Simon Donaldson

Sir Simon Kirwan Donaldson, is an English mathematician known for his work on the topology of smooth four-dimensional manifolds and Donaldson–Thomas theory. He is a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London. Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge, his mother earned a science degree there. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, in 1980 began postgraduate work at Worcester College, Oxford, at first under Nigel Hitchin and under Michael Atiyah's supervision. Still a postgraduate student, Donaldson proved in 1982 a result, he published the result in a paper "Self-dual connections and the topology of smooth 4-manifolds" which appeared in 1983. In the words of Atiyah, the paper "stunned the mathematical world". Whereas Michael Freedman classified topological four-manifolds, Donaldson's work focused on four-manifolds admitting a differentiable structure, using instantons, a particular solution to the equations of Yang–Mills gauge theory which has its origin in quantum field theory.

One of Donaldson's first results gave severe restrictions on the intersection form of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit any smooth structure at all. Donaldson derived polynomial invariants from gauge theory; these were new topological invariants sensitive to the underlying smooth structure of the four-manifold. They made it possible to deduce the existence of "exotic" smooth structures—certain topological four-manifolds could carry an infinite family of different smooth structures. After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983–84 at the Institute for Advanced Study in Princeton, returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998. In 2014, he joined the Simons Center for Geometry and Physics at Stony Brook University in New York, United States.

Donaldson received the Junior Whitehead Prize from the London Mathematical Society in 1985 and in the following year he was elected a Fellow of the Royal Society and in 1986, he received a Fields Medal. He was awarded the 1994 Crafoord Prize. In February 2006, Donaldson was awarded the King Faisal International Prize for science for his work in pure mathematical theories linked to physics, which have helped in forming an understanding of the laws of matter at a subnuclear level. In April 2008, he was awarded the Nemmers Prize in Mathematics, a mathematics prize awarded by Northwestern University. In 2009 he was awarded the Shaw Prize in Mathematics for their contributions to geometry in 3 and 4 dimensions. In 2010, he was elected a foreign member of the Royal Swedish Academy of Sciences. Donaldson was knighted in the 2012 New Year Honours for services to mathematics. In 2012 he became a fellow of the American Mathematical Society. In March 2014, he was awarded the degree "Docteur Honoris Causa" by Université Joseph Fourier, Grenoble.

In 2014 he was awarded the Breakthrough Prize in Mathematics "for the new revolutionary invariants of 4-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties."In January 2017, he was awarded the degree "Doctor Honoris Causa" by the Universidad Complutense de Madrid, Spain. In January 2019, he was awarded the Oswald Veblen Prize in Geometry. Donaldson's work is on the application of mathematical analysis to problems in geometry; the problems concern 4-manifolds, complex differential geometry and symplectic geometry. The following theorems have been mentioned: The diagonalizability theorem: If the intersection form of a smooth, closed connected 4-manifold is positive- or negative-definite it is diagonalizable over the integers; this result is sometimes called Donaldson's theorem. A smooth h-cobordism between connected 4-manifolds need not be trivial; this contrasts with the situation in higher dimensions.

A stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian–Einstein metric. A non-singular, projective algebraic surface can be diffeomorphic to the connected sum of two oriented 4-manifolds only if one of them has negative-definite intersection form; this was an early application of the Donaldson invariant. Any compact symplectic manifold admits a symplectic Lefschetz pencil. Donaldson's recent work centers on a problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "extremal" Kähler metrics those with constant scalar curvature. Donaldson obtained results in the toric case of the problem, he solved the Kähler-Einstein case of the problem in 2012, in collaboration with Chen and Sun. This latest spectacular achievement involved a number of technical papers; the first of these was the paper of Sun on Gromov-Hausdorff limits. The summary of

Kunihiko Kodaira

Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954. Kodaira was born in Tokyo, he graduated from the University of Tokyo in 1938 with a degree in mathematics and graduated from the physics department at the University of Tokyo in 1941. During the war years he worked in isolation, but was able to master Hodge theory as it stood, he obtained his Ph. D. from the University of Tokyo in 1949, with a thesis entitled Harmonic fields in Riemannian manifolds. He was involved in cryptographic work from about 1944, while holding an academic post in Tokyo. In 1949 he travelled to the Institute for Advanced Study in Princeton, New Jersey at the invitation of Hermann Weyl, he was subsequently appointed Associate Professor at Princeton University in 1952 and promoted to Professor in 1955. At this time the foundations of Hodge theory were being brought in line with contemporary technique in operator theory.

