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Intrinsic Flat Distance
In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral currents lying in Euclidean space. Overview The Sormani–Wenger intrinsic flat (SWIF) distance is a distance between compact oriented Riemannian manifolds of the same dimension. More generally it defines the distance between two integral current spaces, (''X'',''d'',''T''), of the same dimension (see below). This class of spaces and this distance were first announced by mathematicians Sormani and Wenger at the Geometry Festival in 2009 and the detailed development of these notions appeared in the ''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 ... [...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] |
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] |
Herbert Federer
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert Federer (1920–2010)'' NAMS 59(5), 622-631. Career Federer was born July 23, 1920, in Vienna, Austria. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley, earning the Ph.D. as a student of Anthony Morse in 1944. He then spent virtually his entire career as a member of the Brown University Mathematics Department, where he eventually retired with the title of Professor Emeritus. Federer wrote more than thirty research papers in addition to his book ''Geometric measure theory''. The Mathematics Genealogy Project assigns him nine Ph.D. students and well over a hundred subsequent descendants. His most productive students include the late Frederick J. Almgren, Jr. (1933–1997), a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Convergence
In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Herbert Federer, Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to Plateau's problem. In 2001 the notion of an integral current was extended to arbitrary metric spaces by Luigi Ambrosio, Ambrosio and Kirchheim. Integral currents A ''k''-dimensional current ''T'' is a linear functional on the space \Omega^k_c(\mathbb^n) of smooth, compactly supported ''k''-forms. For example, given a Lipschitz continuity, Lipschitz map from a manifold into Euclidean space, F: N^k \to \mathbb^n, one has an integral current ''T''(''ω'') defined by integrating the pullback of the differential ''k''-form, ''ω'', over ''N''. Currents have a notion of boundary \partial (which is the usual boundary when ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Current
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a ''k''-current in the sense of Georges de Rham is a functional on the space of compactly supported differential ''k''-forms, on a smooth manifold ''M''. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of ''M''. Definition Let \Omega_c^m(M) denote the space of smooth ''m''-forms with compact support on a smooth manifold M. A current is a linear functional on \Omega_c^m(M) which is continuous in the sense of distributions. Thus a linear functional T : \Omega_c^m(M)\to \R is an ''m''-dimensional current if it is continuous in the following sense: If a sequence \omega_k of smooth forms, all supported in the same compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christina Sormani
Christina Sormani is a professor of mathematics at City University of New York affiliated with Lehman College and the CUNY Graduate Center. She is known for her research in Riemannian geometry, metric geometry, and Ricci curvature, as well as her work on the notion of intrinsic flat distance. Career Sormani received her Ph.D. from New York University in 1996 under Jeff Cheeger. She then took postdoctoral positions at Harvard University (under Shing-Tung Yau) and Johns Hopkins University (under William Minicozzi II). Sormani now works at Lehman College in the City University of New York and at the CUNY Graduate Center. Awards and honors In 2009, Sormani was an invited speaker at the Geometry Festival. in memory of |
Geometry Festival
The Geometry Festival is an annual mathematics conference held in the United States. The festival has been held since 1985 at the University of Pennsylvania, the University of Maryland, Baltimore, University of Maryland, the University of North Carolina, the State University of New York at Stony Brook, Duke University and New York University's Courant Institute of Mathematical Sciences. It is a three day conference that focuses on the major recent results in geometry and related fields. Previous Geometry Festival speakers 1985 at Penn * Marcel Berger * Pat Eberlein * Jost Eschenburg * Friedrich Hirzebruch * H. Blaine Lawson, Blaine Lawson * Leon Simon * Scott Wolpert * Deane Yang 1986 at Maryland * Uwe Abresch, ''Explicit constant mean curvature tori'' * Zhi-yong Gao, ''The existence of negatively Ricci curved metrics'' * David Hoffman (mathematician), David Hoffman, ''New results in the global theory of minimal surfaces'' * Jack Lee (mathematician), Jack Lee, ''Conformal geomet ... [...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] |
Gromov–Hausdorff Convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. Gromov–Hausdorff distance The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two compact metric spaces are from being isometric. If ''X'' and ''Y'' are two compact metric spaces, then ''dGH'' (''X'', ''Y'') is defined to be the infimum of all numbers ''d''''H''(''f''(''X''), ''g''(''Y'')) for all metric spaces ''M'' and all isometric embeddings ''f'' : ''X'' → ''M'' and ''g'' : ''Y'' → ''M''. Here ''d''''H'' denotes Hausdorff distance between subsets in ''M'' and the ''isometric embedding'' is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luigi Ambrosio
Luigi Ambrosio (born 27 January 1963) is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory. Biography Ambrosio entered the Scuola Normale Superiore di Pisa in 1981. He obtained his degree under the guidance of Ennio de Giorgi in 1985 at University of Pisa, and the Diploma at Scuola Normale. He obtained his PhD in 1988. He is currently professor at the Scuola Normale, having taught previously at the University of Rome "Tor Vergata", the University of Pisa, and the University of Pavia. Ambrosio also taught and conducted research at the Massachusetts Institute of Technology, the ETH in Zurich, and the Max Planck Institute for Mathematics in the Sciences in Leipzig. He is the Managing Editor of the scientific journal '' Calculus of Variations and Partial Differential Equations'', and member of the editorial boards of scientific journals. Since May 9, 2019 Ambrosio is the director of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Convergence
In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Herbert Federer, Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to Plateau's problem. In 2001 the notion of an integral current was extended to arbitrary metric spaces by Luigi Ambrosio, Ambrosio and Kirchheim. Integral currents A ''k''-dimensional current ''T'' is a linear functional on the space \Omega^k_c(\mathbb^n) of smooth, compactly supported ''k''-forms. For example, given a Lipschitz continuity, Lipschitz map from a manifold into Euclidean space, F: N^k \to \mathbb^n, one has an integral current ''T''(''ω'') defined by integrating the pullback of the differential ''k''-form, ''ω'', over ''N''. Currents have a notion of boundary \partial (which is the usual boundary when ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
