|
Hurwitz Quaternion Order
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967 by Goro Shimura, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature). Definition Let K be the maximal real subfield of \mathbb(\rho) where \rho is a 7th-primitive root of unity. The ring of integers of K is \mathbbeta/math>, where the element \eta=\rho+ \bar\rho can be identified with the positive real 2\cos(\tfrac). Let D be the quaternion algebra, or symbol algebra :D:=\,(\eta,\eta)_, so that i^2=j^2=\eta and ij=-ji in D. Also let \tau=1+\eta+\eta^2 and j'=\tfrac(1+\eta i + \tau j). Let :\mathcal_=\mathbbetai,j,j']. Then \mathcal_ is a maximal Order (ring theory), order of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (ring Theory)
In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that #''A'' is a finite-dimensional algebra over the field \mathbb of rational numbers #\mathcal spans ''A'' over \mathbb, and #\mathcal is a \mathbb-lattice in ''A''. The last two conditions can be stated in less formal terms: Additively, \mathcal is a free abelian group generated by a basis for ''A'' over \mathbb. More generally for ''R'' an integral domain contained in a field ''K'', we define \mathcal to be an ''R''-order in a ''K''-algebra ''A'' if it is a subring of ''A'' which is a full ''R''-lattice. When ''A'' is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Norm
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''Brauer equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surfaces
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. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Hurwitz Triplet
In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, respectively the Klein quartic and the Macbeath surface). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces. Arithmetic construction Let K be the real subfield of \mathbbrho/math> where \rho is a 7th-primitive root of unity. The ring of integers of ''K'' is \mathbbeta/math>, where \eta=2\cos(\tfrac). Let D be the quaternion algebra, or symbol algebra (\eta,\eta)_. Also Let \tau=1+\eta+\eta^2 and j'=\tfrac(1+\eta i + \tau j). Let \mathcal_\mathrm=\mathbbetai,j,j']. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macbeath Surface
In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface. The automorphism group of the Macbeath surface is the simple group Projective linear group, PSL(2,8), consisting of 504 symmetries. Triangle group construction The surface's Fuchsian group can be constructed as the principal congruence subgroup of the (2,3,7) triangle group in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing the ideal \langle 2 \rangle in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its systolic geometry, systole is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations. It is possible to realize the resulting triangulated surface as a non-convex polyhedron without self-intersections. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to , the second-smallest non-abelian simple group after the alternating group A5. The quartic was first described in . Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see for a survey of properties. Originally, the "Klein quartic" referred specifically to the subset of the complex projective plane defined by an algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systoles Of Surfaces
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The ''systolic area'' of a metric is defined to be the ratio area/sys2. The ''systolic ratio'' SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry. Torus In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by 2/\sqrt, with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice). Real projective plane A similar result is given by Pu's inequality for the real projective plane from 1952, due to Pao Ming Pu, with an upper bound of ''π''/2 for the systolic ratio SR(RP2), also attained in the constant curvature case. Klein bottl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Sarnak
Peter Clive Sarnak (born 18 December 1953) is a South African-born mathematician with dual South-African and American nationalities. Sarnak has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study since 2007. He is also Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics. He is known for his work in analytic number theory. He also sits on the Board of Adjudicators and the selection committee for the Mathematics award, given under the auspices of the Shaw Prize. Education Sarnak is the grandson of one of Johannesburg's leading rabbis and lived in Israel for three years as a child. He graduated from the University of the Witwatersrand (BSc 1975, BSc(Hons) 1976) and Stanford University (PhD 1980), under the direction of Paul Cohen. Sarnak's highly cited work (with A. Lubotzky and R. Phillips) applied deep results in number theory to Ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürg Peter Buser
Jürg Peter Buser, known as Peter Buser, (born 27 February 1946 in Basel) is a Swiss mathematician, specializing in differential geometry and global analysis. Education and career Buser received his doctorate in 1976 from the University of Basel with advisor Heinz Huber and thesis ''Untersuchungen über den ersten Eigenwert des Laplaceoperators auf kompakten Flächen'' (Studies on the first eigenvalue of the Laplace operator on compact surfaces). As a post-doctoral student he was at the University of Bonn, the University of Minnesota. and the State University of New York at Stony Brook, before he habilitated at the University of Bonn with a thesis on the length spectrum of Riemann surfaces. Buser is known for his construction of curved isospectral surfaces (published in 1986 and 1988). His 1988 construction led to a negative solution to Mark Kac's famous 1966 problem '' Can one hear the shape of a drum?''. The negative solution was published in 1992 by Scott Wolpert, David Webb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SL(2,R)
In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics. SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically, :PSL(2, R) = SL(2, R) / , where ''I'' denotes the 2 × 2 identity matrix. It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL±(2, R), the group of real 2 × 2 matrices with determinant ±1; this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
