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L² Cohomology
In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds ''M'' with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on ''M'' gives rise to a norm on differential forms and a volume form. L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker (1978) and Jeff Cheeger (1979). It is closely related to intersection cohomology; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology. Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Eduard Looijenga (1988 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Middle Perversity
In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years. Intersection cohomology was used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to ''L''2 cohomology. Goresky–MacPherson approach The homology groups of a compact, oriented, connected, ''n''-dimensional manifold ''X'' have a fundamental property called Poincaré duality: there is a perfect pairing : H_i(X,\Q) \times H_(X,\Q) \to H_0(X,\Q) \cong \Q. Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of :H_j(X) is represented by a ''j''-dimensional cycle. If an ''i''-dimensional and an (n-i)-dimensional cycle are in general position, then their intersection is a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...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] |
Compositio Mathematica
''Compositio Mathematica'' is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London Mathematical Society in partnership with Cambridge University Press. According to the ''Journal Citation Reports'', the journal has a 2020 2-year impact factor of 1.456 and a 2020 5-year impact factor of 1.696. The editors-in-chief are Jochen Heinloth, Bruno Klingler, Lenny Taelman, and Éric Vasserot. Early history The journal was established by L. E. J. Brouwer in response to his dismissal from ''Mathematische Annalen'' in 1928. An announcement of the new journal was made in a 1934 issue of the ''American Mathematical Monthly''. In 1940 the publication of the journal was suspended due to the German occupation of the Netherlands Despite Dutch neutrality, Nazi Germany invaded the Netherlands on 10 May 1940 as part of Fall Gelb (Case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frances Kirwan
Dame Frances Clare Kirwan, (born 21 August 1959) is a British mathematician, currently Savilian Professor of Geometry at the University of Oxford. Her fields of specialisation are algebraic and symplectic geometry. Education Kirwan was educated at Oxford High School, and studied maths as an undergraduate at Clare College in the University of Cambridge. She took a D.Phil at Oxford in 1984, with the dissertation title ''The Cohomology of Quotients in Symplectic and Algebraic Geometry'', which was supervised by Michael Atiyah. Research Kirwan's research interests include moduli spaces in algebraic geometry, geometric invariant theory (GIT), and in the link between GIT and moment maps in symplectic geometry. Her work endeavours to understand the structure of geometric objects by investigation of their algebraic and topological properties. She introduced the Kirwan map. From 1983 to 1985 she held a junior fellowship at Harvard. From 1983 to 1986 she held a Fellowship at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Goresky
Robert Mark Goresky is a Canadian mathematician who invented intersection homology with his advisor and life partner Robert MacPherson. Career Goresky received his Ph.D. from Brown University in 1976. His thesis, titled ''Geometric Cohomology and Homology of Stratified Objects'', was written under the direction of MacPherson. Many of the results in his thesis were published in 1981 by the American Mathematical Society. He has taught at the University of British Columbia in Vancouver, and Northeastern University. Awards Goresky received a Sloan Research Fellowship in 1981. He received the Coxeter–James Prize in 1984. In 2002, Goresky and MacPherson were jointly awarded the Leroy P. Steele Prize for Seminal Contribution to Research by the American Mathematical Society. In 2012 Goresky became a fellow of the American Mathematical Society. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer , physicist and mathematician , and the son-in-law of physicist Pierre Weiss. Life According to his own words, Henri Cartan was interested in mathematics at a very young age, without being influenced by his family. He moved to Paris with his family after his father's appointment at Sorbonne in 1909 and he attended secondary school at Lycée Hoche in Versailles. available also at In 1923 he started studying mathematics at École Normale Supérieure, receiving an agrégation in 1926 and a doctorate in 1928. His PhD thesis, entitled ''Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires et leurs applications'', was supervised by Paul Montel. Cartan taught at Lycée Malherbe in Caen from 1928 to 1929, at Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. Formal statement Dirichlet's principle states that, if the function u ( x ) is the solution to Poisson's equation :\Delta u + f = 0 on a domain \Omega of \mathbb^n with boundary condition :u=g on the boundary \partial\Omega, then ''u'' can be obtained as the minimizer of the Dirichlet energy :E(x)= \int_\Omega \left(\frac, \nabla v, ^2 - vf\right)\,\mathrmx amongst all twice differentiable functions v such that v=g on \partial\Omega (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet. History The name "Dirichlet's principle" is due to Riemann, who applied it in the study of complex analytic functions. Riemann (and others such as Gauss and Dirichlet) knew that Dirichle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Form
In potential theory (the study of harmonic function) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can be defined on any measure space, without the need for mentioning partial derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds, for example, fractals. The benefit on these spaces is that one can do this without needing a gradient operator, and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form. Definition When working on \mathbb^n , the "classical" Dirichlet form is given by: \mathcal(u, v ) = \int_ \nabla u(x) \cdot \nabla v(x) \; dx where one often discusses \mathcal(u) := \mathcal(u, u) = \, \nabla u\, _2^2 which is often referred to as the "energy" of the function u(x). More generally, a Dirichlet form is a Markovian closed symmetric form on an ''L''2-space.Fukushima, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
