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The adjective abstract has often been applied to differential geometry before, but the abstract differential geometry (ADG) of this article is a form of differential geometry without the calculus notion of smoothness, developed by Anastasios Mallios and Ioannis Raptis from 1998 onwards.[1]

Instead of calculus, an axiomatic treatment of differential geometry is built via sheaf theory and sheaf cohomology using vector sheaves in place of bundles based on arbitrary topological spaces.[2] Mallios says noncommutative geometry can be considered a special case of ADG, and that ADG is similar to synthetic differential geometry.

Applications
ADG Gravity

Mallios and Raptis use ADG to avoid the singularities in general relativity and propose this as a route to quantum gravity.[3]
See also

Discrete differential geometry
Analysis on fractals

References

"Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry", Anastasios Mallios, Springer, 1998, ISBN 978-0-7923-5005-7
"Modern Differential Geometry in Gauge Theories: Maxwell fields", Anastasios Mallios, Springer, 2005, ISBN 978-0-8176-4378-2

Mallios, Anastasios; Raptis, Ioannis (2004). "Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold". arXiv:gr-qc/0411121.

Further reading

Space-time foam dense singularities and de Rham cohomology, A Mallios, EE Rosinger, Acta Applicandae Mathematicae, 2001

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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Index

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