|
Algebraic Geometry (book)
''Algebraic Geometry'' is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.MathSciNet lists more than 2500 citations of this book. Importance It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students. Contents The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ..., with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robin Hartshorne
__NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under the name Robert C. Hartshorne). He received a Ph.D. in mathematics from Princeton University in 1963 after completing a doctoral dissertation titled ''Connectedness of the Hilbert scheme'' under the supervision of John Coleman Moore and Oscar Zariski. He then became a Junior Fellow at Harvard University, where he taught for several years. In 1972, he was appointed to the faculty at the University of California, Berkeley, where he is a Professor Emeritus as of 2020. Hartshorne is the author of the text ''Algebraic Geometry''. Awards In 1979, Hartshorne was awarded the Leroy P. Steele Prize for "his expository research article Equivalence relations on algebraic cycles and subvarieties of small codimension, Proceedings of Symposia in Pure Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Textbook
A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textbooks and other books used in schools. Today, many textbooks are published in both print and digital formats. History The history of textbooks dates back to ancient civilizations. For example, Ancient Greeks wrote educational texts. The modern textbook has its roots in the mass production made possible by the printing press. Johannes Gutenberg himself may have printed editions of ''Ars Minor'', a schoolbook on Latin grammar by Aelius Donatus. Early textbooks were used by tutors and teachers (e.g. alphabet books), as well as by individuals who taught themselves. The Greek philosopher Socrates lamented the loss of knowledge because the media of transmission were changing. Before the invention of the Greek alphabet 2,500 years ago, knowledge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathSciNet
MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links to other MR entries, citations, full journal entries, and links to original articles. It contains almost 3.6 million items and over 2.3 million links to original articles. Along with its parent publication ''Mathematical Reviews'', MathSciNet has become an essential tool for researchers in the mathematical sciences. Access to the database is by subscription only and is not generally available to individual researchers who are not affiliated with a larger subscribing institution. For the first 40 years of its existence, traditional typesetting was used to produce the Mathematical Reviews journal. Starting in 1980 bibliographic information and the reviews themselves were produced in both print and electronic form. This formed the basis of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme Theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with commu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. 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. Conventions 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 fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem). Formulation Let ''k'' be a field (such as the rational numbers) and ''K'' be an algebraically closed field extension (such as the complex numbers). Consider the polynomial ring k _1, \ldots, X_n/math> and let ''I'' be an ideal in this ring. The algebraic set V(''I'') defined by this ideal consists of all ''n''-tuples x = (''x''1,...,''x''''n'') in ''Kn'' such that ''f''(x) = 0 for all ''f'' in ''I''. Hilbert's Nullstellensatz st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Textbooks
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1977 Non-fiction Books
Events January * January 8 – Three bombs explode in Moscow within 37 minutes, killing seven. The bombings are attributed to an Armenian separatist group. * January 10 – Mount Nyiragongo erupts in eastern Zaire (now the Democratic Republic of the Congo). * January 17 ** 49 marines from the and are killed as a result of a collision in Barcelona harbour, Spain. * January 18 ** Scientists identify a previously unknown bacterium as the cause of the mysterious Legionnaires' disease. ** Australia's worst railway disaster at Granville, a suburb of Sydney, leaves 83 people dead. ** SFR Yugoslavia Prime minister Džemal Bijedić, his wife and 6 others are killed in a plane crash in Bosnia and Herzegovina. * January 19 – An Ejército del Aire CASA C-207C Azor (registration T.7-15) plane crashes into the side of a mountain near Chiva, on approach to Valencia Airport in Spain, killing all 11 people on board. * January 20 – Jimmy Carter is sworn in as the 39th President of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
