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Translation Surface
In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory. A particularly interesting subclass is that of Veech surfaces (named after William A. Veech) which are the most symmetric ones. Definitions Geometric definition A translation surface is the space obtained by identifying pairwise by translations the sides of a collection of plane polygons. Here is a more formal definition. Let P_1,\ldots,P_m be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side s_i of any P_k there is a side s_j of some P_l with j\not=i and s_j = s_i + \vec v_i for some nonzero vector \vec v_i (and so that \vec v_j = -\vec v_i. Consider the space obtained by identifying all s_i w ... [...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] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a 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) 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. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuchsian Group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thurston Boundary
In mathematics, the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Thurston boundary can be interpreted as the space of projective measured foliations on the surface. The Thurston boundary of the Teichmüller space of a closed surface of genus g is homeomorphic to a sphere of dimension 6g-7. The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary. Measured foliations on surfaces Let S be a closed surface. A ''measured foliation'' (\mathcal F, \mu) on S is a foliation \mathcal F on S which may admit isolated singularities, together with a ''transverse measure'' \mu, i.e. a function which to each arc \alpha transverse to the foliation \mathcal F associates a positive real number \mu(\alpha). The foliation and the measure must be compatible in the sense that the measure is invariant if the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconformal Mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D'' → ''D''′ be an orientation-preserving homeomorphism between open sets in the plane. If ''f'' is continuously differentiable, then it is ''K''-quasiconformal if the derivative of ''f'' at every point maps circles to ellipses with eccentricity bounded by ''K''. Definition Suppose ''f'' : ''D'' → ''D''′ where ''D'' and ''D''′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of ''f''. If ''f'' is assumed to have continuous partial derivatives, then ''f'' is quasiconformal provided it satisfies the Beltrami equation for some complex valued Lebesgue measurable μ satisfying sup , μ, 0. Then ''f'' satisfies () precisely when it is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (mathematics), compactification of Teichmüller space called the Bers compactification. Maskit slices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Differential
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural interpretation as the cotangent space to the Riemann moduli space, or Teichmüller space. Local form Each quadratic differential on a domain U in the complex plane may be written as f(z) \,dz \otimes dz, where z is the complex variable, and f is a complex-valued function on U. Such a "local" quadratic differential is holomorphic if and only if f is holomorphic. Given a chart \mu for a general Riemann surface R and a quadratic differential q on R, the pull-back (\mu^)^*(q) defines a quadratic differential on a domain in the complex plane. Relation to abelian differentials If \omega is an abelian differential on a Riemann surface, then \omega \otimes \omega is a quadrat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Don Zagier
Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the ''Collège de France'' in Paris from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at the International Centre for Theoretical Physics (ICTP). Background Zagier was born in Heidelberg, West Germany. His mother was a psychiatrist, and his father was the dean of instruction at the American College of Switzerland. His father held five different citizenships, and he spent his youth living in many different countries. After finishing high school (at age 13) and attending Winchester College for a year, he studied for three years at MIT, completing his bachelor's and master's degrees and being named a Putnam Fellow in 1967 at the age of 16. He then wrote a doctoral dissertation on characteristic classes under Friedrich ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Okounkov
Andrei Yuryevich Okounkov (russian: Андре́й Ю́рьевич Окунько́в, ''Andrej Okun'kov'') (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at the University of California, Berkeley and the academic supervisor of HSE International Laboratory of Representation Theory and Mathematical Physics. In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry.""Information about Andrei Okounkov, Fields Medal winner" ICM Press Release Education and career He received his doctorate at |
Alex Eskin
Alex Eskin (born May 19, 1965Alex Eskin, Curriculum Vitae Department of Mathematics, . Accessed 2019-09-07 ) is an American mathematician. He is the Arthur Holly Compton Distinguished Service Professor in the Department of Mathematics at the . His research focuses on rational billiards and |
Anton Zorich
Anton V. Zorich (in Russian: ''Антон Владимирович Зорич''; born 3 September 1962) is a Russian mathematician at the Institut de mathématiques de Jussieu. He is the son of Vladimir A. Zorich. He received his Ph.D. from Moscow State University under the supervision of Sergei Novikov. He was an invited speaker at the 2006 International Congress of Mathematicians in Madrid. The theme was: "Geodesics on flat surfaces". At least two of his papers concern the explanation of mathematical discoveries he made by experimenting with computers. Selected publications * with M. Kontsevich: "Connected components of the moduli spaces of Abelian differentials with prescribed singularities", ''Inventiones mathematicae'' (2003) * "Flat surfaces", ''Frontiers in number theory, physics, and geometry'' (2006) * with M. Kontsevich: "Lyapunov exponents and Hodge theory", (1997) * "Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents", ''An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in 2012, and the Breakthrough Prize in Mathematics in 2014. Academic career and research He was born into the family of Lev Kontsevich, Soviet orientalist and author of the Kontsevich system. After ranking second in the All-Union Mathematics Olympiads, he attended Moscow State University but left without a degree in 1985 to become a researcher at the Institute for Information Transmission Problems in Moscow. While at the institute he published papers that caught the interest of the Max Planck Institute in Bonn and was invited for three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
