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Quillen–Lichtenbaum Conjecture
In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by , who was inspired by earlier conjectures of . and proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of Markus Rost, proved the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes. Statement The conjecture in Quillen's original form states that if ''A'' is a finitely-generated algebra over the integers and ''l'' is prime, then there is a spectral sequence analogous to the Atiyah–Hirzebruch spectral sequence, starting at :E_2^ = H^p_(\textA ell^ Z_\ell(-q/2)), (which is understood to be 0 if ''q'' is odd) and abutting to :K_A\otimes Z_\ell for −''p'' − ''q'' > 1 + dim ''A''. ''K''-theory of the integers Assuming the Quillen–Lichtenbaum conjecture and the Vandiver conjecture, the ''K''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Hirzebruch Spectral Sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, it relates the generalized cohomology groups : E^i(X) with 'ordinary' cohomology groups H^j with coefficients in the generalized cohomology of a point. More precisely, the E_2 term of the spectral sequence is H^p(X;E^q(pt)), and the spectral sequence converges conditionally to E^(X). Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where E=H_. It can be derived from an exact couple that gives the E_1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E. In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandiver Conjecture
Vandiver may refer to: People * Beverly J. Vandiver, American psychologist * Elizabeth Vandiver (born 1956), American classical scholar * Ernest Vandiver (1918–2005), American politician, governor of Georgia * Frank E. Vandiver (1925–2005), American historian and university president * Fuzzy Vandivier (1903–1983), American high school and collegiate basketball player * Harry Vandiver (1882–1973), American mathematician * J. Kim Vandiver (born 1945), American MIT professor and engineer * Jane Kidd (politician) (born 1953), née Vandiver, American retired politician * Jerry Vandiver, American songwriter and musician * Jim Vandiver (1939–2015), American race car driver * Murray Vandiver (1845–1916), American politician and Treasurer of Maryland * Pendleton Vandiver (1869–1932), American fiddler * Robert R. Vandiver (1805–1885), American politician and contractor * Shaun Vandiver (born 1968), American basketball player * Willard Duncan Vandiver (1854–1932), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markus Rost
Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld. He is known for his work on norm varieties (a key part in the proof of the Bloch–Kato conjecture) and for the Rost invariant (a cohomological invariant with values in Galois cohomology of degree 3). Together with J.-P. Serre he is one of the cofounders of the theory of cohomological invariants of linear algebraic groups. He has also made numerous contributions to the theory of torsors, quadratic forms, central simple algebras, Jordan algebras (the Rost-Serre invariant), exceptional groups, and essential dimension. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voevodsky
Vladimir Alexandrovich Voevodsky (, ; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory. Early life and education Vladimir Voevodsky's father, Aleksander Voevodsky, was head of the Laboratory of High Energy Leptons in the Institute for Nuclear Research at the Russian Academy of Sciences. His mother Tatyana was a chemist. Voevodsky attended Moscow State University for a while, but was expelled without a diploma for refusing to attend classes and failing academically. He received his Ph.D. in mathematics from Harvard University in 1992 after being recommended without even applying and without a formal college degree, following several independent pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
