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William Goldman (mathematician)
William Mark Goldman (born 1955 in Kansas City, Missouri) is a professor of mathematics at the University of Maryland, College Park (since 1986). He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980. Research contributions Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis, "Affine manifolds and projective geometry on manifolds", supervised by William Thurston and Dennis Sullivan. This work led to work with Morris Hirsch and David Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex real projective structures on a closed orientable surface of genus g > 1 is homeomorphic to an open cell of dimension 16g-16. With Suhyoung Choi, he proved that this space is a connected component (the "Hitchin component") of the space of equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bar-Ilan University
Bar-Ilan University (BIU, he, אוניברסיטת בר-אילן, ''Universitat Bar-Ilan'') is a public research university in the Tel Aviv District city of Ramat Gan, Israel. Established in 1955, Bar Ilan is Israel's second-largest academic institution. It has about 20,000 students and 1,350 faculty members. Bar-Ilan's mission is to "blend Jewish tradition with modern technologies and scholarship and the university endeavors to ... teach the Jewish heritage to all its students while providing nacademic education." History Bar-Ilan University has Jewish-American roots: It was conceived in Atlanta in a meeting of the American Mizrahi organization in 1950, and was founded by Professor Pinkhos Churgin, an American Orthodox rabbi and educator, who was president from 1955 to 1957 where he was succeeded by Joseph H. Lookstein who was president from 1957 to 1967. When it was opened in 1955, it was described by ''The New York Times'' "as Cultural Link Between the sraeliRepublic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world. In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid ob ... [...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 Riemann surface#Hyperbolic Riemann surfaces, hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a Ball (mathematics), ball of dimension 6g-6 for a surface of genus g \ge 2. In this way Teichmüller space can be viewed as the orbifold, universal covering orbifold of the Moduli of algebraic curves, Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil–Petersson Metric
In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space ''T''''g'',''n'' of genus ''g'' Riemann surfaces with ''n'' marked points. It was introduced by using the Petersson inner product on forms on a Riemann surface (introduced by Hans Petersson). Definition If a point of Teichmüller space is represented by a Riemann surface ''R'', then the cotangent space at that point can be identified with the space of quadratic differentials at ''R''. Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric. Properties stated, and proved, that the Weil–Petersson metric is a Kähler metric. proved that it has negative holomorphic sectional, scalar, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scott A
Scott may refer to: Places Canada * Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec * Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380 * Rural Municipality of Scott No. 98, Saskatchewan United States * Scott, Arkansas * Scott, Georgia * Scott, Indiana * Scott, Louisiana * Scott, Missouri * Scott, New York * Scott, Ohio * Scott, Wisconsin (other) (several places) * Fort Scott, Kansas * Great Scott Township, St. Louis County, Minnesota * Scott Air Force Base, Illinois * Scott City, Kansas * Scott City, Missouri * Scott County (other) (various states) * Scott Mountain, a mountain in Oregon * Scott River, in California * Scott Township (other) (several places) Elsewhere * 876 Scott, minor planet orbiting the Sun * Scott (crater), a lunar impact crater near the south pole of the Moon *Scott Conservation Park, a protected area in South Australia People * Scott (surname), including a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvmanifold
In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.) A special class of solvmanifolds, nilmanifolds, was introduced by Anatoly Maltsev, who proved the first structural theorems. Properties of general solvmanifolds are similar, but somewhat more complicated. Examples * A solvable Lie group is trivially a solvmanifold. * Every nilpotent group is solvable, therefore, every nilmanifold is a solvmanifold. This class of examples includes ''n''-dimensional tori and the quotient of the 3-dimensional real Heisenberg group by its integral Heisenberg subgroup. * The Möbius band and the Klein bottle are solvmanifolds that are not nilmanifolds. * The mapping torus of an Anosov diffeomorphism of the ''n''-torus is a solvmanifold. For n=2, these manifolds belong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilmanifold
In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, the quotient of a nilpotent Lie group ''N'' modulo a closed subgroup ''H''. This notion was introduced by Anatoly Mal'cev in 1951. In the Riemannian category, there is also a good notion of a nilmanifold. A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it. The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization: every homogeneous nilmanifold is isometric to a nilpotent Lie group with left-invariant metric (see Wilson). Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature, almost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Handlebody
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handles are used to particularly study 3-manifolds. Handlebodies play a similar role in the study of manifolds as simplicial complexes and CW complexes play in homotopy theory, allowing one to analyze a space in terms of individual pieces and their interactions. ''n''-dimensional handlebodies If (W,\partial W) is an n-dimensional manifold with boundary, and :S^ \times D^ \subset \partial W (where S^ represents an n-sphere and D^n is an n-ball) is an embedding, the n-dimensional manifold with boundary :(W',\partial W') = ((W \cup( D^r \times D^)),(\partial W - S^ \times D^)\cup (D^r \times S^)) is said to be ''obtained from :(W,\partial W) by attaching an r-handle''. The boundary \partial W' is obtained from \partial W by surgery. As trivial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigory Margulis
Grigory Aleksandrovich Margulis (russian: Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978, a Wolf Prize in Mathematics in 2005, and an Abel Prize in 2020, becoming the fifth mathematician to receive the three prizes. In 1991, he joined the faculty of Yale University, where he is currently the Erastus L. De Forest Professor of Mathematics. Biography Margulis was born to a Russian family of Lithuanian Jewish descent in Moscow, Soviet Union. At age 16 in 1962 he won the silver medal at the International Mathematical Olympiad. He received his PhD in 1970 from the Moscow State University, starting research in ergodic theory under the supervision of Yakov Sinai. Early work with David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor–Wood Inequality
In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after John Milnor and John W. Wood. Flat bundles For linear bundles, flatness is defined as the vanishing of the curvature form of an associated connection. An arbitrary smooth (or topological) ''d''-dimensional fiber bundle is flat if it can be endowed with a foliation of codimension d that is transverse to the fibers. The inequality The Milnor–Wood inequality is named after two separate results that were proven by John Milnor and John W. Wood. Both of them deal with orientable circle bundles over a closed oriented surface \Sigma_g of positive genus ''g''. Theorem (Milnor, 1958) Let \pi\colon E \to \Sigma_g be a flat oriented linear circle bundle. Then the Euler number of the bundle satisfies , e(\pi), \leq g -1. Theorem (Wood, 1971) Let \pi\colon E \to \Sigma_g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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