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K-Stability
In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics (cscK metrics). History In 1954, Eugenio Calabi formulated a conjecture about the existence of Kähler metrics on compact Kähler manifolds, now known as the Calabi conjecture. One formulation of the conjecture is that a compact Kähler manifold X admits a unique Kähler–Einstein metric in the class c_1(X). In the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-stability Of Fano Varieties
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics. K-stability was first defined for Fano manifolds by Gang Tian in 1997 in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition which characterises the existence of a Kähler–Einstein metric on a Fano manifold. It was defined in reference to the ''K-energy functional'' previously introduced by Toshiki Mabuchi. Tian's definition of K-stability was reformulated by Simon Donaldson in 2001 in a purely algebro-geometric way. K-stability has become an important notion in the study and classification of Fano varieties. In 2012 Xiuxiong Che ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler–Einstein Metric
In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat. The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds. This problem can be split up into three cases dependent on the sign of the first Chern class of the Kähler manifold: * When the first Chern class is negative, there is always a Kähler–Einstein metric, as Thierry Aubin and Shing-Tung Yau proved independently. * When the first Chern class is zero, there is always a Kähler–Einstein metric, as Yau proved in the Calabi conjecture. That leads to the name Calabi–Yau manifolds. He was awarded with the Fields Medal partly because of this work. * The third case, the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tian Gang
Tian Gang (; born November 24, 1958) is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis. As of 2020, he is the Vice Chairman of the China Democratic League and the President of the Chinese Mathematical Society. From 2017 to 2019 he served as the Vice President of Peking University. Biography Tian was born in Nanjing, Jiangsu, China. He qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, under the supervision of Shing-Tung Yau. In 1998, he was appointed as a Cheung Kong Scholar professor at Peking University. Later his appointment was changed to Cheung K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York, and a Professor in Pure Mathematics at Imperial College London. Biography Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge, and his mother earned a science degree there. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge, in 1979, and in 1980 began postgraduate work at Worcester College, Oxford, at first under Nigel Hitchin and later under Michael Atiyah's supervision. Still a postgraduate student, Donaldson proved in 1982 a result that would establish his fame. He published the result in a paper "Self-dual connections and the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Invariant Theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group on an algebraic variety (or scheme) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles. Background Invariant theory is concerned with a group action of a group on an algebraic variety (or a scheme) . Classical invariant theory addresses the situation when is a vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Scalar Curvature Kähler Metric
In differential geometry, a constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is Kähler–Einstein metric, and a more general case is extremal Kähler metric. , Tian and Yau conjectured that the existence of a cscK metric on a polarised projective manifold is equivalent to the polarised manifold being K-polystable. Recent developments in the field suggest that the correct equivalence may be to the polarised manifold being ''uniformly'' K-polystable . When the polarisation is given by the (anti)-canonical line bundle (i.e. in the case of Fano or Calabi–Yau manifolds) the notions of K-stability and K-polystability coincide, cscK metrics are precisely Kähler-Einstein metrics and the Yau-Tian-Donaldson conjecture is known to hold . Extremal Kähler metrics Constant scalar curvature Kähler metrics are specific examples of a more general notion of canonica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Variety
In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. Examples * The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of P''n'' over a field ''k'' is ''O''(''n''+1), which is very ample (over the complex numbers, its curvature is ''n+1'' times the Fubini–Study symplectic form). * Let ''D'' be a smooth codimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Lichnerowicz
André Lichnerowicz (January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a noted French differential geometer and mathematical physicist of Polish descent. He is considered the founder of modern Poisson geometry. Biography His grandfather Jan fought in the Polish resistance against the Prussians. Forced to flee Poland in 1860, he finally settled in France, where he married a woman from Auvergne, Justine Faure. Lichnerowicz's father, Jean, held agrégation in classics and was secretary of the Alliance française, while his mother, a descendant of paper makers, was one of the first women to earn the agrégation in mathematics. Lichnerowicz's paternal aunt, Jeanne, was a novelist and translator known under the pseudonym . André attended the Lycée Louis-le-Grand and then the École Normale Supérieure in Paris, gaining agrégation in 1936. After two years, he entered the Centre national de la recherche scientifique (CNRS) as one of the first researchers recr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yozo Matsushima
was a Japanese mathematician. Early life Matsushima was born on February 11, 1921, in Sakai City, Osaka Prefecture, Japan. He studied at Osaka Imperial University (later named Osaka University) and graduated with a Bachelor of Science degree in mathematics in September 1942. At Osaka, he was taught by mathematicians Kenjiro Shoda. After completing his degree, he was appointed as an assistant in the Mathematical Institute of Nagoya Imperial University (later named Nagoya University). These were difficult years for Japanese students and researchers because of World War II. The first paper published by Matsushima contained a proof that a conjecture of Hans Zassenhaus was false. Zassenhaus had conjectured that every semisimple Lie algebra ''L'' over a field of prime characteristic, with 'L'', ''L''= ''L'', is the direct sum of simple ideals. Matsushima constructed a counterexample. He then developed a proof that Cartan subalgebras of a complex Lie algebra are conjugate. However, Jap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications On Pure And Applied Mathematics
''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or mathematical physics. The journal was established in 1948 as the ''Communications on Applied Mathematics'', obtaining its current title the next year. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 3.219. References External links * Mathematics journals Monthly journals Wiley (publisher) academic journals Publications established in 1948 English-lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
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 grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
