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Arithmetic Hyperbolic 3-manifold
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space \mathbb H^3 by an arithmetic Kleinian group. Definition and examples Quaternion algebras A quaternion algebra over a field F is a four-dimensional central simple F-algebra. A quaternion algebra has a basis 1, i, j, ij where i^2, j^2 \in F^\times and ij = -ji. A quaternion algebra is said to be split over F if it is isomorphic as an F-algebra to the algebra of matrices M_2(F); a quaternion algebra over an algebraically closed field is always split. If \sigma is an embedding of F into a field E we shall denote by A \otimes_\sigma E the algebra obtained by extending scalars from F to E where we view F as a subfield of E via \sigma. Arithmetic Kleinian groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bianchi Group
In mathematics, a Bianchi group is a group of the form :PSL_2(\mathcal_d) where ''d'' is a positive square-free integer. Here, PSL denotes the projective special linear group and \mathcal_d is the ring of integers of the imaginary quadratic field \mathbb(\sqrt). The groups were first studied by as a natural class of discrete subgroups of PSL_2(\mathbb), now termed Kleinian groups. As a subgroup of PSL_2(\mathbb), a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space \mathbb^3. The quotient space M_d = PSL_2(\mathcal_d) \backslash\mathbb^3 is a non-compact, hyperbolic 3-fold with finite volume, which is also called ''Bianchi orbifold''. An exact formula for the volume, in terms of the Dedekind zeta function of the base field \mathbb(\sqrt), was computed by Humbert Humbert, Umbert or Humberto (Latinized ''Humbertus'') is a Germanic given name, from ''hun'' "warrior" and ''beraht'' "bright". It also came into use as a surname. Given nam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry & Topology
''Geometry & Topology'' is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sciences Publishers, a nonprofit academic publishing organisation. It was founded in 1997Allyn Jackson The slow revolution of the free electronic journal Notices of the American Mathematical Society, vol. 47 (2000), no. 9, pp. 1053-1059 by a group of topologists who were dissatisfied with recent substantial rises in subscription prices of journals published by major publishing corporations. The aim was to set up a high-quality journal, capable of competing with existing journals, but with substantially lower subscription fees. The journal was open-access for its first ten years of existence and was available free to individual users, although institutions were required to pay modest subscription fees for both online access and for pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Mathematics Research Notices
The ''International Mathematics Research Notices'' is a peer-reviewed mathematics journal. Originally published by Duke University Press and Hindawi Publishing Corporation, it is now published by Oxford University Press. retrieved 2015-02-26. The Executive Editor is ( Tel Aviv University). According to the '' Journal Citation Reports'', the journ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ian Agol
Ian Agol (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of California, San Diego. At UCSD, his advisor was Michael Freedman and his thesis was ''Topology of Hyperbolic 3-Manifolds''. He is a professor at the University of California, Berkeley and a former professor at the University of Illinois at Chicago. Contributions In 2004, Agol proved the Marden tameness conjecture, a conjecture of Albert Marden. It states that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. The conjecture was also independently proven by Danny Calegari and David Gabai, and implies the Ahlfors measure conjecture. In 2012 he announced a proof of the virtually Haken conjecture, which was publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtually Haken Conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is ''virtually Haken''. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968, although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list. A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica. The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure-eight Knot (mathematics)
Figure 8 (figure of 8 in British English) may refer to: * 8 (number), in Arabic numerals Entertainment * ''Figure 8'' (album), a 2000 album by Elliott Smith * "Figure of Eight" (song), a 1989 song by Paul McCartney * '' Figure Eight EP'', a 2008 EP by This Et Al * "Figure 8" (song), a 2012 song by Ellie Goulding from ''Halcyon'' * "Figure Eight", an episode and song from the children's educational series ''Schoolhouse Rock!'' * "Figure of Eight", song by Status Quo from '' In Search of the Fourth Chord'' * "Figure 8", a song by FKA Twigs from the EP ''M3LL155X'' Geography * Figure Eight Island, North Carolina, United States * Figure Eight Lake, Alberta, Canada * Figure-Eight Loops, feature of the Historic Columbia River Highway in Guy W. Talbot State Park Mathematics and sciences * Figure-eight knot (mathematics), in knot theory * ∞, symbol meaning infinity * Lemniscate, various types of mathematical curve that resembles a figure 8 * Figure 8, a two-lobed Lissaj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meyerhoff Manifold
In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by (5,1) surgery on the figure-8 knot complement. It was introduced by as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold turned out to have slightly smaller volume. It has the second smallest volume :V_m = 12\cdot(283)^\zeta_k(2)(2\pi)^ = 0.981368\dots of orientable arithmetic hyperbolic 3-manifolds, where \zeta_k is the zeta function of the quartic field of discriminant -283. Alternatively, : V_m = \Im(\rm_2(\theta)+\ln, \theta, \ln(1-\theta)) = 0.981368\dots where \rm_n is the polylogarithm and , x, is the absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ... of the complex root \theta (with positive imaginary part) of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weeks Manifold
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… () and showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by as well as . Volume Since the Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to Armand Borel: : V_w = \frac = 0.942707\dots where k is the number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ... generated by \theta satisfying \theta^3-\theta+1=0 and \zeta_k is the Dedekind zeta function of k. Alternative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Dehn Surgery
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions. Such an operation is often also called hyperbolic Dehn filling, as Dehn surgery proper refers to a "drill and fill" operation on a link which consists of ''drilling'' out a neighborhood of the link and then ''filling'' back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling". We will generally assume that a hyperbolic 3-manifold is complete. Suppose ''M'' is a cusped hyperbolic 3-manifold with ''n'' cusps. ''M'' can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Zeta Function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at ''s'' = 1, and its values encode arithmetic data of ''K''. The extended Riemann hypothesis states that if ''ζ''''K''(''s'') = 0 and 0 1. In the case ''K'' = Q, this definition reduces to that of the Riemann zeta function. Euler product The Dedekind zeta function of K has an Euler product which is a product over all the prime ideals \mathfrak of \mathcal_K :\zeta_K (s) = \prod_ \frac,\text(s)>1. This is the expression in analytic terms of the uniqueness of prime factorization of ideal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
