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S. S. Chern
Shiing-Shen Chern (; , ; October 26, 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 in Mathematics, 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 cofounded the Mathematical Sciences Research Institute in 1982 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chen (surname)
Chen () is a common Chinese-language surname and one of the most common surnames in Asia. It is the most common surname in Taiwan (2010) and Singapore (2000). Chen is also the most common family name in Guangdong, Zhejiang, Fujian, Macau, and Hong Kong. It is the most common surname in Xiamen, the ancestral hometown of many overseas Hoklo. Chen was listed 10th in the '' Hundred Family Surnames'' poem, in the verse 馮陳褚衛 ''(Féng Chén Chǔ Wèi)''. In Cantonese, it is usually romanized as Chan (e.g., Jackie Chan), most widely used by those from Hong Kong, and also found in Macau and Singapore. It is also sometimes spelled Chun. The spelling Tan usually comes from Southern Min dialects (e.g., Hokkien), while some Teochew dialect speakers use the spelling Tang. In Hakka and Taishanese, the name is spelled Chin. Spellings based on Wu include Zen and Tchen. There are many spellings based on its Hainanese pronunciations, including Dan, Seng, and Sin. In Viet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sidney Martin Webster
Sidney Martin Webster (born 12 November 1945 in Danville, Illinois) is an American mathematician, specializing in multidimensional complex analysis. After military service, Webster attended the University of California, Berkeley as an undergraduate and then as a graduate student, receiving a PhD in 1975 under the supervision of Shiing-Shen Chern with thesis ''Real hypersurfaces in complex space''. Webster was a faculty member at Princeton University from 1975 to 1980 and at the University of Minnesota from 1980 to 1989. In 1989 he became a full professor at the University of Chicago. He has held visiting positions at the University of Wuppertal, Rice University, and ETH Zurich. Webster was a Sloan Fellow for the academic year 1979–1980. In 1994 in Zurich he was an invited speaker of the International Congress of Mathematicians. In 2001 he received, jointly with László Lempert, the Stefan Bergman Prize from the American Mathematical Society The American Mathematical Socie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern's Conjecture For Hypersurfaces In Spheres
Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consider closed minimal submanifolds M^n immersed in the unit sphere S^ with second fundamental form of constant length whose square is denoted by \sigma. Is the set of values for \sigma discrete? What is the infimum of these values of \sigma > \frac? The first question, i.e., whether the set of values for ''σ'' is discrete, can be reformulated as follows: Let M^n be a closed minimal submanifold in \mathbb^ with the second fundamental form of constant length, denote by \mathcal_n the set of all the possible values for the squared length of the second fundamental form of M^n, is \mathcal_n a discrete? Its affirmative hand, more general than the Chern's conjecture for hypersurfaces, sometimes also referred to as the Chern's conjecture and is still, as of 2018, unanswered even with ''M' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 quoti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Weil Homomorphism
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing classes in the de Rham cohomology rings of ''M''. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let ''G'' be a real or complex Lie group with Lie algebra and let \Complex mathfrak g/math> denote the algebra of \Complex-valued polynomials on \mathfrak g (exactly the same argument works if we used \R instead of Let \Complex mathfrak gG be the subalgebra of fixed points in \Complex mathfrak g/math> under the adjoint action of ''G''; that is, the subalgebra consisting of al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Simons Form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose. Definition Given a manifold and a Lie algebra valued Multilinear form, 1-form \mathbf over it, we can define a family of Multilinear form, ''p''-forms: In one dimension, the Chern–Simons Multilinear form, 1-form is given by :\operatorname [ \mathbf ]. In three dimensions, the Chern–Simons 3-form is given by :\operatorname \left[ \mathbf \wedge \mathbf-\frac \mathbf \wedge \mathbf \wedge \mathbf \right] = \operatorname \left[ d\mathbf \wedge \mathbf + \frac \mathbf \wedge \mathbf \wedge \mathbf\right]. In five dimensions, the Chern–Simons 5-form is given by : \begin & \operatorname \left[ \mathbf\wedge\mathbf \wedge \mathbf-\frac \mathbf \wedge\mathbf\wedge\mathbf\wedge\mathbf +\frac \mathbf \wedge \mathbf \w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Simons Theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form. In condensed-matter physics, Chern–Simons theory describes composite fermions and the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial. Particularly, Chern–Simons theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as the ''level'' of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined whe ... [...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 manifold, closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its Curvature of Riemannian manifolds, curvature form (an analytical invariant). It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / Surface (mathematics), 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. The Riemann–Roch theorem and the Atiyah–Singer inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Wai-Kwong Li
Peter Wai-Kwong Li (born 18 April 1952) is an American mathematician whose research interests include differential geometry and partial differential equations, in particular geometric analysis. Mathematical work His most notable work includes the discovery of the Li–Yau differential Harnack inequalities, and the proof of the Willmore conjecture in the case of non-embedded surfaces, both done in collaboration with Shing-Tung Yau. He is an expert on the subject of function theory on complete Riemannian manifolds. Education and career After undergraduate work at California State University, Fresno, he received his Ph.D. at University of California, Berkeley under Shiing-Shen Chern in 1979. Presently he is Professor Emeritus at University of California, Irvine, where he has been located since 1991. Honors He has been the recipient of a Guggenheim Fellowship in 1989 and a Sloan Research Fellowship. In 2002, he was an invited speaker in the Differential Geometry section of the Int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from University of California at Berkeley. Cheng then spent some years as a post-doctoral fellow and assistant professor at Princeton University and the State University of New York at Stony Brook. Then he became a full professor at University of California at Los Angeles. Cheng chaired the Mathematics departments of both the Chinese University of Hong Kong and the Hong Kong University of Science and Technology in the 1990s. In 2004, he became the Dean of Science at HKUST. In 2012, he became a fellow of the American Mathematical Society. He is well known for contributions to differential geometry and partial differential equations, including Cheng's eigenvalue comparison theorem, Cheng's maximal diameter theorem, and a number of works with Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
