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I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of such geometries fits into contemporary tries to generalize geometry?

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2 Answers 2

At the conference on Non-commutative geometry and operator algebras, at Vanderbilt in 2009 (focused on $\mathbb{F}_1$-geometry), Connes gave a series of lectures. In the 4th or 5th lecture he sketched the idea of von Neumann's "continuous geometry" and said how it could fit into the picture of $\mathbb{F}_1$-geometry.

This was long ago and I can't recall what was the exact connection. Possibly the point was that one can see $\mathbb{F}_1$-geometry as limit case of the geometry of Tits buildings at the different primes and von Neumann's geometry was meant to provide a "continuum of geometries" to fill in the space between the primes...

Connes' lectures were filmed by Norihiko Minami, but I know of no place where they appear online. You may ask Minami or the organizers of the conference, who got a copy, I think.

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Skornyakov L.A., Complemented modular lattices and regular rings, 1964,

Ulrich BREHM, Marcus GREFERATH, Stefan E. SCHMIDT, Projective Geometry on Modular Lattices (In: F. Buekenhout, Handbook of incidence geometry. Buildings and foundations, 1995)

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