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Topic: Complex geometry

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	Complex geometry - Wikipedia, the free encyclopedia
		In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables.
		Complex analytical geometry deals with solving geometrical problems, especially those involving angles, by means of complex algebra.
		Complex numbers can therefore be used to advantage to solve problems in elementary geometry.
	en.wikipedia.org /wiki/Complex_geometry (184 words)

	Complex number
		Complex numbers were first introduced in connection with explicit formulas for the roots of cubic[?] polynomials.
		The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
		Complex numbers are used in signal analysis[?] and other fields as a convenient description for periodically varying signals.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Complex_number.html (1524 words)

	PlanetMath: algebraic geometry
		Of course, if the ring is the complex numbers, we can apply the highly succesful theories of complex analysis and complex manifolds to address the problems; many powerful tools are available; de Rham cohomology, singular homology, Hodge theory, spectral sequences and many others.
		This is a problem, as in the complex category, cohomology with constant coefficients (in fact, usually with integer coefficients) determines most of the cohomological invariants that are of interest, such as the Betti numbers.
		This is version 11 of algebraic geometry, born on 2004-03-19, modified 2005-02-26.
	planetmath.org /encyclopedia/AlgebraicGeometry.html (2513 words)

	Algebraic Structure of Complex Numbers
		For, without (1) and (2), the theory of complex numbers would not deliver the closure to the branch of algebra that drove much of its development, viz., the search for the roots of polynomial equations.
		In the complex plane the axes also are referred to as real and imaginary, although both are real enough to the extent that the only way to distinguish between the two is by means of orientation: the rotation from the real to the imaginary axis proceeds counterclockwise.
		Complex numbers for which the real part is 0, i.e., the numbers in the form yi, for some real y, are said to be purely imaginary.
	www.cut-the-knot.org /arithmetic/algebra/ComplexNumbers.shtml (1204 words)

	Présentation
		Complex Analysis and Analytic Geometry belong closely together and are one of the few fields in the center of pure mathematics with many applications to other areas of pure mathematics (algebraic geometry, differential geometry, dynamical systems, P.D.E., topology, number theory, etc.) and applied Mathematics (theoritical physics, geophysics, mathematical economy, tomography).
		Complex Monge-Ampère equation and canonical bundle of a complex manifold.
		Differential geometry of vector bundles on complex manifolds: complex and holomorphic structures, invariants and singularities, gauge theories and moduli, twistor spaces.
	www.math.jussieu.fr /projets/ac/Reseau/presentation.htm (2291 words)

	Geometry of Complex Numbers (Site not responding. Last check: 2007-10-09)
		Complex numbers are ordered pairs of real numbers, so they can be represented by points in the plane.
		In this section we show the effect that algebraic operations on complex numbers have on their geometric representations.
		Addition of complex numbers is analogous to addition of vectors in the plane.
	math.fullerton.edu /mathews/c2003/ComplexGeometryMod.html (475 words)

	Ma 191j, Real and Complex Hyperbolic Geometry
		Hyperbolic space, the geometry which has constant negative curvature, is one of the central examples in Riemannian geometry and has deep connections to the theory of 3-manifolds.
		Complex hyperbolic space is its "complexification", and is a homogeneous geometry of variable negative curvature.
		Shapes of polyhedra and triangulations of the sphere, Geometry and Topology Monographs.
	www.its.caltech.edu /~dunfield/classes/2004/191/index.html (802 words)

	Math Forum - Ask Dr. Math Archives: College Imaginary/Complex Numbers
		Complex Cube Roots of Unity and Simplifying [05/17/2005]
		With w denoting either of the two complex cube roots of unity, find [(2w + 1)/(5 + 3w + w^2)] + [(2w^2 + 1)/(5 + w + 3w^2)], giving your answer as a fraction a/b, where a, b are integers with no factor in common.
		The real numbers and the imaginary numbers are subsets of the complex numbers.
	mathforum.org /library/drmath/sets/college_complex.html (814 words)

	Complex geometry
		Complex analytical geometry deals with solving geometrical problems, especially involving angles, by means of complex algebra.
		Rather than represent a point in the plane as a pair of Cartesian coordinates, it can simply be represented as a complex number, which in turn can be written in either rectangular or polar form.
		Complex numbers can thus be used to solve problems in elementary geometry[?] in simple and clear way.
	www.fastload.org /co/Complex_geometry.html (142 words)

	Constructive solid geometry - Wikipedia, the free encyclopedia
		Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects.
		Often CSG presents a model or surface that appears visually complex, but is actually little more than cleverly combined or decombined objects.
		When CSG is procedural or parametric, the user can revise their complex geometry by changing the position of objects or by changing the Boolean operation used to combine those objects.
	en.wikipedia.org /wiki/Constructive_solid_geometry (480 words)