Kodaira became involved in exploiting the tools it opened up in algebraic geometry, adding sheaf theory as it became available. This work was influential, for example on Friedrich Hirzebruch. In a second research phase, Kodaira wrote a long series of papers in collaboration with Donald C. Spencer, founding the deformation theory of complex structures on manifolds; this gave the possibility of constructions of moduli spaces, since in general such structures depend continuously on parameters. It identified the sheaf cohomology groups, for the sheaf associated with the holomorphic tangent bundle, that carried the basic data about the dimension of the moduli space, obstructions to deformations; this theory is still foundational, had an influence on the scheme theory of Grothendieck. Spencer continued this work, applying the techniques to structures other than complex ones, such as G-structures. In a third major part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds.

This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically. He provided detailed studies of elliptic fibrations of surfaces over a curve, or in other language elliptic curves over algebraic function fields, a theory whose arithmetic analogue proved important soon afterwards; this work included a characterisation of K3 surfaces as deformations of quartic surfaces in P4, the theorem that they form a single diffeomorphism class. Again, this work has proved foundational.. Kodaira left Princeton University and the Institute for Advanced Study in 1961, served as chair at the Johns Hopkins University and Stanford University. In 1967, returned to the University of Tokyo, he was awarded a Wolf Prize in 1984/5. He died in Kofu on 26 July 1997. Morrow, James. J. ISBN 978-0-691-08158-8, MR 0366598 Kodaira, Baily, Walter L. ed. Kunihiko Kodaira: collected works, II, Iwanami Shoten, Tokyo. J. ISBN 978-0-691-08163-2, MR 0366599 Kodaira, Baily, Walter L. ed. Kunihiko Kodaira: collected works, III, Iwanami Shoten, Tokyo.

J. ISBN 978-0-691-08164-9, MR 0366600 Kodaira, Complex manifolds and deformation of complex structures, Classics in Mathematics, New York: Springer-Verlag, ISBN 978-3-540-22614-7, MR 0815922, review by Andrew J. Sommese Kodaira, Complex analysis, Cambridge Studies in Advanced Mathematics, 107, Cambridge University Press, ISBN 978-0-521-80937-5, MR 2343868 Bochner–Kodaira–Nakano identity Spectral theory of ordinary differential equations Kodaira vanishing theorem Kodaira–Spencer mapping Kodaira dimension Kodaira embedding theorem Enriques–Kodaira classification Kodaira's classification of singular fibers O'Connor, John J.. Donald C. Spencer, "Kunihiko Kodaira", Notices of the AMS, 45: 388–389. Friedrich Hirzebruch, "Kunihiko Kodaira: Mathematician and Teacher", Notices of the American Mathematical Society, 45: 1456–1462. "Special Issue to Honor Professor Kunihiko Kodaira on his 85th birthday", Asian Journal of Mathematics, 4, 2000

Mathematics

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

Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.

For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia

W. V. D. Hodge

Sir William Vallance Douglas Hodge was a British mathematician a geometer. His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more to Kähler manifolds—has been a major influence on subsequent work in geometry, he was born in Edinburgh in 1903, the son of Archibald James Hodge, searcher of public records, his wife, Jane Vallance. They lived at 1 Church Hill Place in the Morningside district, he attended George Watson's College, studied at Edinburgh University, graduating MA in 1923. With help from E. T. Whittaker, whose son J. M. Whittaker was a college friend, he took the Cambridge Mathematical Tripos. At Cambridge he fell under the influence of the geometer H. F. Baker, he gained a second MA in 1925. In 1926 he took up a teaching position at the University of Bristol, began work on the interface between the Italian school of algebraic geometry problems posed by Francesco Severi, the topological methods of Solomon Lefschetz.

