Simply connected space
In topology, a topological space is called connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be connected: a path-connected topological space is connected if and only if its fundamental group is trivial. A topological space X is called connected if it is path-connected and any loop in X defined by f: S1 → X can be contracted to a point: there exists a continuous map F: D2 → X such that F restricted to S1 is f. Here, S1 and D2 closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is connected if and only if it is path-connected, whenever p: → X and q: → X are two paths with the same start and endpoint p can be continuously deformed into q while keeping both endpoints fixed. Explicitly, there exists a continuous homotopy F: × → X such that F = F = q.
A topological space X is connected if and only if X is path-connected and the fundamental group of X at each point is trivial, i.e. consists only of the identity element. X is connected if and only if for all points x, y ∈ X, the set of morphisms Hom Π in the fundamental groupoid of X has only one element. In complex analysis: an open subset X ⊆ C is connected if and only if both X and its complement in the Riemann sphere are connected; the set of complex numbers with imaginary part greater than zero and less than one, furnishes a nice example of an unbounded, open subset of the plane whose complement is not connected. It is simply connected, it might be worth pointing out that a relaxation of the requirement that X be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a open set has connected extended complement when each of its connected components are connected. Informally, an object in our space is connected if it consists of one piece and does not have any "holes" that pass all the way through it.
For example, neither a doughnut nor a coffee cup is connected, but a hollow rubber ball is connected. In two dimensions, a circle is not connected, but a disk and a line are. Spaces that are connected but not connected are called non-simply connected or multiply connected; the definition only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point though it has a "hole" in the hollow center; the stronger condition, that the object has no holes of any dimension, is called contractibility. The Euclidean plane R2 is connected, but R2 minus the origin is not. If n > 2 both Rn and Rn minus the origin are connected. Analogously: the n-dimensional sphere Sn is connected if and only if n > 2. Every convex subset of Rn is connected. A torus, the cylinder, the Möbius strip, the projective plane and the Klein bottle are not connected; every topological vector space is connected. For n ≥ 2, the special orthogonal group SO is not connected and the special unitary group SU is connected.
The one-point compactification of R is not connected. The long line L is connected, but its compactification, the extended long line L* is not. A surface is connected if and only if it is connected and its genus is 0. A universal cover of any space X is a connected space which maps to X via a covering map. If X and Y are homotopy equivalent and X is connected so is Y; the image of a connected set under a continuous function need not be connected. Take for example the complex plane under the exponential map: the image is C -, not connected; the notion of simple connectedness is important in complex analysis because of the following facts: The Cauchy's integral theorem states that if U is a connected open subset of the complex plane C, f: U → C is a holomorphic function f has an antiderivative F on U, the value of every line integral in U with integrand f depends only on the end points u and v of the path, can be computed as F - F. The integral thus does not depend on the particular path connecting u and v.
The Riemann mapping theorem states that any non-empty open connected subset of C is conformally eq
Miles Anthony Reid FRS is a mathematician who works in algebraic geometry. Reid studied the Cambridge Mathematical Tripos at Trinity College and obtained his Ph. D. in 1973 under the supervision of Peter Swinnerton-Dyer and Pierre Deligne. Reid was a research fellow of Christ's College, Cambridge from 1973 to 1978, he became a lecturer at the University of Warwick in 1978 and was appointed professor there in 1992. He has written two well known books: Undergraduate Algebraic Geometry and Undergraduate Commutative Algebra. Reid was elected a Fellow of the Royal Society in 2002. Reid was awarded the Senior Berwick Prize in 2006 for his paper with Alessio Corti and Aleksandr Pukhlikov, "Fano 3-fold hypersurfaces", which made a big advance in the study of 3-dimensional algebraic varieties. Reid has given lectures in Japanese, his most famous book is Undergraduate Algebraic Geometry, Cambridge University Press 1988 doi:10.1017/CBO9781139163699Other books Undergraduate commutative algebra, Cambridge University Press 1995, doi:10.1017/CBO9781139172721 with Balazs Szendroi: Geometry and topology, Cambridge University Press 2007His most famous translation is the 2-vols book by Shafarevich Basic Algebraic Geometry 1 Basic Algebraic Geometry 2
Tian Gang is a Chinese mathematician. He is an academician of the Chinese Academy of Sciences and of the American Academy of Arts and Sciences, he is known for his contributions to geometric analysis and quantum cohomology Gromov-Witten invariants, among other fields. He has been Vice President of Peking University since February 2017. Tian was born in Nanjing, China, he qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated from Nanjing University in 1982, received a master's degree from Peking University in 1984. In 1988, he received a Ph. D. in mathematics from Harvard University, under the supervision of Shing-Tung Yau. This work was so exceptional. In 1998, he was appointed as a Cheung Kong Scholar professor at Peking University, his appointment was changed to Cheung Kong Scholar chair professorship. He was awarded the Alan T. Waterman Award in 1994, the Veblen Prize in 1996. In 2004 Tian was elected a fellow of the American Academy of Sciences, he was a professor of mathematics at the Massachusetts Institute of Technology from 1995 to 2006.
