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Bers Slice
In the mathematical theory of Kleinian groups, Bers slices and Maskit slices, named after Lipman Bers and Bernard Maskit, are certain slices through the moduli space of Kleinian groups. Bers slices For a quasi-Fuchsian group, the limit set is a Jordan curve whose complement has two components. The quotient of each of these components by the groups is a Riemann surface, so we get a map from marked quasi-Fuchsian groups to pairs of Riemann surfaces, and hence to a product of two copies of Teichmüller space. A Bers slice is a subset of the moduli space of quasi-Fuchsian groups for which one of the two components of this map is a constant function to a single point in its copy of Teichmüller space. The Bers slice gives an embedding of Teichmüller space into the moduli space of quasi-Fuchsian groups, called the Bers embedding, and the closure of its image is a compactification of Teichmüller space called the Bers compactification. Maskit slices A Maskit slice is similar to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex number, complex matrix (mathematics), matrices of determinant 1 by their center (group theory), 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 transformation, Möbius transformations is also related as the non-orientation-preserving isometry group of , . So, a Kleinian group can be regarded as a discrete subgroup group action, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipman Bers
Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also known for his work in human rights activism.. Biography Bers was born in Riga, then under the rule of the Russian Czars, and spent several years as a child in Saint Petersburg; his family returned to Riga in approximately 1919, by which time it was part of independent Latvia. In Riga, his mother was the principal of a Jewish elementary school, and his father became the principal of a Jewish high school, both of which Bers attended, with an interlude in Berlin while his mother, by then separated from his father, attended the Berlin Psychoanalytic Institute. After high school, Bers studied at the University of Zurich for a year, but had to return to Riga again because of the difficulty of transferring money from Latvia in the international f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Maskit
Bernard (Bernie) Maskit (27 May 1935 – 15 March 2024) was an American mathematician who worked on Kleinian groups, low dimensional geometry and topology, and related topics. Life and Work Maskit studied for both his bachelors and doctoral degrees at New York University, earning his Ph.D. in 1964 under the supervision of Lipman Bers,with a thesis entitled ''On Klein's Combination Theorem''. After postdoctoral studies at the Institute for Advanced Study, he held an assistant professorship at the Massachusetts Institute of Technology from 1965 to 1972. He then moved to the mathematics department at Stony Brook University, where he retired in 2008 and was then a professor emeritus until his death. In 2012, he became one of the inaugural fellows of the American Mathematical Society. Maskit’s main area of mathematical expertise was the study of Kleinian groups acting on low dimensional hyperbolic spaces, where he made fundamental contributions. His works include thplanarity theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus (topology), genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Fuchsian Group
In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that .... If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two. Some authors use "quasi-Fuchsian group" to mean "quasi-Fuchsian group of type 1", in other words the limit set is the whole Jordan curve. This terminology is incompatible with the use of the terms "type 1" and "type 2" for Kleinian groups: all quasi-Fuchsian groups are Kleinian groups of type 2 (even if they are quasi-Fuchsian groups of type 1), as their limit sets are proper subsets of the Riemann sphere. The special case when t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include Graph of a function, graphs of Multivalued function, multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a Surface (topology), surface: a two-dimensional real manifold, but it contains more structure (specifically a Complex Manifold, complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and Metrizabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller Space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T(S) may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6g-6 for a surface of genus g \ge 2. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of researc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Function
In mathematics, a constant function is a function whose (output) value is the same for every input value. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, the function is the specific constant function where the output value is . The domain of this function is the set of all real numbers. The image of this function is the singleton set . The independent variable does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely , , , and so on. No matter what value of is input, the output is . The graph of the constant function is a ''horizontal line'' in the plane that passes through the point . In the context of a polynomial in one variable , the constant function is called ''non-zero constant function'' because it is a polynomial of degree 0, and its general form is , where is nonzero. This function has no intersection point with the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactification (mathematics)
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape". An example Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞. The resulting compactification is homeomorphism, homeomorphic to a circle in the plane (which, as a closed and bounded subset of the Euclidean plane, is compact). Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification. The direction in whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, Academic journal, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, Technology, technical, and engineering subject areas, abandoning its literary interests. Wiley's son Joh ... [...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] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