This led to some initial scepticism on the part of Lefschetz. According to Atiyah's memoir and Hodge in 1931 had a meeting in Max Newman's rooms in Cambridge, to try to resolve issues. In the end Lefschetz was convinced. In 1928 he was elected a Fellow of the Royal Society of Edinburgh, his proposers were Sir Edmund Taylor Whittaker, Ralph Allan Sampson, Charles Glover Barkla, Sir Charles Galton Darwin. He was awarded the Society's Gunning Victoria Jubilee Prize for the period 1964 to 1968. In 1930 Hodge was awarded a Research Fellowship at Cambridge, he spent a year 1931–2 at Princeton University, where Lefschetz was, visiting Oscar Zariski at Johns Hopkins University. At this time he was assimilating de Rham's theorem, defining the Hodge star operation, it would allow him to so refine the de Rham theory. On his return to Cambridge, he was offered a University Lecturer position in 1933, he became the Lowndean Professor of Astronomy and Geometry at Cambridge, a position he held from 1936 to 1970.

He was the first head of DPMMS. He was the Master of Pembroke College, Cambridge from 1958 to 1970, vice-president of the Royal Society from 1959 to 1965, he was knighted in 1959. Amongst other honours, he received the Adams Prize in 1937 and the Copley Medal of the Royal Society in 1974, he died in Cambridge on 7 July 1975. The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the signature of the corresponding quadratic form; this result was sought by the Italian school of algebraic geometry, but was proved by the topological methods of Lefschetz. The Theory and Applications of Harmonic Integrals summed up Hodge's development during the 1930s of his general theory; this starts with the existence for any Kähler metric of a theory of Laplacians – it applies to an algebraic variety V because projective space itself carries such a metric. In de Rham cohomology terms, a cohomology class of degree k is represented by a k-form α on V.

There is no unique representative. This has the important, immediate consequence of splitting up Hkinto subspaces Hp,qaccording to the number p of holomorphic differentials dzi wedged to make up α; the dimensions of the subspaces are the Hodge numbers. This Hodge decomposition has become a fundamental tool. Not only do the dimensions hp,q refine the Betti numbers, by breaking them into parts with identifiable geometric meaning. In broad terms, Hodge theory contributes both to the discrete and the continuous classification of algebraic varieties. Further developments by others led in particular to an idea of mixed Hodge structure on singular varieties, to deep analogies with étale cohomology; the Hodge conjecture on the'middle' spaces Hp,p is still unsolved, in general. It is one of the seven Millennium Prize Problems set up by the Clay Mathematics Institute. Hodge wrote, with Daniel Pedoe, a three-volume work Methods of Algebraic Geometry, on classical algebraic geometry, with much concrete content – illustrating though what Élie Cartan called'the debauch of indices' in its component notation.

According to Atiyah, this was intended to replace H. F. Baker's Principles of Geometry. In 1929 he married Kathleen Anne Cameron. Hodge, W. V. D; the Theory and Applications of Harmonic Integrals, Cambridge University Press, ISBN 978-0-521-35881-1, MR 0003947 Hodge, W. V. D.. Methods of Algebraic Geometry, Volume I, Cambridge University Press, ISBN 978-0-521-46900-5 Hodge, W. V. D.. Book IV: Quadrics and Grassmann varieties. Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-46901-2, MR 0048065 Hodge, W. V. D..

