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Chern's Conjecture (affine Geometry)
Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem. Chern's conjecture states that the Euler characteristic of a compact affine manifold vanishes. Details In case the connection ∇ is the Levi-Civita connection of a Riemannian metric, the Chern–Gauss–Bonnet formula: : \chi(M) = \left ( \frac \right )^n \int_M \operatorname(K) implies that the Euler characteristic is zero. However, not all flat torsion-free connections on T M admit a compatible metric, and therefore, Chern–Weil theory cannot be used in general to write down the Euler class in terms of the curvature. History The conjecture is known to hold in several special cases: * when a compact affine manifold is 2-dimensional (as shown by Jean-Paul Benzécri in 1955, and later by John Milnor in 1957) * when a compact affine manifold is complete (i.e., affinely diffeomorphic to a quotien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shiing-Shen Chern
Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geometry" and is widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize. In memory of Shiing-Shen Chern, the International Mathematical Union established the Chern Medal in 2010 to recognize "an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics". Chern worked at the Institute for Advanced Study (1943–45), spent about a decade at the University of Chicago (1949-1960), and then moved to University of California, Berkeley, where he co-founded the Mathematical Sciences Research Institute in 1982 and was the institute's found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformations
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can be repres ... [...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''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, he r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morris Hirsch
Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley. A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of Edwin Spanier and Stephen Smale. His thesis was entitled ''Immersions of Manifolds''. In 2012 he became a fellow of the American Mathematical Society. Hirsch had 23 doctoral students, including William Thurston, William Goldman, and Mary Lou Zeeman. Selected works *with Stephen Smale and Robert L. Devaney: ''Differential equations, dynamical systems and an introduction to chaos'', Academic Press 2004 (2nd edition3rd edition, 2013*with Stephen Smale: ''Differential equations, dynamical systems and linear algebra'', Academic Press 1974 *Differential Topology, Springer 1976, 1997 *with Barry MazurSmoothings of piecewise linear manifolds Princeton University Press 1974 *with Charles C. Pugh, Michael Shub: Invariant Manifolds, Springer 1977 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Gauss–Bonnet Theorem
In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant). It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry. Riemann–Roch and Atiyah–Singer are other generalizations of the Gauss–Bonnet theorem. Statement One useful form o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be embedded as a smooth submanifold of R2''n'', whereas it is "rare" for a complex manifold to have a holomorphic embedding into C''n''. Consider for example any compact connected complex manifold ''M'': any holomorphic function on it is cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-Riemannian Manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Surface
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markus Conjecture
Marcus, Markus, Márkus or Mărcuș may refer to: * Marcus (name), a masculine given name * Marcus (praenomen), a Roman personal name Places * Marcus, a main belt asteroid, also known as (369088) Marcus 2008 GG44 * Mărcuş, a village in Dobârlău Commune, Covasna County, Romania * Marcus, Illinois, an unincorporated community * Marcus, Iowa, a city * Marcus, South Dakota, an unincorporated community * Marcus, Washington, a town * Marcus Island, Japan, also known as Minami-Tori-shima * Mărcuș River, Romania * Marcus Township, Cherokee County, Iowa Other uses * Markus, a beetle genus in family Cantharidae * ''Marcus'' (album), 2008 album by Marcus Miller * Marcus (comedian), finalist on ''Last Comic Standing'' season 6 * Marcus Amphitheater, Milwaukee, Wisconsin * Marcus Center, Milwaukee, Wisconsin * Marcus & Co., American jewelry retailer * Marcus by Goldman Sachs, an online bank * USS ''Marcus'' (DD-321), a US Navy destroyer (1919-1935) See also * Marcos (disambiguatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsachik Gelander
Tsachik Gelander (צחיק גלנדר) is an Israeli mathematician working in the fields of Lie groups, topological groups, symmetric spaces, lattices and discrete subgroups (of Lie groups as well as general locally compact groups). He is a professor in Northwestern University. Gelander earned his PhD from the Hebrew University of Jerusalem in 2003, under the supervision of Shahar Mozes. His doctoral dissertation, ''Counting Manifolds and Tits Alternative'', won the Haim Nessyahu Prize in Mathematics, awarded by the Israel Mathematical Union for the best annual doctoral dissertations in mathematics. After holding a Gibbs Assistant Professorship at Yale University, and faculty positions at the Hebrew University of Jerusalem and the Weizmann Institute of Science, Gelander joined Northwestern where he is currently a professor of mathematics. He contributed to the theory of lattices, Fuchsian groups and local rigidity, and the work on Chern's conjecture and the Derivation Probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
