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Arxiv (Free subscription) | 10/28/2008
Noncommutative differential geometry over the Moyal algebra is developed following an algebraic approach. It is then applied to investigate embedded noncommutative spaces. We explicitly construct the projective modules corresponding to the tangent bundles of the noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvature introduced in...
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Arxiv (Free subscription) | 10/15/2008
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic geometry, etc.). These algebras have finite presentations and are completely characterized and classified by their (noncommutative) volume forms.
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The Geomblog (Free subscription) | 06/13/2008
A Discrete Laplace-Beltrami Operator Alexander Bobenko gave the first invited talk at the conference. As with Mathieu Desbrun's talk from a few years ago, this talk was on the area of 'discrete differential geometry', particularly focusing on a discrete Laplace-Beltrami operator, and how to compute Delaunay triangulations on polyhedral surfaces. The general principle behind efforts in discrete differential...
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Arxiv (Free subscription) | 05/15/2008
There ought to exist a reformulation of quantum mechanics which does not refer to an external classical spacetime manifold. Such a reformulation can be achieved using the language of noncommutative differential geometry. A consequence which follows is that the `weakly quantum, strongly gravitational' dynamics of a relativistic particle whose mass is much greater than Planck mass is dual to the `strongly...
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Dartmouth News Headlines (Free subscription) | 03/18/2008
Craig Sutton, a Dartmouth College assistant professor of mathematics who specializes in differential geometry, has been awarded a prestigious research fellowship from the Woodrow Wilson Foundation for the 2008-2009 academic year.
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Arxiv (Free subscription) | 03/14/2008
Space-Time in general relativity is a dynamical entity because it is subject to the Einstein field equations. The space-time metric provides different geometrical structures: conformal, volume, projective and linear connection. A deep understanding of them has consequences on the dynamical role played by geometry. We present a unified description of those geometrical structures, with a standard criterion...
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What's new (Free subscription) | 12/26/2007
I’m continuing my series of articles for the Princeton Companion to Mathematics through the holiday season with my article on “Differential forms and integration“. This is my attempt to explain the concept of a differential form in differential geometry and several variable calculus; which I view as an extension of the concept of the [...]
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core77.com (Free subscription) | 11/19/2007
Polygon is a group show highlighting work by recent graduates and final year students from Central Saint Martins, London College of Communication, University of Brighton, Denmark's Design School and The Royal College of Art. "Each piece of work takes its inspiration from a different geometric shape and will be interpreted through a range of different mediums." Polygon Show Private View: Friday, November...
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Arxiv (Free subscription) | 11/12/2007
We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan's investigations started by analyzing Einsteins general relativity theory and by taking recourse to the theory of Cosserat continua. In these continua, the points...
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Arxiv (Free subscription) | 10/19/2007
The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. This year, GRworkbench has been significantly extended to facilitate numerical experimentation. The numerical differential geometric engine has been rewritten using functional programming techniques, enabling fundamental concepts to be directly represented as variables...
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Arxiv (Free subscription) | 10/09/2007
We show that the differential-geometric description of matter by differential structures of spacetime leads to a unifying model of the three types of energy in the cosmos: matter, dark matter and dark energy. Using this model we are able to calculate the ratio of dark energy to the total energy of the cosmos.
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Shifting Pixel (Free subscription) | 09/20/2007
ƒ/1.4, 1/350 s, 400 ISO I have been liking a bit of a greener aesthetic lately for some reason. I can't quite put my finger on it. Regardless, I really like all of the different geometry going on in this photo. It's always a joy when simple angles and ellipses form an interesting image.
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Arxiv (Free subscription) | 07/26/2007
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces...
