Ya’ gotta’ learn it all
Inverse spectral problem
M. Kac (1966) “Can you hear the shape of a drum?”
Wave equation: [pic] u = 0 on bd(M)
Solution by separation of variables: u(x,t) = f(t)g(x), g = 0 on bd(M)
This implies g must be an eigenfunction of ( - the eigenvalues determine the frequencies of vibration:
[pic]
So the eigenvalues determine the frequencies (overtones) the drum can produce.
PROBLEM: Does the sequence of (’s determine the drum? (In other words, are there non-congruent “isospectral” plane domains?)
Motivation/evidence: Rayleigh-Jeans/ Weyl-Courant [pic].
Later asymptotic improvements give L(bd M) etc..
Generalize to (M,g) – Riemannian manifolds (like the torus), and ask the same question:
Minakshisundaram/Pleijel ; McKean/Singer:
Heat equation asymptotics give dim(M), vol(M,g), vol(bd(M),g), total scalar curvature, etc..
Duistermaat/Guillemin etc:
Wave equation asymptotics give lengths of closed geodesics, billiard paths, etc.
Early examples of isospectral manifolds:
Flat 16-dimensional tori (Milnor 1964)
Riemann surfaces (genus=201,601 – Vigneras 1980)
Domains in three sphere and 4-space (Urakawa 1982)
Spherical space forms (Ikeda 1982)
Isospectral deformations of solv- and nil-manifolds (Gordon-Wilson 1984)
General technique for constructing isospectral manifolds with common finite covers (Sunada 1985)
[pic]
Sunada’s condition: If there is a bijection [pic] such that for each [pic], we have [pic], then M1 is isospectral to M2.
Now we’re in the realm of group theory!
For a finite group to satisfy Sunada’s condition, need [G; [pic]] > 7. And there is an example for this index:
[pic]
(which has order 168). The subgroups are order 24 and given by
[pic] and [pic]
Now G is generated by [pic] and [pic]
And we can get an idea of how to construct manifolds by looking at the Cayley graphs of the coset spaces with respect to these generators.
[pic]
How can you use this to construct isospectral manifolds?
The basic piece of the (thickened) Cayley graph of G looks like this:
[pic]
We can identify edges as shown to make a space whose fundamental group has two generators:
[pic]
We won’t draw the covering space whose group of deck transformations is G !
But we can draw the two intermediate ones corresponding to our subgroups:
[pic][pic]
By Sunada’s construction, these are two isospectral surfaces in three-space!
Gordon, Webb, Wolpert: Squish ‘em (i.e., take orbifold quotient) to get plane domains:
[pic] [pic]
We can give a simple proof of isospectrality of these domains without using Sunada’s theorem:
[pic] [pic]
Here is a pair of simpler domains (octagons!) based on the same construction:
[pic] [pic]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- why is it important to learn english
- all children can learn philosophy
- learn all recipes minecraft command
- all students can learn philosophy
- why is it important to learn history
- through it all gospel song
- i did it all song
- i did it all lyrics
- gotta have faith limp bizkit
- know it all person
- alabina ya mama ya mama
- through it all song hymnal