	Coformal theories, curved phase spaces, relativistic wavelets and the geometry of complex domains
		We investigate some aspects of complex geometry in relation with possible applications to quantization, relativistic phase spaces, conformal field theories, general relativity and the music of two and three-dimensional spheres.
		Complex manifolds and in particular classical domains have been studied for many years by mathematicians and theoretical physicists.
		For instance, we cannot say that the study of complex domains (and in particular Cartan classical domains) belongs more to the realm of analysis than to the one of algebra or of geometry but, it is clear that most mathematical articles dealing with the subject fall into one of these three families.
	quantumfuture.net /quantum_future/conformal_theories.htm (1465 words)

	Complex Geometry Continued (Site not responding. Last check: 2007-10-09)
		You will see that this complex exponential has all the properties of real exponentials that you studied in earlier mathematics courses.
		Euler's formula is of tremendous use in establishing important algebraic and geometric properties of complex numbers.
		It also allows you to express a polar form of the complex number z in a more compact way.
	math.fullerton.edu /mathews/c2003/ComplexGeometryContinuedMod.html (680 words)

	The Reference Frame: Generalized geometry (Site not responding. Last check: 2007-10-09)
		This bihermitean geometry is an older concept than generalized geometry, but we could not resist the temptation to think that they're equivalent or at least closely related.
		The language of generalized geometry should be good for T-duality, but we had a feeling that the integrity and topology of the base space of the bundle is preserved anyway, whatever one does with the bundles, so it does not treat the T-dual backgrounds on equal footing.
		About the generalized stuff and bihermitian geometry, there is a theorem by Gualtieri (in his thesis) who proved that a bihermititan structure and a generalized Kaehler structure (what you get if you have two commuting generalized complex structures) are equivalent.
	motls.blogspot.com /2005/04/generalized-geometry.html (1856 words)

	Complex Numbers and Geometry
		For any complex number, c = x + y i, where x and y are real numbers, we refer to x as the "real part of c" and y as the "complex part of c".
		We plot a complex number by associating the real part with the horizontal axis and the complex part with the vertical axis.
		The geometric rule for addition is implicit in the very way we plot complex numbers, since a general complex number is naturally written as a sum.
	campus.northpark.edu /math/PreCalculus/Transcendental/Trigonometric/Complex/index.html (2082 words)

	EDGE
		The study of Riemannian geometry in the complex setting often yields strong and interesting results that can have an impact both on Riemannian geometry and algebraic geometry.
		Symplectic geometry is a part of geometry where `almost-complex' methods already play a large role, and this area forms an integral part of the proposed research.
		Twistor methods give a correspondence between holomorphic geometry and low-dimensional conformal geometry, and also allow the use of complex methods in the study of quaternionic geometry, (parts of) gauge theory and integrable systems, all of which are subjects included in this research proposal.
	edge.imada.sdu.dk (525 words)

	Making Mathematics: Mathematics Tools: The Geometry of Complex Numbers
		Dave's Short Course on Complex Numbers provides a thorough and accessible introduction to complex numbers, their meaning, geometry, and operations.
		Geometry and Complex Numbers offers a text with exercises on complex numbers and trigonometry.
		Complex Applet provides an interactive tool for studying the geometry of complex numbers.
	www2.edc.org /makingmath/mathtools/complex/complex.asp (326 words)

	51: Geometry
		Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.
		Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
		A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	Plane Graphic Calculator Gallery 2 : Complex Geometry
		Therefore, raising a complex number to the power of n has the effect of simultaneously "multiplying its phase angle by n" and "raising its magnitude to the nth power", for all real values of n.
		Move point c up and down with the mouse to explore how each complex number along a horizontal line (red) is transformed by the complex exponential to generate a new line (purple).
		Move point c left and right with the mouse to explore how each complex number along a vertical line (red) is transformed by the complex exponential to generate a circle (purple).
	www.accesscom.com /~lillge/pgc/pgc-gallery2.shtml (1111 words)

	University of Michigan Geometry Seminar, Winter 2006
		A fake projective plane is a smooth compact complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane.
		An interesting problem in complex algebraic geometry is to determine all fake projective planes and study their geometric (and arithmetic) properties.
		The resulting geometry, which was investigated in the affine hypersurface theory by affine geometers, has recently found applications in theoretical statistics.
	www.math.lsa.umich.edu /seminars/geometry/index.html (1059 words)

	The geometry of complex multiplication (Site not responding. Last check: 2007-10-09)
		The problems we just did suggest that complex multiplcation is easily described in the polar representation.
		When we multiply complex numbers the length of the product is the product of the lengths of the numbers being multiplied.
		When we multiply complex numbers the polar angle of the product is the sum of the polar angles of the numebrs being multiplied.
	www.math.pitt.edu /~sparling/23012/23012complex1/node4.html (215 words)

