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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]

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