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Topic: Complex projective space

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  Complex projective space - Wikipedia, the free encyclopedia
Complex projective space is a complex manifold that may be described by n+1 complex coordinates as
It is a special case of a Grassmannian, and is a homogeneous space for various Lie groups.
It is BU(1), the classifying space of U(1), in the sense of homotopy theory, and so classifies complex line bundles; equivalently it accounts for the first Chern class.
en.wikipedia.org /wiki/Complex_projective_space   (410 words)

 NationMaster - Encyclopedia: Projective space
Projective space is basic in algebraic geometry, through the rich field of projective geometry developed in the nineteenth century but also in the constructions of the modern theory (based on graded algebras).
Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.
Projective spaces are essential to algebraic geometry through the rich field of projective geometry developed in the nineteenth century, but also in the constructions of the modern theory (based on graded algebras).
www.nationmaster.com /encyclopedia/Projective-space   (1799 words)

 PlanetMath: projective space
Projective space is defined to be the set of the corresponding equivalence classes.
A projective automorphism, also known as a projectivity, is a bijective transformation of projective space that preserves all incidence relations.
This is version 8 of projective space, born on 2001-12-21, modified 2006-07-27.
planetmath.org /encyclopedia/ProjectiveSpace.html   (414 words)

 Projective space - Wikipedia, the free encyclopedia
In mathematics, a projective space is a fundamental construction, obtained from a vector space over an arbitrary division ring, in particular over a field.
It generalises the notion of projective plane, which is constructed from a three-dimensional vector space.
The basic construction, given a vector space V over a division ring K, is to form the set of equivalence classes of non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to w if v = cw with c in K non-zero.
en.wikipedia.org /wiki/Projective_space   (792 words)

 Quaternionic projective space - Wikipedia, the free encyclopedia
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H.
It is a homogeneous space for a Lie group action, in more than one way.
Its direct construction is as a special case of the projective space over a division algebra.
en.wikipedia.org /wiki/Quaternionic_projective_space   (263 words)

 Hilbert Space
Hilbert space is not a space of simple points, rather it is a space of functions at a higher level of mathematical abstraction.
Rather, Hilbert space is a mathematical device for arranging pieces of information, with each complex coordinate representing a possibility, or probability amplitude, for a given quantum state that might correspond to a definite eigenvalue for energy, or position, or momentum, or spin, etc. Note, that not all of these observable properties can be definite simultaneously.
The angular momentum of a particle relative to a direction in space is incompatible with the angle of rotation of that particle in a plane perpendicular to that direction in space.
www.qedcorp.com /pcr/pcr/hilberts.html   (2670 words)

 Projective Space -- Recommendations and Resources   (Site not responding. Last check: 2007-11-06)
In mathematics, projective spaces are a fundamental construction, obtained from a vector space over an arbitrary division ring, in particular over a field.
The Teacher in Space Project (TISP) is a NASA program designed to educate students and spur excitement in math, science, and space exploration.
Complex projective space is a complex manifold that may be described by ''n+1'' complex coordinates as :
www.becomingapediatrician.com /health/121/projective-space.html   (1142 words)

 Senior Thesis Topics
Classically, enumerative, or counting, questions have been answered in complex projective space where the answers are precise.
In this project, we ask these enumerative questions in real affine space; in real space, the answers may no longer be precise.
The goal of invariant theory is to describe the algebra of invariants for a vector space under a given group action.
www.williams.edu /go/math/tgarrity/thesis.html   (1628 words)

 Math Colloquia
For example the complex projective space, CP^n is the moduli space of complex lines through the origin in the complex vector space C^{n+1}.
Closely related to these spaces are spaces of holomorphic maps from Riemann surfaces to other complex manifolds and moduli spaces of of complex vector bundles over Riemann surfaces and other complex manifolds.
For example the moduli space of linear control systems with $n$ inputs and $m$ outputs is identified with the mapping space Hol(P^1, G_{n,m}), where P^1 is the Riemann sphere, and G_{n,m} is the complex Grassmannian of complex n-planes in C^{n+m}.
www.math.ucsc.edu /seminars/past/f97.collo.html   (1555 words)

 Tutorial Topics, Fall and Spring (2006-2007)   (Site not responding. Last check: 2007-11-06)
Generally, the projective setting for geometric objects is more regular than the linear setting, because the structure of a projective space is in a sense more uniform than that of a linear space.
Another important feature is the compactness of projective spaces (meaning that there is no infinity), which disallows much of the diversity of, say, smooth functions on them.
Projective spaces and varieties are a good compromise between richness of structure and plentitude of methods.
www.math.harvard.edu /undergrad/Pamphlets/tutorial.html   (1319 words)

