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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 14th 2018

added this to the list of references:

Speculative remarks on the possible role of maps from spacetime to the 4-sphere in some kind of quantum gravity via spectral geometry are in

reviewed in

• Alain Connes, section 4 of Geometry and the Quantum, Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham, 2018. 159-196 (arXiv:1703.02470)

• Alain Connes, from 58:00 to 1:25:00 in Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG, talk at IHES, 2017 (video recording)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 27th 2019

added pointer to this preprint from Friday

• Selman Akbulut, Homotopy 4-spheres associated to an infinite order loose cork (arXiv:1901.08299)
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 31st 2019

added a note on coset space structures (here)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 15th 2019

The other day I was re-reading Chamseddine-Connes-Mukhanov 14. While it’s fun, I don’t get how the core of the statement is not put in by hand:

Under the standard translation to the language of spectral geometry, they literally say: Let there be a smooth function from 4d spacetime to the 4-sphere, or a pair of such, and impose a kind of integrality condition on the pullback of the noncommutative volume form. It seems not so surprising to me that with this assumption, the main result follows.

I gather they mean to suggest that what on the face of it is an kind of integrality condition on an NCG volume form is “really” a quantum gravity variant of the Heisenberg uncertainty relation. While I can see how that is a fun perspective on the situation, it doesn’t serm to follow from anything, or to be more than vaguely suggestive at this point. Also, it’s unclear to me how that suggestion for quantized geometry is supposed to play nicely with the Connes-Lott-Barrett model.

From the public lectures and announcements I gather the group is very excited about their QG insight here, regarding maps to the 4-sphere. I am trying to follow what’s really going on. I do follow the maths, but am missing something else.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 5th 2019
• (edited May 5th 2019)

added pointer now also to the original announcement article

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 30th 2019

Edit pointer to

from a few days back.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeDec 1st 2021

Added short section giving volume form and the resulting volume on a 4-sphere of radius $R$.