Simple Applications of Noncommutative Geometry?

[LukeD]

Noncommutative Geometry has many simple applications. For instance, it can be used to do differential geometry on a lattice, on a finite set, over Brownian motions, or in quantum phase space. I'm much more interested in learning the applications of noncommutative geometry than its particulars in the most abstract settings.

Unfortunately, most of the information I've found on NCG is very high level. Except for a single paper I found that treated Brownian motions, I've yet to find any information on simple applications that is written for people who don't understand much of algebra or geometry.

Does anyone know of any papers on applications of noncommutative geometry written for, say... a crowd of engineers or computer scientists?

 

[LukeD]

No one? Does anyone know of another active math forum where I might get a helpful response?

 

[LukeD]

Lots of views, but no responses. I guess that means that people are interested, but no one has an answer.
Maybe someone knows some answers to elementary questions that I have (though I don't expect that the answers will all be so elementary!)

1. How do I form a differentials & do integrals for functions on a lattice? on a graph?
2. How do the ideas used relate to the things I can already calculate with my unrefined ideas of differences and sums?
3. Other examples of of forming differentials & doing integrals in non-classical situations? stochastic calculus? quantum phase space? a geometric algebra?
4. Have you yet answered the question "Just what is \sqrt{dx^2 + dy^2}?" I won't consider NCG a success until I have the answer to that question! (I asked about the geometric algebra for this reason)

 

[Monocles]

I am confused by your question because you ask for something aimed at people that do not know much algebra or geometry, but then go on to mention geometric algebra, quantum phase space, etc.

This thread might be helpful: http://mathoverflow.net/questions/14518/applications-of-noncommutative-geometry

Though, the above thread is mostly about NC algebraic geometry, and it sounds like you're asking about NC differential geometry. Unfortunately, I do not know much about NCG other than you start with the Gelfand representation theorem and then remove the assumption of commutativity on the algebra to obtain the notion of a noncommutative space.