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Maximal Surface
In the mathematical field of differential geometry, a maximal surface is a certain kind of submanifold of a Lorentzian manifold. Precisely, given a Lorentzian manifold , a maximal surface is a spacelike submanifold of whose mean curvature is zero. As such, maximal surfaces in Lorentzian geometry are directly analogous to minimal surfaces in Riemannian geometry. The difference in terminology between the two settings has to do with the fact that small regions in maximal surfaces are local maximizers of the area functional, while small regions in minimal surfaces are local minimizers of the area functional. In 1976, Shiu-Yuen Cheng and Shing-Tung Yau resolved the "Bernstein problem" for maximal hypersurfaces of Minkowski space which are properly embedded, showing that any such hypersurface is a plane. This was part of the body of work for which Yau was awarded the Fields medal in 1982. The Bernstein problem was originally posed by Eugenio Calabi in 1970, who proved some special cases o ... [...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] |
De Sitter Space
In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe. More specifically, de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant \Lambda (corresponding to a positive vacuum energy density and negative pressure). There is cosmological evidence that the universe itself is asymptotically de Sitter, i.e. it will evolve like the de Sitter universe in the far future when dark energy dominates. de Sitter space and anti-de Sitter space are named after Willem de&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Field Equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergiu Klainerman
Sergiu Klainerman (born May 13, 1950) is a mathematician known for his contributions to the study of hyperbolic differential equations and general relativity. He is currently the Eugene Higgins Professor of Mathematics at Princeton University, where he has been teaching since 1987. Biography He was born in 1950 in Bucharest, Romania, into a Jewish family. After attending the Petru Groza High School, he studied mathematics at the University of Bucharest from 1969 to 1974. For graduate studies he went to New York University, obtaining his Ph.D. in 1978. His thesis, written under the direction of Fritz John and Louis Nirenberg, was titled ''Global Existence for Nonlinear Wave Equations''. From 1978 to 1980 Klainerman was a Miller Research Fellow at the University of California, Berkeley, while from 1980 to 1987 he was a faculty member at New York University's Courant Institute of Mathematical Sciences, rising in rank to Professor in 1986. Klainerman is a member of the U.S. Natio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demetrios Christodoulou
Demetrios Christodoulou ( el, Δημήτριος Χριστοδούλου; born 19 October 1951) is a Greek mathematician and physicist, who first became well known for his proof, together with Sergiu Klainerman, of the nonlinear stability of the Minkowski spacetime of special relativity in the framework of general relativity. Christodoulou is a 1993 MacArthur Fellow. Early life and education Christodoulou was born in Athens and received his doctorate in physics from Princeton University in 1971 under the direction of John Archibald Wheeler. After temporary positions at Caltech, CERN, and the Max Planck Institute for Physics, he became Professor of Mathematics, first at Syracuse University, then at the Courant Institute, and at Princeton University, before taking up his last position as Professor of Mathematics and Physics at the ETH Zurich in Switzerland. He is Emeritus Professor since January 2017. He holds dual Greek and U.S. citizenship. Achievements In 1993, he publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Simon
Leon Melvyn Simon , born in 1945, is a Leroy P. Steele PrizeSee announcemen retrieved 15 September 2017. and Bôcher Memorial Prize, Bôcher Prize-winningSee . mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently Professor Emeritus in the Mathematics Department at Stanford University. Biography Academic career Leon Simon, born 6 July 1945, received his BSc from the University of Adelaide in 1967, and his PhD in 1971 from the same institution, under the direction of James H. Michael. His doctoral thesis was titled ''Interior Gradient Bounds for Non-Uniformly Elliptic Equations''. He was employed from 1968 to 1971 as a Tutor in Mathematics by the university. Simon has since held a variety of academic positions. He worked first at Flinders University as a lecturer, then at Australian National University as a professor, at the University of Melbourne, the University of Mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Bartnik
Robert Bartnik (1956-2022) was an Australian mathematician based at Monash University. He is known for his contributions to the rigorous mathematical study of general relativity. He received his bachelor's and master's degrees from Melbourne University and a PhD in mathematics from Princeton University in 1983, where his advisor was Shing-Tung Yau. In 2004 he was elected to the Australian Academy of Science, with citation: His work with John McKinnon has been widely studied in the physics literature. They show that there is a discrete set of static solutions to the coupled Einstein/Yang-Mills equations which are geodesically complete and asymptotically flat. This is interesting since such solutions are known not to exist in the cases of the Einstein vacuum equations, the coupled Einstein/Maxwell equations, and the Yang-Mills equations. Although Bartnik and McKinnon's work was numerical, their observed phenomena has been mathematically justified by Joel Smoller, Arthur Wasserman, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications. Academic career Calabi was a Putnam Fellow as an undergraduate at the Massachusetts Institute of Technology in 1946. He received his PhD in mathematics from Princeton University in 1950 after completing a doctoral dissertation, titled "Isometric complex analytic imbedding of Kahler manifolds", under the supervision of Salomon Bochner. He later obtained a professorship at the University of Minnesota. In 1964, Calabi joined the mathematics faculty at the University of Pennsylvania. Following the retirement of the German-born American mathematician Hans Rademacher, he was appointed to the Thomas A. Scott Professorship of Mathematics at the University of Pennsylvania in 1967. He won the Steele Prize from the America ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class ''C''''r'' for a fixed , and all morphisms are differentiable of class ''C''''r''. Immersed submanifolds An immersed submanifold of a manifold ''M'' is the image ''S'' of an immersion map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset ''S'' together with a topology and differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields. The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics. The prize includes a monetary award which, since 2006, has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
