How was this infinite sequence of numbers found? (non-commutative geometry )

In summary, the discovery of the infinite sequence of numbers in the context of non-commutative geometry involves exploring mathematical structures that extend traditional geometric concepts. This field examines how these numbers relate to quantum physics and spacetime, revealing deep connections between algebra, topology, and physics. The sequence emerges from the study of operator algebras and their representations, highlighting the interplay between geometry and number theory in understanding complex systems.
  • #1
Heidi
418
40
Hi Pfs,
I read these slides:
https://indico.math.cnrs.fr/event/782/attachments/1851/1997/Connes.pdf
It is about non commutative geometry (Alain Connes)
After Shapes II, you see a the plots of the square roots of a sequence of numbers given below:
5/4, 2, 5/2, 13/4 ....
I think that they are the solutions of laplacian equation related to the shape of the "drums" above. that is to say the frequencies one can get when hitting this drums.
How to retrieve this infinite sequence?
It is at page 50
thanks
 
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  • #2
In this article Connes gives an example of two isospectral drums (which are non connex).
One drum with a triangle and a quare and the other with a rectangle and a different triangle.
I found this
https://math.libretexts.org/Bookshe...ons/6.01:_Vibrations_of_Rectangular_Membranes
explaining how to get the possible frequencies emitted by a square or rectangular drum.
Do you know what are the frequencies ommitted by a trangular drum (half a square)?
 
  • #4
Thread closed for Moderation...
 
  • #5
Since the OP has found their answer, this thread can be closed now.
 

FAQ: How was this infinite sequence of numbers found? (non-commutative geometry )

What is non-commutative geometry?

Non-commutative geometry is a branch of mathematics that generalizes geometric concepts to spaces where the coordinates do not commute, meaning the order in which you multiply them matters. This framework is particularly useful in quantum physics and string theory, where traditional geometric intuition breaks down.

How does non-commutative geometry relate to infinite sequences of numbers?

In non-commutative geometry, infinite sequences of numbers can arise naturally when studying operators on Hilbert spaces or other algebraic structures. These sequences often encode geometric and topological information about non-commutative spaces, similar to how coordinates describe points in classical geometry.

What methods are used to find infinite sequences in non-commutative geometry?

Various mathematical tools are employed to find infinite sequences in non-commutative geometry. These include spectral analysis of operators, K-theory, cyclic cohomology, and non-commutative versions of differential geometry and topology. Techniques from functional analysis and operator algebras are also crucial.

Can you give an example of an infinite sequence found in non-commutative geometry?

An example is the sequence of eigenvalues of a Dirac operator on a non-commutative space. These eigenvalues form an infinite sequence that encodes important geometric information about the space, such as its curvature and topological invariants.

What are the applications of these infinite sequences in physics or other fields?

Infinite sequences found in non-commutative geometry have applications in various fields, particularly in theoretical physics. They are used in the study of quantum field theory, string theory, and the standard model of particle physics. These sequences help in understanding the geometry of space-time at the quantum level and in formulating physical theories in non-commutative spaces.

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