Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Idempotence is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra and functional programming; the term was introduced by Benjamin Peirce in the context of elements of algebras that remain invariant when raised to a positive integer power, means " the same power", from idem + potence. An element x of a magma is said to be idempotent if: x • x = x. If all elements are idempotent with respect to • • is called idempotent; the formula ∀x, x • x = x is called the idempotency law for •. The natural number 1 is an idempotent element with respect to multiplication, so is 0, but no other natural number is. For the latter reason, multiplication of natural numbers is not an idempotent operation. More formally, in the monoid, idempotent elements are just 0 and 1. In a magma, an identity element e or an absorbing element a, if it exists, is idempotent.
Indeed, e • e = e and a • a = a. In a group, the identity element e is the only idempotent element. Indeed, if x is an element of G such that x • x = x x • x = x • e and x = e by multiplying on the left by the inverse element of x. Taking the intersection x∩y of two sets x and y is an idempotent operation, since x∩x always equals x; this means that the idempotency law ∀ x ∩ x = x is true. Taking the union of two sets is an idempotent operation. Formally, in the monoids and of the power set of the set E with the set union ∪ and set intersection ∩ all elements are idempotent. In the monoids and of the Boolean domain with the logical disjunction ∨ and the logical conjunction ∧ all elements are idempotent. In a Boolean ring, multiplication is idempotent. In the monoid of the functions from a set E to a subset F of E with the function composition ∘, idempotent elements are the functions f: E → F such that f ∘ f = f, in other words such that for all x in E, f = f. For example: Taking the absolute value abs of an integer number x is an idempotent function for the following reason: abs = abs is true for each integer number x.
This means that abs ∘ abs = abs holds, that is, abs is an idempotent element in the set of all functions with respect to function composition. Therefore, abs satisfies the above definition of an idempotent function. Other examples include: the identity function is idempotent. If the set E has n elements, we can partition it into k chosen fixed points and n − k non-fixed points under f, kn−k is the number of different idempotent functions. Hence, taking into account all possible partitions, ∑ k = 0 n k n − k is the total number of possible idempotent functions on the set; the integer sequence of the number of idempotent functions as given by the sum above for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, … starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, …. Neither the property of being idempotent nor that of being not is preserved under function composition; as an example for the former, f = x mod 3 and g = max are both idempotent, but f ∘ g is not, although g ∘ f happens to be. As an example for the latter, the negation function ¬ on the Boolean domain is not idempotent, but ¬ ∘ ¬ is.
Unary negation − of real numbers is not idempotent, but − ∘ − is. In computer science, the term idempotence may have a different meaning depending on the context in which it is applied: in imperative programming, a subroutine with side effects is idempotent if the system state remains the same after one or several calls, in other words if the function from the system state space to itself associated to the subroutine is idempotent in the mathematical sense given in the definition; this is a useful property in many situations, as it means that an operation can be repeated or retried as as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was performed or not. A function looking up a customer's name and address in a database is idempotent, since this will not cause the database to change. Changing a customer's address is idempotent, because the final address will be the same no matter how many times it is submitted.
However, placing an order for a cart for the customer is not idempotent, since running the call several t
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Sir Michael Francis Atiyah was a British-Lebanese mathematician specialising in geometry. Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at University of Oxford and University of Cambridge, in the United States at the Institute for Advanced Study, he was the President of the Royal Society, founding director of the Isaac Newton Institute, master of Trinity College, chancellor of the University of Leicester, the President of the Royal Society of Edinburgh. From 1997 until his death, he was an honorary professor at the University of Edinburgh. Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch and Isadore Singer, his students included Graeme Segal, Nigel Hitchin and Simon Donaldson. Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, informally speaking, describes ways in which spaces can be twisted, his best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations.
Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Atiyah was born on 22 April 1929 in Hampstead, England, the son of Jean and Edward Atiyah, his mother was Scottish and his father was a Lebanese Orthodox Christian. He had two brothers and Joe, a sister, Selma. Atiyah went to primary school at the Diocesan school in Khartoum, Sudan and to secondary school at Victoria College in Cairo and Alexandria, he returned to England and Manchester Grammar School for his HSC studies and did his national service with the Royal Electrical and Mechanical Engineers. His undergraduate and postgraduate studies took place at Cambridge, he was a doctoral student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry. During his time at Cambridge, he was president of The Archimedeans.
Atiyah spent the academic year 1955–1956 at the Institute for Advanced Study, Princeton returned to Cambridge University, where he was a research fellow and assistant lecturer a university lecturer and tutorial fellow at Pembroke College, Cambridge. In 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherine's College, he became Savilian Professor of Geometry and a professorial fellow of New College, from 1963 to 1969. He took up a three-year professorship at the Institute for Advanced Study in Princeton after which he returned to Oxford as a Royal Society Research Professor and professorial fellow of St Catherine's College, he was president of the London Mathematical Society from 1974 to 1976. Atiyah was president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002, he contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies, the European Mathematical Society.
Within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director. He was President of the Royal Society, Master of Trinity College, Chancellor of the University of Leicester, president of the Royal Society of Edinburgh. From 1997 until his death in 2019 he was an honorary professor in the University of Edinburgh, he was a Trustee of the James Clerk Maxwell Foundation. Atiyah collaborated with many mathematicians, his three main collaborations were with Raoul Bott on the Atiyah–Bott fixed-point theorem and many other topics, with Isadore M. Singer on the Atiyah–Singer index theorem, with Friedrich Hirzebruch on topological K-theory, all of whom he met at the Institute for Advanced Study in Princeton in 1955, his other collaborators included. Manin, Nick S. Manton, Vijay K. Patodi, A. N. Pressley, Elmer Rees, Wilfried Schmid, Graeme Segal, Alexander Shapiro, L. Smith, Paul Sutcliffe, David O. Tall, John A. Todd, Cumrun Vafa, Richard S. Ward and Edward Witten.