 ZeroDivisor Algebras, Charles Muses
the space of one-dimensional linear subspaces of Va)...
Acccording to the theory of (complex projective) quadric surfaces...
Split number-algebras, such as split complex numbers, split quaternions, Split Octonions, and split sedenions, are number-algebras of signature (p-1, p+1) with p dimensions of signature -1 plus a real 1 and p dimensions, other than the real axis, of signature +1.
www.valdostamuseum.org /hamsmith/NDalg.html   (3990 words)

 Introduction to Complex Manifolds   (Site not responding. Last check: 2007-11-06)
Complex manifolds play a great role in many areas of modern mathematics.
Complex manifold are of importance also in the transcendental approach to algebraic geometry in higher dimension; a pioneer here was Hodge.
A basic question is then to find conditions for a complex manifold to admit imbedding in a complex projective space of sufficiently high dimension -- an answer to this question is provided by the Kodaira imbedding theorem.
www.maths.lth.se /matematiklu/personal/jaak/Complex-Manifolds.html   (218 words)

Projective geometry goes way back to the Renaissance painters and their interest in perspective, and axiomatic projective geometry was very fashionable in the 19th century, but here we are seeing it in a more modern light, because we're seeing its relation to quantum logic.
Acting on this space we have two symmetry groups: SL(2,C), which is the double cover of the Lorentz group in 4d spacetime, and SU(2), which is the double cover of the rotation group in 3d space.
For starters, each complex simple Lie group G is the symmetries of a kind of generalized projective geometry called an "incidence geometry".
math.ucr.edu /home/baez/week178.html   (3470 words)

 Department of Mathematics at MIT | Graduate Study : Spring 2002 Thesis Defenses
In particular, we describe spaces in the genus of infinite quaternionic projective space which occur as targets of essential maps from infinite complex projective space, and we compute explicitly the homotopy classes of maps in these cases.
The most straightforward applications of the methods developed are to enumeration of rational curves with a cusp of specified nature in projective spaces.
The applications described include enumeration of rational curves with a (3,4)-cusp, genus-two and genus-three curves with complex structure fixed in the two-dimensional complex projective space, and genus-two curves with complex structure fixed in the three-dimensional complex projective space.
www-math.mit.edu /graduate/thesis-defense-spring-2002.html   (1282 words)

 Bounded Complex Domains
This 78-52 = 26 dimensional symmetric space is the set of OP2 in (CxO)P2, and corresponds to the traceless 26-dimensional Jordan subalgebra J3(O)o of the 27-dimensional exceptional Jordan algebra J3(O), and to the 26-dimensional representation of F4.
This 248-120 = 128 dimensional symmetric space is the octooctonionic projective plane (OxO)P2, and corresponds to a half-spinor representation of Spin(16).
This 52-36 = 16 dimensional symmetric space is the octonionic projective plane OP2, and corresponds to the spinor representation of Spin(9).
www.valdostamuseum.org /hamsmith/cdomain.html   (8754 words)

 ::2005 Incentive Fund Winners::
It is known that a curve invariant under self-maps of the complex projective space of degree greater than one must be rational or elliptic.
In a previous paper (Self-maps of the complex projective plane with invariant elliptic curves), Marius Dabija and the P.I., give a criterion to identify in which cases a regular self-map g of a (possibly singular) elliptic curve C admits a regular (resp.
The main goal of this project is to design and construct spectrograph imager, which would allow recording the whole fluorescence spectra simultaneously in fast time scale (miliseconds).
autocrat.uri.edu /index.php?id=2127   (958 words)

 No Title   (Site not responding. Last check: 2007-11-06)
It emerges that the two projective spaces are naturally dual, the duality corresponding to the incidence.
If the conformal structure of Minkowski space is described by means of a duality operator on two-forms, then the restrictions of all self-dual two-forms vanish on one kind of surface and the restrictions of all anti-self-dual two-forms vanish on the other.
If the information of the scale of such a spinor is added in a suitable way to the twistor surface, then the resulting space is the Hopf bundle over the projective space, which may also be viewed as the complement of the origin in a four dimensional complex vector space.
www.math.pitt.edu /~sparling/abbanew/abnew41/abnew41.html   (1964 words)

 [No title]
The space BgSp(m) is the classifying space for symplectic vector bundle* *s of real geometric dimension m.
In this range, the fiber of q is the * *stable stunted real projective space P16p-8= RP 1=RP 16p-9, whose homotopy groups in t* *his range are displayed in [10, Table 8.9].
In fact, he* * wrote in his proof on page 166 that in several cases, including this one, it can be s* *hown that the bundle over the quaternionic projective space is stably equivalent to a bun* *dle of the desired dimension.
hopf.math.purdue.edu /DavisD/CPcrabb4.txt   (3616 words)