His employment at Princeton started from 2003, was appointed the Higgins Professor of Mathematics. Starting 2005, he has been the director of the Beijing International Center for Mathematical Research, he and John Milnor are Senior Scholars of the Clay Mathematics Institute. In 2011, Tian became director of the Sino-French Research Program in Mathematics at the Centre national de la recherche scientifique in Paris. In 2010, he became scientific consultant for the International Center for Theoretical Physics in Trieste, Italy. Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifolds under the direction of Yau. In particular he solved the existence question for Kähler–Einstein metrics on compact complex surfaces with positive first Chern class, showed that hypersurfaces with a Kähler–Einstein metric are stable in the sense of geometric invariant theory, he proved that a Kähler manifold with trivial canonical bundle has trivial obstruction space, known as the Bogomolov–Tian–Todorov theoremTian found an explicit formula for Weil-Petersson metric on moduli space of polarized Calabi-Yau manifolds.
Tian made foundational contributions to Gromov-Witten theory. He constructed virtual fundamental cycles of the moduli spaces of maps from curves in both algebraic geometry and symplectic geometry, he showed that the quantum cohomology ring of a semi-positive symplectic manifold is associative. He introduced the Analytical Minimal Model program, known as Tian-Song program in birational geometry. In Kähler geometry he has a new theory, known as Cheeger-Colding-Tian's theory. Tian's alpha-invariant was introduced by him and was given an algebraic interpretation by János Kollár and Jean-Pierre Demailly, he proposed the Yau-Tian-Donaldson conjecture. It was proven by Xiuxiong Chen and Song Sun in 2014. Tian gave a proof electronically published on September 16, 2015. In 2006, together with John Morgan and others, Tian helped verify the proof of the Poincaré conjecture given by Grigori Perelman. Tian served as one of the five members of the Abel Prize Committee, he was one of the five members of the Ramanujan Prize selection committee.
In 2012, he became a member of Leroy P. Steele Prize Committee in AMS. Gang Tian is member of the editorial boards of a number of journals in Mathematics. 1. Annals of Mathematics2. Annali della Scuola Normale Superiore3. Journal of Symplectic Geometry4. Journal of the American Mathematical Society,1995-1998.5. Geometry & Topology6; the Journal of Geometric Analysis7. Geometric and Functional Analysis8. Advances in Mathematics 9. International Mathematics Research Notices10. Pacific Journal of Mathematics, 1994-1998. 11. Communications in Analysis and Geometry, 1994-2000. 12. Acta Mathematica Sinica,13. Mathematics Revista Matemática Complutense 14. Communications in Mathematics and Statistics,15. Communication in Contemporary Mathematics, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory, 629—646, Adv. Ser. Math. Phys. 1, World Sci. Publishing, Singapore, 1987. Tian, Gang. On Kähler-Einstein metrics on certain Kähler manifolds with C 1 > 0.
