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In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa. The properties that are meaningful in projective geometry are those that are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations); the first issue for geometers is what kind of geometric language would be adequate to the novel situation. It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.
The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).
Overview
Projective geometry is an elementary nonmetrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was seen as projective geometry was developed by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the principle of duality. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a straightedge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy. It was realised that the theorems that do hold in projective geometry are simpler statements. For example the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be seen as special cases of these general theorems.
In the early 19th century the work of Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics . Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri
The Algebra Project is a national U.S.mathematics literacy effort aimed at helping lowincome students and students of color successfully achieve mathematical skills that are a prerequisite for a college preparatory mathematics sequence in high school. Partially, the Project's mission is to ensure "full citizenship in today's technological society." Founded by Civil Rights activist and Math educator Robert Parris Moses in the 1980s, the Algebra Project has developed curricular materials, trained teachers and teachertrainers, and provided ongoing professional development support and community involvement activities to schools seeking to achieve a systemic change in mathematics education.
The Algebra Project reaches approximately 10,000 students and approximately 300 teachers per year in 28 local sites across 10 states.
About
The Algebra Project focuses on the Southern U.S., where the Southern Initiative of the Algebra Project is directed by David J. Dennis, Sr., and on the Young Peoples' Project (YPP), which recruits, trains and deploys high school and college age "Math Literacy Workers" to work with their younger peers in a variety of math learning opportunities and engage "the demand side" of mathematics education reform. The YPP is directed by Omowale Moses.
Increased student performance in mathematics, as well as greater numbers of students enrolling in college preparatory mathematics classes, is a well documented outcome of the project's work.
History
The Algebra Project was born out of one parent's concern with the mathematics education of his children in the public schools of Cambridge, Massachusetts. In 1982, Bob Moses was invited by Mary Lou Mehrling, his daughter's eighth grade teacher, to help several students with the study of algebra. Moses, who had taught secondary school mathematics in New York City and Tanzania, decided that an appropriate goal for those students was to have enough skills in algebra to qualify for honors math and science courses in high school. His success in producing the first students from the Open Program of the Martin Luther King School who passed the citywide algebra examination and qualified for ninth grade honors geometry was a testament to his skill as a teacher. It also highlighted a serious problem: Most students in the Open Program were expected not to do well in mathematics.
Moses approached the problem at the Open Program in a similar manner to problems he and others had faced in the early sixties in helping the black community of Mississippi seek political power through the vote. While on the surface the problem of the acquisition of political power looked like a simple issue of enticing people to vote, the problem would involve answering an interrelated set of questions. "What is the vote for?" "Why do we want it in the first place?" What must we do right now to ensure that when we have the vote, it will work for us to benefit our communities? Answers to these questions eventually resulted in an important context in which to ask people to vote. This context was the Mississippi Freedom Democratic Party, a community based political party.
Similarly, the everyday issues of students failing at mathematics in the Open Program would require a more complex set of issues and community of individuals. Moses, the parentasorganizer in the program, instinctively used the lesson he had learned in Mississippi transforming the everyday issues into a broader political question for the Open Program community to consider: What is algebra for? Why do we want children to study it? What do we need to include in the mathematics education of every middle school student, to provide each of them with access to the college preparatory mathematics curriculum in high school? Why is it important to gain such access? Within these questions, a context for understanding the problems of mathematics education emerged, and a possible solution and effort at community organizing represented by the Algebra Project began to take shape.
The answers to the questions, "What is algebra for?" and "Why do we want children to study it?", play an important role in the Algebra Project. The project assumes that there is a new standard in assessing mathematics education, a standard of mathematical literacy. In this not so far future, a broad range of mathematical skills will join traditional skills in reading and writing in the definition of literacy. These mathematical skills will not only be important in gaining access to college and math and science related careers, but will also be necessary for full participation in the economic life of this society. In this context, the Algebra Project has as a goal that schools embrace a standard of mathematics education that requires that children be mathematically literate. This will require a community of educators including parents, teachers and school administrators who understand the paramount importance of mathematics education in providing access to the economic life of this society. An answer to the question "What do we need to include in the mathematics education of every middle school student?" also frames the Algebra Project.
Student strike
From March 1, 2006 to March 4, 2006, Baltimore City Public School System students led by the Baltimore City Algebra Project and coming from high schools across Baltimore City held a threeday student strike to oppose an imminent plan to "consolidate" many area high schools into fewer buildings. The school system claims these buildings are underutilized, but the students and other advocates counter that the only reason there is extra space in these buildings is because class sizes often are about 40 students per class. Mayor O'Malley apparently gave an ear to the students' demands in this latest round of strike actions, fearing it could affect his status with the general public in a gubernatorial election year.
