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Calabi Conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswald Veblen Prize in part for his proof. His work, principally an analysis of an elliptic partial differential equation known as the complex Monge–Ampère equation, was an influential early result in the field of geometric analysis. More precisely, Calabi's conjecture asserts the resolution of the prescribed Ricci curvature problem within the setting of Kähler metrics on closed complex manifolds. According to Chern–Weil theory, the Ricci form of any such metric is a closed differential 2-form which represents the first Chern class. Calabi conjectured that for any such differential form , there is exactly one Kähler metric in each Kähler class whose Ricci form is . (Some compact complex manifolds admit no Kähler classes, in which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi–Yau Manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by , after who first conjectured that such surfaces might exist, and who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used. Definitions The motivational definition given by Shing-Tung Yau is of a compact Kähle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences Of The United States Of America
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2021 impact factor of 12.779. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the mass media, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are ava ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
North-Holland Publishing Company
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit margin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thierry Aubin
Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry and non-linear partial differential equations. His fundamental contributions to the theory of the Yamabe equation led, in conjunction with results of Trudinger and Schoen, to a proof of the Yamabe Conjecture: every compact Riemannian manifold can be conformally rescaled to produce a manifold of constant scalar curvature. Along with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class of known examples of compact Einstein manifolds. Aubin was the first mathematician to propose the Cartan–Hadamard conjecture. Aubin was a visiting scholar at the Institute for Advanced Study in 1979. He was elected to the Académie des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Priori Bound
In the theory of partial differential equations, an ''a priori'' estimate (also called an apriori estimate or ''a priori'' bound) is an estimate for the size of a solution or its derivatives of a partial differential equation. ''A priori'' is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an ''a priori'' estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem. ''A priori'' estimates were introduced and named by , who used them to prove existence of solutions to second order nonlinear elliptic equations in the plane. Some other early influential examples of ''a priori'' estimates include the Schauder estimates given by , and the estimates given by De Giorgi and Nash for second order elliptic or parabolic equations in many variables, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ddbar Lemma
In complex geometry, the \partial \bar \partial lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The \partial \bar \partial-lemma is a result of Hodge theory and the Kähler identities on a Compact space, compact Kähler manifold. Sometimes it is also known as the dd^c-lemma, due to the use of a related operator d^c = -\frac(\partial - \bar \partial), with the relation between the two operators being i\partial \bar \partial = dd^c and so \alpha = dd^c \beta. Statement The \partial \bar \partial lemma asserts that if (X,\omega) is a compact Kähler manifold and \alpha \in \Omega^(X) is a complex differential form of bidegree (p,q) (with p,q\ge 1) whose class [\alpha] \in H_^(X,\mathbb) is zero in de Rham cohomology, then there exists a form \beta\in \Omega^(X) of bidegree (p-1,q-1) such that \alpha = i\partial \bar \partial \beta, where \partial and \bar \partial are the Dolbeault operators of the complex mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Priori Estimate
In the theory of partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...s, an ''a priori'' estimate (also called an apriori estimate or ''a priori'' bound) is an estimate for the size of a solution or its derivatives of a partial differential equation. ''A priori'' is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an ''a priori'' estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem. ''A priori'' estimates were introduced and named by , who used them to prove existence of solutions to second order nonlinear elli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