His research on gauge field theories Yang–Mills theory, stimulated important interactions between geometry and physics, most notably in the work of Edward Witten. Atiyah's students included.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.
In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.
In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co
Condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is large and the interactions between the constituents are strong; the most familiar examples of condensed phases are solids and liquids, which arise from the electromagnetic forces between atoms. Condensed matter physicists seek to understand the behavior of these phases by using physical laws. In particular, they include the laws of quantum mechanics and statistical mechanics; the most familiar condensed phases are solids and liquids while more exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, the Bose–Einstein condensate found in ultracold atomic systems. The study of condensed matter physics involves measuring various material properties via experimental probes along with using methods of theoretical physics to develop mathematical models that help in understanding physical behavior.
The diversity of systems and phenomena available for study makes condensed matter physics the most active field of contemporary physics: one third of all American physicists self-identify as condensed matter physicists, the Division of Condensed Matter Physics is the largest division at the American Physical Society. The field overlaps with chemistry, materials science, nanotechnology, relates to atomic physics and biophysics; the theoretical physics of condensed matter shares important concepts and methods with that of particle physics and nuclear physics. A variety of topics in physics such as crystallography, elasticity, etc. were treated as distinct areas until the 1940s, when they were grouped together as solid state physics. Around the 1960s, the study of physical properties of liquids was added to this list, forming the basis for the new, related specialty of condensed matter physics. According to physicist Philip Warren Anderson, the term was coined by him and Volker Heine, when they changed the name of their group at the Cavendish Laboratories, Cambridge from Solid state theory to Theory of Condensed Matter in 1967, as they felt it did not exclude their interests in the study of liquids, nuclear matter, so on.
Although Anderson and Heine helped popularize the name "condensed matter", it had been present in Europe for some years, most prominently in the form of a journal published in English and German by Springer-Verlag titled Physics of Condensed Matter, launched in 1963. The funding environment and Cold War politics of the 1960s and 1970s were factors that lead some physicists to prefer the name "condensed matter physics", which emphasized the commonality of scientific problems encountered by physicists working on solids, liquids and other complex matter, over "solid state physics", associated with the industrial applications of metals and semiconductors; the Bell Telephone Laboratories was one of the first institutes to conduct a research program in condensed matter physics. References to "condensed" state can be traced to earlier sources. For example, in the introduction to his 1947 book Kinetic Theory of Liquids, Yakov Frenkel proposed that "The kinetic theory of liquids must accordingly be developed as a generalization and extension of the kinetic theory of solid bodies.
As a matter of fact, it would be more correct to unify them under the title of'condensed bodies'". One of the first studies of condensed states of matter was by English chemist Humphry Davy, in the first decades of the nineteenth century. Davy observed that of the forty chemical elements known at the time, twenty-six had metallic properties such as lustre and high electrical and thermal conductivity; this indicated that the atoms in John Dalton's atomic theory were not indivisible as Dalton claimed, but had inner structure. Davy further claimed that elements that were believed to be gases, such as nitrogen and hydrogen could be liquefied under the right conditions and would behave as metals. In 1823, Michael Faraday an assistant in Davy's lab liquefied chlorine and went on to liquefy all known gaseous elements, except for nitrogen and oxygen. Shortly after, in 1869, Irish chemist Thomas Andrews studied the phase transition from a liquid to a gas and coined the term critical point to describe the condition where a gas and a liquid were indistinguishable as phases, Dutch physicist Johannes van der Waals supplied the theoretical framework which allowed the prediction of critical behavior based on measurements at much higher temperatures.
By 1908, James Dewar and Heike Kamerlingh Onnes were able to liquefy hydrogen and newly discovered helium, respectively. Paul Drude in 1900 proposed the first theoretical model for a classical electron moving through a metallic solid. Drude's model described properties of metals in terms of a gas of free electrons, was the first microscopic model to explain empirical observations such as the Wiedemann–Franz law. However, despite the success of Drude's free electron model, it had one notable problem: it was unable to explain the electronic contribution to the specific heat and magnetic properties of metals, the temperature dependence of resistivity at low temperatures. In 1911, three years after helium was first liquefied, Onnes working at University of Leiden discovered superconductivity in mercury, when he observed the electrical resistivity of mercury to vanish at temperatures below a certain value; the phenomenon surprised the best theoretical physicists of the time, it remain
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.:58Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility; the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot; the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex number coefficients is determined by the set of its roots in the complex plane.
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory; this correspondence is a defining feature of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way; the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. For an algebraically closed field K and a natural number n, let An be affine n-space over K; the polynomials f in the ring K can be viewed as K-valued functions on An by evaluating f at the points in An, i.e. by choosing values in K for each xi.
For each set S of polynomials in K, define the zero-locus Z to be the set of points in An on which the functions in S vanish, to say Z =. A subset V of An is called an affine algebraic set if V = Z for some S.:2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.:3 An irreducible affine algebraic set is called an affine variety.:3 Affine varieties can be given a natural topology by declaring the closed sets to be the affine algebraic sets. This topology is called the Zariski topology.:2Given a subset V of An, we define I to be the ideal of all polynomial functions vanishing on V: I =. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.:4 Let k be an algebraically closed field and let Pn be the projective n-space over k. Let f in k be a homogeneous polynomial of degree d, it is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
However, because f is homogeneous, meaning that f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: Z =. A subset V of Pn is called a projective algebraic set if V = Z for some S.:9 An irreducible projective algebraic set is called a projective variety.:10Projective varieties are equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.:10A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety. In classical algebraic geometry, a