Here's a quick overview from John Baez [sci.physics.research, 13 Dec 1996]:

In Kac's classic "can you hear the shape of drum?" puzzle you are trying to reconstruct the geometry of a n-dimensional manifold (possibly with boundary) from the eigenvalues of the Laplacian on that manifold.

In lowbrow lingo, you're trying to figure out a shape knowing the resonant frequencies at which it vibrates.

It's been known ever since Weyl that you can figure out a bunch of interesting stuff about the shape, which is why the puzzle is interesting. But recently two 2-dimensional regions in the plane with different shapes were discovered that have the same eigenvalues. (See Gorden, Webb, and Wolpert, Bull. AMS 27 (1992), 134.) One says they are "isospectral".

So no, you can't hear the shape of a drum! What about higher dimensions? Well, actually some counterexamples in higher dimensions were known before the 2d counterexamples. But it seems clear that from the 2d counterexamples you can get counterexamples in all higher dimensions. Just take the 2d shapes and take their cross product with a fixed n-dimensional shape, like an n-dimensional torus. By separation of variables the eigenvalues of the Laplacian on a 2d shape determine the eigenvalues of the Laplacian on its cross product with an n-dimensional torus so we obtain two different (n+2)-dimensional shapes whose Laplacians have the same eigenvalues.

[Someone subsequently asked "what other interesting stuff can be deduced from the spectrum?"]

There will be lots of eigenvalues of the Laplacian, or resonant frequencies, so let N(x) be the number of eigenvalues less than x. N(x) grows asymptotically like x^n/2 where n is the dimension of the drumhead. So the easiest thing to spot is the dimension of the drumhead. Say that

N(x) = A x ^n/2 + smaller error terms.

N(x) = A x^n/2 + B x^(n-1)/2 + smaller error terms

N(x) = A x^n/2 + B x^D/2 + smaller error terms

My colleague here at UCR, Michel Lapidus, works on this sort of thing. One of the subtleties is that there are different concepts of "fractal dimension". The most well-known one, the Hausdorff dimension, does not make the Weyl-Berry conjecture come out to be universally valid: there are counterexamples. The Minkowski dimension seems to work better, but as far as I know, the whole subject of drums with fractal boundary becomes more and more complicated, the closer and closer you look at it...

What else can you figure out? Well, if you generalize the problem a bit and look at the eigenvalues of the Laplacian not on functions but on p-forms, you can figure out a lot about the topology of the drum, using de Rham theory. To be precise, you can figure out its Betti numbers.

There is probably a lot more you can figure out, but I don't know exactly what it is! For example, what do the still smaller correction terms in the asymptotic formula above mean?...And what about information obtained, not from the behavior N(x) as x tends to infinity, but the from the first few eigenvalues? I'm pretty sure they study this a bit in the subject of "spectral geometry"...

The "Weyl conjecture" first appeared in the following article:

H. Weyl, "Uber die Abhangigkeit der Eigenschwingungen einer Membran von deren Begrenzung", J. Angew. Math. 141 (1912) 1-11.

It concerned the asymptotic behaviour of the spectrum of eigenvalues of the Laplacian on an open bounded subset of Rⁿ with "sufficiently smooth" boundary.

Here is an excerpt from M. Lapidus's survey article "Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture":

Berry, as a physicist motivated in part by the study of porous media and the scattering of light from fractal surfaces, then extended this to what became known as the Weyl-Berry conjecture in the following papers:

M.V. Berry, "Distribution of modes in fractal resonators", from Structural Stability in Physics (ed. W. Guttinger and H. Eikemeier, Springer, Berlin, 1979) 51-53.

M.V. Berry, "Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals", Geometry of the Laplace Operator (eds. R. Osserman and A. Weinstein) Proceedings of Symposia in Pure Mathematics 36 (AMS, 1980) 13-38.

This deals with the analogous problem of the spectral asymptotics of the Laplacian, but on an open subset with fractal rather than smooth boundary. Here is a further excerpt from the Lapidus survey:

The Weyl-Berry conjecture, in its original form, was ultimately disproved (by counterexample) in:

J. Brossard and R. Carmona, "Can one hear the dimension of a fractal", Communications in Mathematical Physics 104 (1986) 103-122

The authors suggested that the Minkowski dimension was more appropriate than the Hausdorff dimension for the boundary.

Lapidus reformulated the Weyl-Berry conjecture in

M.L. Lapidus, "Fractal drum, inverse spectral problems of elliptic operators and a partial resolution of the Weyl-Berry conjecture", Transactions of the American Mathematical Society 325 (1991) 465-529.

In this paper, Lapidus obtains a partial resolution of the Weyl-Berry conjecture by establishing (in any dimension n >1) remainder estimates for the asymptotics of the eigenvalue counting function, expressed in terms of the Minkowski (or box) dimension of the boundary. These estimates are valid for Laplacians with Dirichlet or (under suitable hypotheses) Neumann boundary conditions, and are also extended to higher order elliptic operators (with possibly nonsmooth coefficients). Further, families of examples are provided to show that these estimates are sharp in any possible "fractal" (i.e., Minkowski) dimension.