The Young People's Project
Founded in 1996, the Young Peopleâ€™s Project (YPP) is an outgrowth of the Algebra Project. YPP has established sites in Jackson, MS, Chicago, IL, and the Greater Boston area of Massachusetts, and is developing programs in Miami, FL, Petersburg, VA, Los Angeles, CA, Ann Arbor, MI, and Mansfield, OH. Through Math Literacy Worker trainings and development, workshops and community events, YPP promotes math literacy as a tool for young people to demand of themselves, their
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While 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. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Background 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. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere. Important results There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order. Riemann's existence theorem Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complexanalytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field. The Lefschetz principle In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic. This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0. Chow's theorem Chow's theorem, proved by W. L. Chow. is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complexanalytic methods within the classical parts of algebraic geometry. Serre's GAGA Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was GÃ©ometrie AlgÃ©brique et GÃ©omÃ©trie Analytique by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves. Nowadays the phrase GAGAstyle result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a welldefined subcategory of analytic geometry objects and holomorphic mappings. Formal statement of GAGA Let (X,\mathcal O_X) be a scheme of finite type over C. Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map Î»X: Xan â†’ X. The topology on Xan is called the "complex topology" (and is very different from the subspace topology). Suppose Ï†: X â†’ Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map Ï†an: Xan â†’ Yan such Î»Y Â° Ï†an = Ï† Â° Î»Y. There is a sheaf \mathcal O_X^{an} on Xan such that (X^{an}, \mathcal O_X^{an}) is a ringed space and Î»X: Xan â†’ X becomes a map of ringed spaces. The space (X^{an}, \mathcal O_X^{an}) is called the "analytifiction" of (X,\mathcal O_X) and is an analytic space. For every Ï†: X â†’ Y the map Ï†an defined above is a mapping of analytic spaces. Furthermore, the map Ï† â†¦ Ï†an maps open immersions into open immersions. If X = C[x1,...,xn] then Xan = Cn and \mathcal O_X^{an}(U) for every polydisc U is a suitable quotient of the space of holomorphic functions on U. For every sheaf \mathcal F on X (called algebraic sheaf) there is a sheaf \mathcal F^{an} on Xan (called analytic sheaf) and a map of sheaves of \mathcal O_X modules \lambda_X^*: \mathcal F\rightarrow (\lambda_X)_* \mathcal F^{an} . The sheaf \mathcal F^{an} is defined as \lambda_X^{1} \mathcal F \otimes_{\lambda_X^{1} \mathcal O_X} \mathcal O_X^{an} . The correspondence \mathcal F \mapsto \mathcal F^{an} defines an exact functor from the category of sheaves over (X, \mathcal O_X) to the category of sheaves of (X^{an}, \mathcal O_X^{an}) . The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.) If f: X â†’ Y is an arbitrary morphism of schemes of finite type over C and \mathcal F is coherent then the natural map (f_* \mathcal F)^{an}\rightarrow f_*^{an} \mathcal F^{an} is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves (R^i f_* \mathcal F)^{an} \cong R^i f_*^{an} \mathcal F^{an} in this case. Now assume that Xan is hausdorff and compact. If \mathcal F, \mathcal G are two coherent algebraic sheaves on (X, \mathcal O_X) and if f: \mathcal F^{an} \rightarrow \mathcal G^{an} is a map of sheaves of \mathcal O_X^{an} modules then there exists a unique map of sheaves of \mathcal O_X modules \varphi: \mathcal F\rightarrow \mathcal G with f = Ï†an. If \mathcal R is a coherent analytic sheaf of \mathcal O_X^{an} modules over Xan then there exists a coherent algebraic sheaf \mathcal F of \mathcal O_X modules and an isomorphism \mathcal F^{an} \cong \mathcal R . Moishezon manifolds A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions on M has transcendence degree equal to the complex dimension of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic spaces. In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a KÃ¤hler metric.
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Answers:You find it here: http://www.cbsemath.com/cbsemathsproject.htm http://www.ct4me.net/math_projects.htm http://www.creativeteensclub.org/ctc/node/21
Answers:Um... it's called Homework for a reason.
Answers:F: Figure, face. N: Nonagon Q: Quadrilateral, Quadratic equation U: Undecagon, unit, union. W: wedge Hope that helped=)
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