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Quillen Spectral Sequence
In the area of mathematics known as K-theory, the Quillen spectral sequence, also called the Brown–Gersten–Quillen or BGQ spectral sequence (named after Kenneth Brown, Stephen Gersten, and Daniel Quillen), is a spectral sequence converging to the sheaf cohomology of a type of topological space that occurs in algebraic geometry. It is used in calculating the homotopy properties of a simplicial group. References * * External links A spectral sequence of Quillenat the Stacks Project The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them". , the book consists of 115 chapters (excluding the license and index ... {{algebra-stub Algebraic K-theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenneth Brown (mathematician)
Kenneth Stephen Brown (born 1945) is a professor of mathematics at Cornell University, working in category theory and cohomology theory. Among other things, he is known for Ken Brown's lemma in the theory of model category, model categories. He is also the author of the book ''Cohomology of Groups'' (Graduate Texts in Mathematics 87, Springer, 1982). Brown earned his Doctor of Philosophy, Ph.D. in 1971 from the Massachusetts Institute of Technology, under the supervision of Daniel Quillen, with thesis ''Abstract Homotopy Theory and Generalized Sheaf Cohomology''. He was an invited speaker at the International Congress of Mathematicians in 1978 in Helsinki. In 2012 he became a fellow of the American Mathematical Society. retrieved 2013-11-16. Referenc ...
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Stephen Gersten
Stephen M. Gersten (born 2 December 1940) was an American mathematician, specializing in finitely presented groups and their geometric properties. Gersten graduated in 1961 with an AB from Princeton University and in 1965 with a PhD from Trinity College, Cambridge. His doctoral thesis was ''Class Groups of Supplemented Algebras'' written under the supervision of John R. Stallings. In the late 1960s and early 1970s he taught at Rice University. In 1972–1973 he was a visiting scholar at the Institute for Advanced Study. In 1973 he became a professor at the University of Illinois at Urbana–Champaign. In 1974 he was an Invited Speaker at the International Congress of Mathematicians in Vancouver. At the University of Utah he became a professor in 1975 and is now semi-retired there. His PhD students include Roger C. Alperin and Edward W. Formanek. Gersten's conjecture has motivated considerable research. Gersten's theorem If is an automorphism of a finitely generated free group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Education and career Quillen was born in Orange, New Jersey, and attended Newark Academy. He entered Harvard University, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott, with a thesis in partial differential equations. He was a Putnam Fellow in 1959. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf (mathematics), sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the direct image of a sheaf, pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science & Business Media
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 internationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Group
In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. In fact it can be shown that any simplicial abelian group A is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, \prod_ K(\pi_iA,i). A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring. discusses a simplicial analogue of the fact that a cohomology class on a Kähler manifold has a unique harmonic representative and deduces Kirchhoff's circuit laws from these observations. References * * * Charles Weibel Charles Alexander Weibel (born October 28, 1950 in Terre Haute, Indiana) is an A ... [...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] |
Stacks Project
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them". , the book consists of 115 chapters (excluding the license and index chapters) spreading over 7500 pages. The maintainer of the project, who reviews and accepts the changes, is Aise Johan de Jong. See alsoKerodona Stacks project inspired online textbook on categorical homotopy theory maintained by Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science ... References External linksProject website*Latest from the Stacks Project(as of 2013) (Accessed 2020-04-01) Mathematics textbooks {{mathematics-lit-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