Invent. Math. 89, no. 2, 225—246. Tian, G.. Complete Kähler manifolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3, no. 3, 579—609. Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101, no. 1, 101—172. Tian, Gang. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32, no. 1, 99—130. Ruan, Yongbin. A mathematical theory of quantum cohomology. J. Differential Geom. 42, no. 2, 259—367. Tian, Gang. Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, no. 1, 1--37. Ruan, Yongbin. Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130, no. 3, 455—516. Li, Jun. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc. 11, no. 1, 119—174. Liu, Gang. Floer homology and Arnold conjecture. J. Differential Geom. 49, no. 1, 1--74. Liu, Xiao
Mirror symmetry (string theory)
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look different geometrically but are equivalent when employed as extra dimensions of string theory. Mirror symmetry was discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously. Today, mirror symmetry is a major research topic in pure mathematics, mathematicians are working to develop a mathematical understanding of the relationship based on physicists' intuition.
Mirror symmetry is a fundamental tool for doing calculations in string theory, it has been used to understand aspects of quantum field theory, the formalism that physicists use to describe elementary particles. Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, Eric Zaslow. In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings; these strings look like small loops of ordinary string. String theory describes how strings propagate through interact with each other. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its mass and other properties determined by the vibrational state of the string. Splitting and recombination of strings correspond to particle emission and absorption, giving rise to the interactions between particles.
There are notable differences between the world described by the everyday world. In everyday life, there are three familiar dimensions of space, there is one dimension of time. Thus, in the language of modern physics, one says. One of the peculiar features of string theory is that it requires extra dimensions of spacetime for its mathematical consistency. In superstring theory, the version of the theory that incorporates a theoretical idea called supersymmetry, there are six extra dimensions of spacetime in addition to the four that are familiar from everyday experience. One of the goals of current research in string theory is to develop models in which the strings represent particles observed in high energy physics experiments. For such a model to be consistent with observations, its spacetime must be four-dimensional at the relevant distance scales, so one must look for ways to restrict the extra dimensions to smaller scales. In most realistic models of physics based on string theory, this is accomplished by a process called compactification, in which the extra dimensions are assumed to "close up" on themselves to form circles.
In the limit where these curled up dimensions become small, one obtains a theory in which spacetime has a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions. Compactification can be used to construct models in which spacetime is four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics, the compact extra dimensions must be shaped like a Calabi–Yau manifold. A Calabi–Yau manifold is a special space, taken to be six-dimensional in applications to string theory, it is named after mathematicians Eugenio Shing-Tung Yau. After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions, many physicists began studying these manifolds.
In the late 1980s, Lance Dixon, Wolfgang Lerche, Cumrun Vafa, Nick Warner noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold. Instead, two different versions of string theory called type IIA string theory and type IIB can be compactified on different Calabi–Yau manifolds giving rise to the same physics. In this situation, the manifolds are called mirror manifolds, the relationship between the two physical theories is called mirror symmetry; the mirror symmetry relationship is a particular example of. In general, the term duality refers to a situation where two different physical theories turn out to be equivalent in a nontrivial way. If one theory can be transformed so it looks just like another theory, the two are said to be dual under that transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena; such dualities play an important role in modern physics in string theory.
Regardless of whether Calabi–Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between differe
In mathematics, topology, a fiber bundle is a space, locally a product space, but globally may have a different topological structure. The similarity between a space E and a product space B × F is defined using a continuous surjective map π: E → B that in small regions of E behaves just like a projection from corresponding regions of B × F to B; the map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, F the fiber. In the trivial case, E is just B × F, the map π is just the projection from the product space to the first factor; this is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself to E is called a section of E. Fiber bundles can be specialized in a number of ways, the most common of, requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber F. In topology, the terms fiber and fiber space appeared for the first time in a paper by Herbert Seifert in 1933, but his definitions are limited to a special case; the main difference from the present day conception of a fiber space, was that for Seifert what is now called the base space of a fiber space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space was given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau, Norman Steenrod, Charles Ehresmann, Jean-Pierre Serre, others. Fiber bundles became their own object of study in the period 1935–1940; the first general definition appeared in the works of Whitney. Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, a fiber bundle whose fiber is a sphere of arbitrary dimension. A fiber bundle is a structure, where E, B, F are topological spaces and π: E → B is a continuous surjection satisfying a local triviality condition outlined below; the space B is called the base space of the bundle, E the total space, F the fiber. The map π is called the projection map. We shall assume in. We require that for every x in E, there is an open neighborhood U ⊂ B of π such that there is a homeomorphism φ: π−1 → U × F in such a way that π agrees with the projection onto the first factor.
