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Geometry and physics today. (English) Zbl 1111.83019
This essay represents a kind of meta-theoretical comparison study, between different geometrization methods in theoretical physics: the old ones, belonging to classical differential geometry must (in the author’s opinion) be replaced by some new methods, in the framework of “abstract” differential geometry (described as a general gauge theory on topological manifolds and fiber bundles).

MSC:
83C45Quantization of the gravitational field
81P05General and philosophical topics in quantum theory
83C05Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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References:
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[12] Mallios, A. (1998a). On an axiomatic treatment of differential geometry via vector sheaves. Applications. Math. Japonica (Int. Plaza) 48, 93--180. · Zbl 0910.53013
[13] Mallios, A. (1998b). Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry, Vols. I (Chapts I--V), II (Chapts VI--XI). Kluwer, Dordrecht. [This is still quoted in the text, as VS].
[14] Mallios, A. (2002). Remarks on ”singularities.” gr-qc/0202028. · Zbl 02201337
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[16] Mallios, A. (2006) Quantum gravity and ”singularities.” Note Mat. 25, 57--76. · Zbl 1115.83016
[17] Mallios, A. (2005). Modern Differential Geometry in Gauge Theories. Vol.I: Maxwell Fields, Vol.II: Yang-Mills Fields. Birkhäuser, Boston, (2005/2006). · Zbl 1116.18006
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[20] Mallios, A. and Raptis, I. (2003). Finitary, causal, and quantal vacuum Einstein gravity. Ibid. 42, 1479--1620. · Zbl 1033.83018
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