|
Lars Ahlfors
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Background Ahlfors was born in Helsinki, Finland. His mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology. The Ahlfors family was Swedish-speaking, so he first attended the private school Nya svenska samskolan where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and Rolf Nevanlinna. He assisted Nevanlinna in 1929 with his work on Denjoy's conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem. It states that the number of asymptotic values approached by an entire function of order ρ along c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helsinki
Helsinki ( or ; ; sv, Helsingfors, ) is the capital, primate, and most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of Uusimaa in southern Finland, and has a population of . The city's urban area has a population of , making it by far the most populous urban area in Finland as well as the country's most important center for politics, education, finance, culture, and research; while Tampere in the Pirkanmaa region, located to the north from Helsinki, is the second largest urban area in Finland. Helsinki is located north of Tallinn, Estonia, east of Stockholm, Sweden, and west of Saint Petersburg, Russia. It has close historical ties with these three cities. Together with the cities of Espoo, Vantaa, and Kauniainen (and surrounding commuter towns, including the eastern neighboring municipality of Sipoo), Helsinki forms the Greater Helsinki metropolitan area, which has a population of over 1.5 million. Of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Capacity
In the mathematical discipline of complex analysis, the analytic capacity of a compact subset ''K'' of the complex plane is a number that denotes "how big" a bounded analytic function on C \ ''K'' can become. Roughly speaking, ''γ''(''K'') measures the size of the unit ball of the space of bounded analytic functions outside ''K''. It was first introduced by Lars Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions. Definition Let ''K'' ⊂ C be compact. Then its analytic capacity is defined to be :\gamma(K) = \sup \ Here, \mathcal^\infty (U) denotes the set of bounded analytic functions ''U'' → C, whenever ''U'' is an open subset of the complex plane. Further, : f'(\infty):= \lim_z\left(f(z)-f(\infty)\right) : f(\infty):= \lim_f(z) Note that f'(\infty) = g'(0), where g(z) = f(1/z). However, usually f'(\infty)\neq \lim_ f'(z). If ''A'' ⊂ C is an arbitrary set, then we define :\gamma(A) = \sup \. Removable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leroy P
Leroy or Le Roy may refer to: People * Leroy (name), a given name and surname * Leroy (musician), American musician * Leroy (sailor), French sailor Places United States * Leroy, Alabama * Le Roy, Illinois * Le Roy, Iowa * Le Roy, Kansas * Le Roy, Michigan * Le Roy, Minnesota * Le Roy (town), New York ** Le Roy (village), New York * Leroy, Indiana * Leroy, Texas * LeRoy, Wisconsin, a town * LeRoy (community), Wisconsin, an unincorporated community * Leroy Township, Calhoun County, Michigan * Leroy Township, Ingham County, Michigan * LeRoy Township, Lake County, Ohio * Leroy Township, Pennsylvania * LeRoy, West Virginia Elsewhere * Leroy, Saskatchewan, Canada * Rural Municipality of Leroy No. 339, Saskatchewan, Canada * 93102 Leroy Year 931 ( CMXXXI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. Events By place Europe * Spring – Hugh of Provence, king of Italy, cedes Lower Burgundy to Rudolp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolf Prize In Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. According to a reputation survey conducted in 2013 and 2014, the Wolf Prize in Mathematics is the third most prestigious international academic award in mathematics, after the Abel Prize and the Fields Medal. Until the establishment of the Abel Prize, it was probably the closest equivalent of a "Nobel Prize in Mathematics", since the Fields Medal is awarded every four years only to mathematicians under the age of 40. Laureates Laureates per country Below is a chart of all laureates per country (updated to 2022 laureates). Some laureates are counted more than once if have multiple citizenship. Notes See also * List of mathematics awards References External links * * * Israel-Wolf-Prizes 2015Jerusalempost Wolf Prizes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wihuri International Prize
The Wihuri International Prize is a prize awarded by the Wihuri Foundation for International Prizes to people who have furthered the cultural or economic development in Finland. Founded in 1953 by Antti Wihuri, The Wihuri Foundation for International Prizes also awards the music prize Wihuri Sibelius Prize The Wihuri Sibelius Prize is a music prize awarded by the Wihuri Foundation for International Prizes to prominent composers who have become internationally known and acknowledged. The Wihuri Sibelius Prize is one of the biggest and most prestig .... The awarded prizes range from EUR 30,000 to 150,000, and a prize must be awarded at least every three years. The prize may be awarded to private individuals or organizations regardless of nationality, religion, race or language. The International Prize has been awarded to mostly Finnish and foreign scientists. The first Wihuri International Prize was awarded on October 9, 1958, the birthday of Antti Wihuri, to Mathematics Profess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields. The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics. The prize includes a monetary award which, since 2006, has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Function Theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem. Topics in geometric function theory The following are some of the most important topics in geometric function theory: Conformal maps A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map, : f: U \rightarrow V\qquad with U,V \subset \mathbb^n is called conformal (or angle-preserving) at a point u_0 if it preserves oriented angles between curves through u_0 with respect to their orientation (i.e., not just the magnitude of the angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. Quasiconformal maps In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry. Conformal manifolds A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics ''g'' and ''h'' are equivalent if and only if :h = \lambda^2 g , where ''λ'' is a real-valued smooth function defined on the manifold and is called the conformal factor. An equivalence clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ahlfors Theory
Ahlfors theory is a mathematical theory invented by Lars Ahlfors as a geometric counterpart of the Nevanlinna theory. Ahlfors was awarded one of the two very first Fields Medals for this theory in 1936. It can be considered as a generalization of the basic properties of covering maps to the maps which are "almost coverings" in some well defined sense. It applies to bordered Riemann surfaces equipped with conformal Riemannian metrics. Preliminaries A ''bordered Riemann surface'' ''X'' can be defined as a region on a compact Riemann surface whose boundary ∂''X'' consists of finitely many disjoint Jordan curves. In most applications these curves are piecewise analytic, but there is some explicit minimal regularity condition on these curves which is necessary to make the theory work; it is called the ''Ahlfors regularity''. A ''conformal Riemannian metric'' is defined by a length element ''ds'' which is expressed in conformal local coordinates ''z'' as ''ds'' = ''ρ''(''z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by . has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball in . The group of Möbius transformations is also related as the non-orientation-preserving isometry group of , . So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces. History The theory of general Kleinian groups was founded by and , who named them after Felix Klein. The special case of Schottky groups had been studied a few years earlier, in 1877, by Schottky. Definitions One modern definition of Kleinian group is as a group which acts on the 3-ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ahlfors Finiteness Theorem
In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by . The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed. Bers area inequality The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with ''N'' generators and with region of discontinuity Ω, then :Area(Ω/Γ) ≤ with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for Fuchsian group In math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