	Undecidable Problems in Fractal Geometry
		In this paper, the intuitive fact that fractals are complex objects is made precise from a computational point of view by proving that there do not exist algorithms to answer simple questions about fractals.
		Fractal geometry was pioneered by Mandelbrot who showed that many natural objects and phenomena have fractal characteristics [9].
		Since the fractal geometry has strong links with the theory of complex systems, one is pleased to see these three scientifically rich disciplines conveying the same basic message, that finitely describable and seemingly simple systems have complex and provably unpredictable behaviour and such systems occur everywhere.
	www.complexity.org.au /ci/vol02/undecide/undecide.html (1961 words)

	complex geometry (Site not responding. Last check: 2007-10-09)
		Abacci > Abaccipedia > co > complex geometry
		In mathematics, complex geometry is the application of complex numbers to plane geometry.
		Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete.
	www.abacci.com /wikipedia/topic.aspx?cur_title=complex_geometry (216 words)

	Complex geometry - meaning of word
		Rather than represent a point in the plane as a pair of Cartesian coordinate system, it can be represented as a single complex number, which can be written at will in either rectangular or polar form.
		Complex analytical geometry deals with solving geometrical problems, especially those involving angles, by means of complex number.
		The geometry of complex numbers is already explained in complex number.
	www.wordsonline.org /Complex_geometry (230 words)

	Graduate Study in Algebraic Geometry
		Algebraic Geometry is a new emphasis area in the department, whose purpose is the geometric study of solutions of systems of polynomial equations in several variables.
		A transcendental algebraic geometry course, following Griffith's and Harris', Principles of Algebraic Geometry, this course focuses on the study of compact complex manifolds, and in the process lays the foundation for further study in algebraic geometry.
		The foundational results frequently have counterparts in a purely algebraic formulation of algebraic geometry as is typically taught in Math 511, while the techniques are frequently very different.
	www.math.uiuc.edu /ResearchAreas/AlgebraicGeometry/gradag.html (404 words)

	Amazon.com: Geometry of Complex Numbers: Books: Hans Schwerdtfeger (Site not responding. Last check: 2007-10-09)
		Schwerdtfeger's nice little book starts at the beginning with geometry of circles, Moebius transformations (a third of the book), and it covers some selected aspects of complex function theory, but the emphasis is on elementary geometry.
		It is suitable as a supplement in a standard course in complex function theory, at the late undergraduate level, or perhaps at beginning graduate.
		The problem with complex functions is they are hard to visualize because the input is a plane and the output is another plane.
	www.amazon.com /Geometry-Complex-Numbers-Hans-Schwerdtfeger/dp/0486638308 (1213 words)

	Algebra, Analytic Geometry: 8 Complex Numbers I
		New (August 3, 2001): Two webpages on Complex Numbers (this one) and on Distributive Law for Complex Numbers offer a short way to reach and explain trigonometry, the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem.
		Here is a geometric story which describes the complex numbers, or what mathematicians since Gauss in the 1840's have regarded as the complex numbers.
		The angle of a purely imaginary complex number z = a+ib = 0+ib = (0,b) is 90 degrees or 270 degrees (modulo 360 degrees), depending on the sign of the imaginary part b.
	whyslopes.com /Analytic-Geometry-Functions/analGeo07a_Complex_Number_Intro.html (2258 words)

	Digital Form-Finding: A Case Study in Complex Geometry (Site not responding. Last check: 2007-10-09)
		The development of sophisticated digitally based environments has led directly to a preoccupation with their use in developing forms and shapes involving highly complex geometric forms and manipulations.
		The complex external shape is a priori determined.
		Sectional planes are passed at regular intervals through the whole shape; and the geometry of the intersection is used to define the shapes of structural members.
	www.pubs.asce.org /WWWdisplay.cgi?0603664 (238 words)

	MAT 545 - Complex Geometry -- Spring 2005
		Griffiths' and J. Harris' textbook Principles of Algebraic Geometry and part of Claire Voisin's textbook Hodge Theory and Complex Algebraic Geometry I.
		Complex analytic and algebraic geometry, Jean-Pierre Demailly (a compressed PostScript file of the current version).
		Complex differential geometry: complex vector bundles, hermitian metrics, connections, hermitian vector bundles, the metric connection, sub-bundles and quotient bundles, tensor bundles, curvature, Chern classes (differential geometric approach), positivity.
	www.math.sunysb.edu /~sorin/545-2005 (491 words)

	The Book Pl@ce: Title Detail
		"...focused mainly on complex differential geometry and holomorphic bundle theory.
		Manin is an outstanding mathematician, and writer as well, perfectly at ease in the most abstract and complex situation.
		Methods of Complex Geometry, in Particular Sheaf Cohomology, are Used Throughout.
	www.thebookplace.com /bookplace/display.asp?K=181554515004533&aub=Y&m=214&dc=3201 (281 words)

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