 Bezout's Theorem
Statement of the Theorem: The intersection of a variety of degree m with a variety of degree n in complex projective space is either a common component or it has mn points when the intersection points are counted with the appropriate mulitiplicity.
A "point" in projective space is a line through the origin, but not including the origin.
Hence in the complex projection plane the two circles have four intersection points as required by Bezout's Theorem.
www2.sjsu.edu /faculty/watkins/bezout.htm   (795 words)

 projective geometry in theoretical physics
But this is just another way of saying that the\ntrue state space is the projectivization of the Hilbert space,\nresulting in a complex projective space.\n\nWe can do calculations in this complex projective space by doing\ncalculations in the underlying Hilbert space as long as we always keep\nthe identification between complex-parallel vectors in mind.
See the literature\nfrom the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.\n\n2b) From the perspective of projective geometry, the difference between\nMinkowski (affine) geometry and Euclidean (affine) geometry is that the\nformer is characterized by a real number (c=1) and the latter by an\nimaginary one (c=i).
See the\nliterature\nandgt; from the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.\nandgt;\nandgt; 2b) From the perspective of projective geometry, the difference\nbetween\nandgt; Minkowski (affine) geometry and Euclidean (affine) geometry is that\nthe\nandgt; former is characterized by a real number (c=1) and the latter by an\nandgt; imaginary one (c=i).
www.physicsforums.com /showthread.php?p=399667   (2630 words)

 Re: Symmetries of complex projective space
Baez] >There are a number of groups associated with the complex >projective space CP^n.
[Previous] >Now, CP^n is a PROJECTIVE SPACE in the usual axiomatic >sense, meaning that it has a concept of "lines" satisfying >the axioms given here: > >http://math.ucr.edu/home/baez/Octonions/node8.html..................................................................................
One > is that it's the space of pure states associated to > the Hilbert space C^{n+1}.
www.lns.cornell.edu /spr/2003-09/msg0053710.html   (948 words)

 Hilbert space setting for quantum mechanics
For math folks, we are in effect working in Complex projective space, normalizing to 1 so that the probabilities make sense.
be the projection operator onto the subspace spanned by the eigenvectors, and the probability of observing
Most projection operators do not commute with each other, and are not invertible.
astarte.csustan.edu /~tom/MISC/qc-article/node2.html   (316 words)

 Quantization on the Complex Projective Space   (Site not responding. Last check: 2007-11-06)
This paper derives bounds on the distortion rate function for quantization on the complex projective space denoted as CP^n-1.
In essence the problem of quantization in an Euclidean space with constraints can be posed as an unconstrained problem on an appropriate manifold.
CP^n-1 is a non-linear manifold that represents the constraints that arise in areas such as communication with multiple antennas at the transmitter and receiver.
www.ece.utexas.edu /~rheath/papers/2006/dcc1   (115 words)

 110.611 Complex Geometry   (Site not responding. Last check: 2007-11-06)
In a nutshell, the course is about analysis on complex manifolds.
  Most of the complex manifolds studied in complex analysis and geometry are "complex submanifolds" of (1) and (2).
  A complex submanifold is given locally by the vanishing of holomorphic functions.
www.math.jhu.edu /~shiffman/611   (166 words)

 Springer Online Reference Works
Elements (points, straight lines, planes, etc.), generated by extending a given affine space to a compact space.
The continuous connection of the finite and the infinite is manifested by the fact that infinitely-distant elements are meaningful only in as far as they are considered in the context of some concrete compactification of a given  "finite"  space.
The types of infinitely-distant elements resulting from the most frequent methods of compactification of finite-dimensional Euclidean spaces are described below.
eom.springer.de /i/i050900.htm   (292 words)

 Adam Coffman --- Grants and Talks
"CR singularities of real submanifolds in complex space," at the Analysis Seminar, University of Western Ontario, Sept. 21, 2005.
Several Complex Variables Seminar, University of Illinois - Urbana-Champaign, Nov. 5, 2003.
Coffman*, Complex projections of real Veronese varieties, Abstracts of Papers Presented to the AMS (3) 22 (2001), p.
www.ipfw.edu /math/coffman/linx5.html   (643 words)

 Illustrations1   (Site not responding. Last check: 2007-11-06)
This means that the symmetries of the empty Minkowski space are lost as are lost also the corresponding conservation laws, in particular the conservation of energy.
By physical constraints (elementary particle spectrum) the space S must be CP, the complex projective space of two complex (four real) dimensions.
The shadow (projection!) of a nondynamical solid object (< --> metric and spinor connection of H) with time-independent size and shape to a surface (<--> 3-surface) changing its size and shape is dynamical.
www.helsinki.fi /~matpitka/illua.html   (3259 words)

 Polynomials, symmetry, and dynamics: An undertaking in aesthetics   (Site not responding. Last check: 2007-11-06)
Since it takes two real numbers to specify a complex number (for example, 3 + 2 i where i is a square root of -1), a complex space has two times the number of "real dimensions" as the corresponding real space.
An operation that takes each point in a space A and associates it with a point in another space B is called a mapping (or map) from A to B.
Each intersection is a real projective line as well as an "equatorial slice" of the associated complex projective line-a sphere.
www.mi.sanu.ac.yu /vismath/crass/index.html   (5557 words)

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