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David Gabai
David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects. Biography David Gabai received his B.S. in mathematics from MIT in 1976 and his Ph.D. in mathematics from Princeton University in 1980. Gabai completed his doctoral dissertation, titled "Foliations and genera of links", under the supervision of William Thurston. After positions at Harvard and University of Pennsylvania, Gabai spent most of the period of 1986–2001 at Caltech, and has been at Princeton since 2001. Gabai was the Chair of the Department of Mathematics at Princeton University from 2012 to 2019. Honours and awards In 2004, David Gabai was awarded the Oswald Veblen Prize in Geometry, given every three years by the American Mathematical Society. He was an invited speaker in the International Congress of Mathematicians 2010, Hyderabad on the top ... [...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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyderabad, India
Hyderabad ( ; , ) is the capital and largest city of the Indian state of Telangana and the '' de jure'' capital of Andhra Pradesh. It occupies on the Deccan Plateau along the banks of the Musi River, in the northern part of Southern India. With an average altitude of , much of Hyderabad is situated on hilly terrain around artificial lakes, including the Hussain Sagar lake, predating the city's founding, in the north of the city centre. According to the 2011 Census of India, Hyderabad is the fourth-most populous city in India with a population of residents within the city limits, and has a population of residents in the metropolitan region, making it the sixth-most populous metropolitan area in India. With an output of 74 billion, Hyderabad has the fifth-largest urban economy in India. Muhammad Quli Qutb Shah established Hyderabad in 1591 to extend the capital beyond the fortified Golconda. In 1687, the city was annexed by the Mughals. In 1724, Asaf Jah I, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3–manifolds of finite volume have a particular importance in 3–dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory. Importance in topology Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far f ... [...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 generated by \theta satisfying \theta^3-\theta+1=0 and \zeta_k is the Dedekind zeta function of k. Alternatively, : V_w = \Im(\rm_2(\theta)+\ln, \theta, \ln(1-\theta)) = 0.942707\dots where \rm_n is the polylogarithm and , x, is the absolute value of the complex root \theta (with positive imaginary part) of the cubic. Related manifolds The cusped hyperboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marden Conjecture
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure conjecture. History Topological tameness may be viewed as a property of the ends of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, that is, for surfaces. However, as the example of Alexander horned sphere shows, there are wild embeddings among 3-manifolds, so this property is not automatic. The conjecture was raised in the form of a question by Albert Marden, who proved that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smale Conjecture
The Smale conjecture, named after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in 1983 by Allen Hatcher. Equivalent statements There are several equivalent statements of the Smale conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3-space has the homotopy-type of the round circles, equivalently, O(3). Interestingly, this statement is not equivalent to the generalized Smale Conjecture, in higher dimensions. Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible. Yet another equivalent statement is that the space of constant-curvature Riemann metrics on the 3-sphere is contractible. Higher dimensions The (false) statement that the inclusion O(n+1) \to \text(S^n) is a weak equivalence for all n is sometimes meant whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genuine Lamination
{{disambiguation ...
Genuine may refer to: Companies *Genuine Parts Company, a Fortune 1000 company that was founded in 1928 *Genuine Scooters, a Chicago-based scooter manufacturer *Genuine Games, a video game company founded in early 2002 Music * ''Genuine'' (Stacie Orrico album), 2000 * ''Genuine'' (Fayray album), 2001 * "Genuine" (song), a 2000 song by Stacie Orrico *"Genuine", a 1995 song by Canadian singer-songwriter Mae Moore Other uses * ''Genuine'' (film), a 1920 silent film by Robert Wiene *Genuine, a difficulty rating in ''Dance Dance Revolution'' *Genuine (horse), a Japanese Thoroughbred racehorse *Authenticity (philosophy) Authenticity is a concept of personality in the fields of psychology, existential psychotherapy, existentialist philosophy, and aesthetics. In existentialism, authenticity is the degree to which a person's actions are congruent with his or her v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert Fiber Space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture. Definition A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of circles (called fibers) such that each fiber has a tubular neighborhood that forms a standard fibered torus. A standard fibered torus corresponding to a pair of coprime integers (a,b) with a>0 is the surface bundle of the automorphism of a disk given by rotation by an angle of 2\pi b/a (with the natural fibering by circles). If a=1 the middle fiber is called ordinary, while if a>1 the middle fiber is called exceptional. A compact Seifert fiber space has only a finite number of exceptional fibers. The set of fibers forms a 2-dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Laminations
Essential or essentials may refer to: Biology *Essential amino acid, one that the organism cannot produce by itself Groups and organizations * EQ Media Group, formerly Essential Media Group, a global television production company * Essential Media Communications, an Australian PR and polling company *Essential Products, a smart device company led by Andy Rubin *Essential Records (London), a subsidiary of London Records *Essential Records (Christian), a Christian subsidiary of Sony BMG Music Entertainment *The Essentials (band), a Canadian a cappella group formed in 1993 *Essentials (PlayStation), a budget package of PlayStation games Albums * ''Essential'' (Divinyls album), a compilation album, 1991 * ''Essential'' (Pet Shop Boys album), 1998 * ''Essential'' (Ramones album), 2007 * ''Essential'' (CeCe Peniston album), 2000 * ''Essential'' (Jethro Tull album) * Essential (Kate Ryan album), 2008 * ''Essential'' (Praga Khan album), 2005 * ''Essentials'' (Nate Dogg album), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Property R Conjecture
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, redefine, rent, mortgage, pawn, sell, exchange, transfer, give away or destroy it, or to exclude others from doing these things, as well as to perhaps abandon it; whereas regardless of the nature of the property, the owner thereof has the right to properly use it under the granted property rights. In economics and political economy, there are three broad forms of property: private property, public property, and collective property (also called cooperative property). Property that jointly belongs to more than one party may be possessed or controlled thereby in very similar or very distinct ways, whether simply or complexly, whether equally or unequally. However, there is an expectation that each party's will (rather discretion) with regard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taut Foliation
A taut object is one under tension. Taut is also a surname, and may refer to: * Bruno Taut (1880–1938), prolific German architect, urban planner and author * Max Taut (1884–1967), German architect Taut may also refer to: * Tauț, a commune in Arad County, Romania * Tăut, a village in Batăr Commune, Bihor County, Romania * Taut International, sports drink company acquired by A.G. Barr TAUT, an acronym, may refer to: * '' Tramways and Urban Transit'', a monthly magazine published in the UK * The complement of the SAT-problem; testing if a formula is a tautology, known to be co-NP In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP precisely ...-complete. {{surname Surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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