In the following papers, the authors prove the case for n = 1 and establish unexpected and intruiging connections with the Riemann zeta function:

M.L. Lapidus and C. Pomerance, "Fonction zeta de Riemann et conjecture de Weyl-Berry pour les tambours fractals", C. R. Acad. Sci. Paris Ser. I Math. 310 (1990) 343-348.

M.L. Lapidus and C. Pomerance, "The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums", Proceedings of the London Mathematical Society (3) 66 (1993) 41-69.

"Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M.V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl's conjecture for the asymptotics of the eigenvalues of the Laplacian on a bounded open subset of Rⁿ. We solve here in the one-dimensional case (that is n = 1) this 'modified Weyl-Berry conjecture'. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. We therefore show that one can 'hear' (that is, recover from the spectrum) not only Minkowski fractal dimension of the boundary - as was established previously by the first author - but also, under the stronger assumptions of the conjecture, its Minkowski content (a 'fractal' analogue of its 'length')."

In the following paper, the authors disprove the modified Weyl-Berry conjecture for n > 1:

M.L. Lapidus and C. Pomerance, "Counterexamples to the modified Weyl-Berry conjecture on fractal drums", Mathematical Proceedings of the Cambridge Philosophical Society 119 (1996) 167-178.

The two families of counterexamples given suggest that to determine the spectrum, more geometry of the drum is needed than just its volume and the Minkowski dimension and content of its boundary.

M. Kac, "Can one hear the shape of a drum?", American Mathematical Monthly (Slaught Memorial Papers No. 11) 73 (1966) 1-23.

B.D. Sleeman and Chen Hua, "An analogue of Berry's conjecture for the phase in fractal obstacle scattering", IMA Journal of Applied Mathematics 49 (1992) 193-200

B.D. Sleeman and Chen Hua, "The modified Weyl-Berry conjecture" (preprint, 1991)

The following extensive survey contains probably the best introduction to the history of these conjectures:

M.L. Lapidus, "Vibrations of fractal drums, the Riemann Hypothesis, waves in fractal media, and the Weyl-Berry conjecture", in: Ordinary and Partial Differential Equations (B.D. Sleeman and R.J. Jarvis, eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series 289 (Longman Scientific and Technical, 1993) 126-209.

This also documents an intriguing analogy between dynamical zeta functions on hyperbolic manifolds and spectral zeta functions on regions with fractal boundary.

In Part II of the above survey paper, are also formulated various forms of the WBL conjectures in the important special cases of drums with (approximately or possibly random) self-similar boundaries. In particular, one has the following dichotomy: In the "nonlattice case" (the generic case), the asymptotic second term of the eigenvalue counting function N(x) is conjectured to be monotonic (and proportional to x^D/2), whereas in the "lattice case" (roughly speaking, when the self-similar boundary has a large set of symmetries), it is expected to be oscillatory, in a specific sense. (A sample of references on work towards the above conjectures for drums with self-similar boundary can be found, for example, in various places in the book Fractal Geometry and Number Theory by M.L. Lapidus and M. van Frankenhuysen.)

In that same survey paper by Lapidus, parallel conjectures are formulated for drums with self-similar membranes, that is, for Laplacians on self-similar fractals themselves rather on bounded open sets with self-similar boundaries. These conjectures are specified (by means of a suitable notion of "spectral dimension") and established in the following paper:

J. Kigami and M.L. Lapidus, "Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals", Commun. Math. Phys. 158 (1993), 93-125.

The partial analogue of Weyl's asymptotic formula for Laplacians on self-similar fractals obtained in the above paper was significantly strengthened in the following two papers:

M.L. Lapidus, "Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions", Topological Methods in Nonlinear Analysis 4 (1994), 137-195.

J. Kigami and M.L. Lapidus, "Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals", Commun. Math. Phys. 217 (2001), 165-180.

In the first of these papers, were constructed, by means of methods from operator algebras and noncommutative geometry, (spectral) "volume measures" for fractals, viewed in an important special case as an analogue on fractals of Riemannian volume measure. In the second paper, for a smaller class of self-similar fractals, theses volume measures were precisely identified and shown to be self-similar. In particular, for the homogeneous Sierpinski gasket, this measure coincides with the natural Hausdorff measure on the fractal; in general, however, it is different from it.

An important question: Lapidus and Pomerance unexpectedly related the (modified) Weyl-Berry conjecture to the Riemann zeta function. Berry's best known work links the zeta function and quantum chaos, and isn't obviously linked to the Weyl-Berry conjecture. Is this further Berry-zeta connection merely coincidental, or does it all stem from the same root (spectral asymptotics)?

Lapidus provided the following answer:

"It is somewhat amusing that there does not seem to be any direct connection between Michael Berry's work on the statistical distribution of the Riemann zeros and the link with the Riemann zeta function discovered in the process of resolving the modified Weyl-Berry conjecture in dimension 1. In particular, the connection with the Riemann hypothesis discovered in my work with Helmut Maier is completely independent. Indeed, the study of the statistical distribution usually requires as a prerequisite that the Riemann hypothesis is true.