Algebraic variety

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

OCLC

OCLC Online Computer Library Center, Incorporated d/b/a OCLC is an American nonprofit cooperative organization "dedicated to the public purposes of furthering access to the world's information and reducing information costs". It was founded in 1967 as the Ohio College Library Center. OCLC and its member libraries cooperatively produce and maintain WorldCat, the largest online public access catalog in the world. OCLC is funded by the fees that libraries have to pay for its services. OCLC maintains the Dewey Decimal Classification system. OCLC began in 1967, as the Ohio College Library Center, through a collaboration of university presidents, vice presidents, library directors who wanted to create a cooperative computerized network for libraries in the state of Ohio; the group first met on July 5, 1967 on the campus of the Ohio State University to sign the articles of incorporation for the nonprofit organization, hired Frederick G. Kilgour, a former Yale University medical school librarian, to design the shared cataloging system.

Kilgour wished to merge the latest information storage and retrieval system of the time, the computer, with the oldest, the library. The plan was to merge the catalogs of Ohio libraries electronically through a computer network and database to streamline operations, control costs, increase efficiency in library management, bringing libraries together to cooperatively keep track of the world's information in order to best serve researchers and scholars; the first library to do online cataloging through OCLC was the Alden Library at Ohio University on August 26, 1971. This was the first online cataloging by any library worldwide. Membership in OCLC is based on use of services and contribution of data. Between 1967 and 1977, OCLC membership was limited to institutions in Ohio, but in 1978, a new governance structure was established that allowed institutions from other states to join. In 2002, the governance structure was again modified to accommodate participation from outside the United States.

As OCLC expanded services in the United States outside Ohio, it relied on establishing strategic partnerships with "networks", organizations that provided training and marketing services. By 2008, there were 15 independent United States regional service providers. OCLC networks played a key role in OCLC governance, with networks electing delegates to serve on the OCLC Members Council. During 2008, OCLC commissioned two studies to look at distribution channels. In early 2009, OCLC negotiated new contracts with the former networks and opened a centralized support center. OCLC provides bibliographic and full-text information to anyone. OCLC and its member libraries cooperatively produce and maintain WorldCat—the OCLC Online Union Catalog, the largest online public access catalog in the world. WorldCat has holding records from private libraries worldwide; the Open WorldCat program, launched in late 2003, exposed a subset of WorldCat records to Web users via popular Internet search and bookselling sites.

In October 2005, the OCLC technical staff began a wiki project, WikiD, allowing readers to add commentary and structured-field information associated with any WorldCat record. WikiD was phased out; the Online Computer Library Center acquired the trademark and copyrights associated with the Dewey Decimal Classification System when it bought Forest Press in 1988. A browser for books with their Dewey Decimal Classifications was available until July 2013; until August 2009, when it was sold to Backstage Library Works, OCLC owned a preservation microfilm and digitization operation called the OCLC Preservation Service Center, with its principal office in Bethlehem, Pennsylvania. The reference management service QuestionPoint provides libraries with tools to communicate with users; this around-the-clock reference service is provided by a cooperative of participating global libraries. Starting in 1971, OCLC produced catalog cards for members alongside its shared online catalog. OCLC commercially sells software, such as CONTENTdm for managing digital collections.

It offers the bibliographic discovery system WorldCat Discovery, which allows for library patrons to use a single search interface to access an institution's catalog, database subscriptions and more. OCLC has been conducting research for the library community for more than 30 years. In accordance with its mission, OCLC makes its research outcomes known through various publications; these publications, including journal articles, reports and presentations, are available through the organization's website. OCLC Publications – Research articles from various journals including Code4Lib Journal, OCLC Research, Reference & User Services Quarterly, College & Research Libraries News, Art Libraries Journal, National Education Association Newsletter; the most recent publications are displayed first, all archived resources, starting in 1970, are available. Membership Reports – A number of significant reports on topics ranging from virtual reference in libraries to perceptions about library funding. Newsletters – Current and archived newsletters for the library and archive community.

Presentations – Presentations from both guest speakers and OCLC research from conferences and other events. The presentations are organized into five categories: Conference presentations, Dewey presentations, Distinguished Seminar Series, Guest presentations, Research staff