists, and non-commutative algebraists, among others, find it useful.
		Math 615 is a follow-up on the first course in commutative algebra (614), and our first order of business will be to continue with the topics basic to such a study.
	www.calpoly.edu /~brichert/teaching/oldclass/w2002615/615.html (247 words)

	Analytic Geometry - Mathematics and the Liberal Arts
		Analytic geometry and calculus were invented in part to better understand motion.
		The rise and decline of Greek geometry, the logical structure of Greek proofs.
		The interaction of Islamic algebra with algebra and geometry.
	math.truman.edu /~thammond/history/AnalyticGeometry.html (953 words)

	algebraic geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05
		branch of geometry, based on analytic geometry, that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates).
		In plane geometry an algebraic curve is the locus of all points satisfying the polynomial equation f(x,y)=0; in three dimensions the polynomial equation f(x,y,z)=0 defines an algebraic surface.
		The intersection of two or more algebraic hypersurfaces defines an algebraic set, or variety, a concept of particular importance in algebraic geometry.
	www.bartleby.com /65/al/algebGeo.html (186 words)

	Real Algebraic and Analytic Geometry - Preprint Server
		Lev Birbrair, Alexandre Fernandes: Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities.
		Sérgio Alvarez, Lev Birbrair, João Costa, Alexandre Fernandes: Topological K-Equivalence of Analytic Function-Germs.
		Aleksandra Nowel, Zbigniew Szafraniec: On trajectories of analytic gradient vector fields on analytic manifolds.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG (2160 words)

	Arithmetic Summary
		Euclid rejected analytic geometry (numerical measurement of length and area) and relied instead on purely synthetic (non-numerical) constructions, largely because Greek mathematics could not adequately denote and compute with real numbers, especially irrationals such as √2.
		Conversely, the flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.
		Since the introduction of the electronic calculator, which can perform the algorithms far more efficiently than humans, an influential school of educators has argued that mechanical mastery of the standard arithmetic algorithms is no longer necessary.
	www.bookrags.com /Arithmetic (3476 words)

	Clay Mathematics Institute: Workshop on Algebraic Statistics and Computational Biology
		By one count, there seems to be six independent, essentially simultaneous constructions currently underway, of parametrized (p-adic) spaces of automorphic forms attached to algebraic groups, and their concomitant Galois representations.
		These parameter spaces are called "eigenvarieties," or "Hecke varieties," and are being constructed by different people, in different but sometimes overlapping contexts: for unitary groups of higher rank, for symplectic groups of high rank, for general linear groups over number fields.
		Eigenvarieties are a unifying force for classical and modern aspects of number theory, algebraic geometry, analytic geometry (p-adic, mainly) and the theory of group representations (both automorphic representations and Galois representations).
	www.claymath.org /programs/cmiworkshops/eigenvarieties (338 words)

	Algebra, Analytic Geometry: Analytic Geometry Introduction
		This is the simplest area topic for students able to solve two equations in two unknowns using exact arithmetic with integers and whole numbers, efficiently.
		Assumption that the plane can be coordinates using lines parallel to a horizontal and vertical axis and real numbers as coordinates leads to a powerful numerical or algebraic model of work with points, lines, circles and further geometric objects in the plane.
		By connecting geometry with numbers, exact computations properties of numbers can be used to arrive at conclusions about geometry via chains of reasons with coordinates which provide a precision missing in useful but suggestive and approximately drawn diagrams.
	whyslopes.com /Analytic-Geometry-Functions (700 words)

	Publisher description for Library of Congress control number 85011691 (Site not responding. Last check: 2007-10-09)
		In addition to being an interesting and deep subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytic geometry.
		The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject.
		Consequently it can be used by graduate students taking courses in commutative algebra, algebraic geometry or complex analytic geometry, but will also serve as an introduction to these subjects for non-specialists.
	www.loc.gov /catdir/description/cam031/85011691.html (207 words)

	AllRefer.com - Cartesian coordinates (Mathematics) - Encyclopedia
		Analogous systems may be defined for describing points in abstract spaces of four or more dimensions.
		Many of the curves studied in classical geometry can be described as the set of points (x,y) that satisfy some equation f(x,y)=0.
		In this way certain questions in geometry can be transformed into questions about numbers and resolved by means of analytic geometry.
	reference.allrefer.com /encyclopedia/C/Cartes-coo.html (279 words)

	Find in a Library Algebraic geometry and analytic geometry : proceedings of a conference held in Tokyo, Japan, August ...
		Find in a Library Algebraic geometry and analytic geometry : proceedings of a conference held in Tokyo, Japan, August 13-17, 1990
		Algebraic geometry and analytic geometry : proceedings of a conference held in Tokyo, Japan, August 13-17, 1990
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/68b3cdc2eb11ca8fa19afeb4da09e526.html (89 words)

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