That is, the following diagram should commute: where proj1: U × F → U is the natural projection and φ: π−1 → U × F is a homeomorphism. The set of all is called a local trivialization of the bundle, thus for any p in B, the preimage π−1 is homeomorphic to F and is called the fiber over p. Every fiber bundle π: E → B is an open map, since projections of products are open maps; therefore B carries the quotient topology determined by the map π. A fiber bundle is denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category of smooth manifolds; that is, E, B, F are required to be smooth manifolds and all the functions above are required to be smooth maps. Let E = B × F and let π: E → B be the projection onto the first factor. E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle.
Any fiber bundle over a contractible CW-complex is trivial. The simplest example of a nontrivial bundle E is the Möbius strip, it has the circle that runs lengthwise along the center of the strip as a base B and a line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood U of π ∈ B is an arc; the preimage π − 1 in the picture is a slice of the strip one long. A homeomorphism exists that maps the preimage of U to a slice of a cylinder: curved, but not twisted; this pair locally trivializes the strip. The corresponding trivial bundle B × F would be a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally.
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize and predict natural phenomena. This is in contrast to experimental physics; the advancement of science depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations. For example, while developing special relativity, Albert Einstein was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was uninterested in the Michelson–Morley experiment on Earth's drift through a luminiferous aether. Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect an experimental result lacking a theoretical formulation. A physical theory is a model of physical events, it is judged by the extent. The quality of a physical theory is judged on its ability to make new predictions which can be verified by new observations.
A physical theory differs from a mathematical theorem in that while both are based on some form of axioms, judgment of mathematical applicability is not based on agreement with any experimental results. A physical theory differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms. A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces. Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that energy are not continuously variable. Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ empirical formulas to agree with experimental results without deep physical understanding.
"Modelers" appear much like phenomenologists, but try to model speculative theories that have certain desirable features, or apply the techniques of mathematical modeling to physics problems. Some attempt to create approximate theories, called effective theories, because developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, reinterpret or generalise extant theories, or create new ones altogether. Sometimes the vision provided by pure mathematical systems can provide clues to how a physical system might be modeled. Theoretical problems that need computational investigation are the concern of computational physics. Theoretical advances may consist in setting aside old, incorrect paradigms or may be an alternative model that provides answers that are more accurate or that can be more applied. In the latter case, a correspondence principle will be required to recover the known result. Sometimes though, advances may proceed along different paths. For example, an correct theory may need some conceptual or factual revisions.
However, an exception to all the above is the wave–particle duality, a theory combining aspects of different, opposing models via the Bohr complementarity principle. Physical theories become accepted if they are able to make correct predictions and no incorrect ones; the theory should have, at least as a secondary objective, a certain economy and elegance, a notion sometimes called "Occam's razor" after the 13th-century English philosopher William of Occam, in which the simpler of two theories that describe the same matter just as adequately is preferred. They are more to be accepted if they connect a wide range of phenomena. Testing the consequences of a theory is part of the scientific method. Physical theories can be grouped into three categories: mainstream theories, proposed theories and fringe theories. Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy, continued by Plato and Aristotle, whose views held sway for a millennium. During the rise of medieval universities, the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar and rhetoric and of the Quadrivium like arithmetic, geometry and astronomy.
During the Middle Ages and Renaissance, the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon. As the Scientific Revolution gathered pace, the concepts of matter, space and causality began to acquire the form we know today, other sciences spun off from the rubric of natural philosophy, thus began the modern era of theory with the Copernican paradigm shift in astronomy, soon followed by Johannes Kepler's expressions for planetary orbits, which summarized the meticulous observations of Tycho Brahe.
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.
As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.
Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.
This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.
Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic