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.

 

[blue_raver22]

Thank you for your interest in noncommutative geometry and its applications. As a scientist in this field, I can assure you that there are many simple and practical applications of noncommutative geometry that are relevant to engineers and computer scientists.

One example is in signal processing, where noncommutative geometry has been used to develop efficient algorithms for image and signal compression. This has practical applications in areas such as video and audio compression, which are widely used in engineering and computer science.

Noncommutative geometry also has applications in quantum information theory, where it is used to study quantum entanglement and quantum error correction codes. These have important practical applications in areas such as quantum computing and cryptography.

In computer science, noncommutative geometry has been used to develop efficient algorithms for data mining and machine learning. This is because noncommutative geometry provides a natural framework for analyzing complex data sets and identifying underlying patterns and structures.

Additionally, noncommutative geometry has been applied to problems in robotics and control theory, where it has been used to model and analyze the behavior of complex systems with noncommutative group symmetries.

I would recommend looking into specific applications such as these for more information and accessible resources. Noncommutative geometry may seem abstract at first, but its practical applications are numerous and diverse. I hope this helps in your understanding and exploration of this fascinating field.