Of course, there may be some interesting links between the general problem of understanding the oscillations occurring in my work with Helmut Maier and in the theory of complex dimensions developed in my joint book with Machiel van Frankenhuysen. Michael Berry and I have discussed this issue on several occasions and I have also thought about this for other reasons. (Most of my heuristic conclusions about this very difficult problem are unpublished but some comments are made about the possible connections between our theory of complex dimensions of fractals and a (yet to be defined precisely, in dimension larger than 2) dynamical system associated with fractal drums; see section 10.4.3 (as well as 10.4.2) entitled "spectrum and periodic orbits" of my joint book (with M-vF) "Fractal Geometry and Number Theory".

In summary, the answer to your question is "no". However, in the future, one may discover a nice dynamical interpretation of our 'explicit formulas' and of my (joint) reformulation of the Riemann Hypothesis. Hence, potentially, one may eventually establish a link with the random matrix theoretic approach to the statistical distribution of the Riemann zeros. For the moment, however, this is out of reach and it is fair to say that the subjects are totally distinct (and motivated by very different problems) even though they both attempt to gain new information about different aspects of the Riemann zeros."

[later note] "...some discussion of possible relations between the theory of complex dimensions developed in the above-mentioned book Fractal Geometry and Number Theory and some of the work on quantum chaos (including that by Michael V. Berry) is given in Chapter 10 of that book (especially, Section 10.4). I now have some further ideas on how these connections might occur but they are yet to be written up..."

M.L. Lapidus and H. Maier, "Hypothese de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiee", C. R. Acad. Sci Paris Ser. I Math. 313 (1991) 19-24.

(Abstract) "Jointly with C. Pomerance, the first author has recently proved in dimension one the "modified Weyl-Berry conjecture" formulated in his earlier work on the vibrations of fractal drums. Here, we show, in particular, that (still in dimension one) the converse of this conjecture is not true in the "midfractal" case and that it is true everywhere else if and only if the Riemann hypothesis is true. We thus obtain a new characterization of the Riemann hypothesis by means of a inverse spectral problem."

"As was the first author's hope at the beginning of his investigations connecting fractal and spectral geometry with aspects of analytic number theory, the above spectral characterization of the Riemann hypothesis may shed new light on aspects of the theory of the Riemann zeta-function."

See also the longer, related, paper:

M.L. Lapidus and H. Maier, "The Riemann hypothesis and inverse spectral problems for fractal strings", Journal of the London Mathematical Society (2) 52 (1995), 15-34.

By means of a suitable notion of generalised Minkowski content, many of the above results (by Lapidus, Lapidus and Pomerance, as well as Lapidus and Maier) were extended in the following papers to the situation where the fractality of the boundary is governed by more general "gauge functions" than the usual power laws:

C.Q. He and M.L. Lapidus, "Generalized Minkowski content and the vibrations of fractal drums and strings", Mathematical Research Letters 3 (1996) 31-40.

C.Q. He and M.L. Lapidus, "Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta function", Memoirs of the AMS No. 608, 127 (1997), 1-97.

Moreover, mainly in the one-dimensional case, partly motivated by the above joint works of Lapidus with Pomerance and with Maier, a detailed theory of complex dimensions of fractal strings (defined as the poles of suitably defined 'geometric zeta functions') was recently developed in the following book:

M.L. Lapidus and M. van Frankenhuysen, Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta funtions, research monograph, (Birkhauser, Boston, 2000).

In particular, a precise description of the geometric and spectral oscillations of fractal strings is obtained in terms of the underlying complex dimensions via suitable 'explicit formulas' (in the sense of number theory, but more general). Further, the characterization of the Riemann hypothesis obtained by Lapidus and Maier is placed in a broader and more conceptual framwork by means of the notion of complex dimension, and is extended to a class of Dirichlet series including all those for which Riemann's conjecture is expected to be true. Also, some new results concerning the vertical ditribution of the zeros of Dirichlet series (including the Riemann zeta function) are obtained via a detailed study of the interplay between the geometric and spectral oscillations of certain generalized Cantor strings. Further references related to this subject include:

M.L. Lapidus and M. van Frankenhuysen, "Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic", in: Spectral Problems in Geometry and Arithmetic (T. Branson, ed.), Contemporary Mathematics 237, (AMS, 1999) 87-105.

M.L. Lapidus and M. van Frankenhuysen, "A prime number theorem for self-similar flows and Diophantine approximation", in: Dynamical, Spectral and Arithmetic Zeta Functions (M.L. Lapidus and M. van Frankenhuysen, eds.), Contemporary Mathematics (AMS, 2001) (in press).

M.L. Lapidus and M. van Frankenhuysen, "Complex dimensions of self-similar fractal strings and Diophantine approximation", preprint, June 2001.