From: David.L.Webb@Dartmouth.edu (David L. Webb)
Newsgroups: sci.math
Subject: Re: Can you hear the shape of a drum?
Date: 13 Sep 1995 19:51:01 GMT

In article <430mk3$qf9@aggedor.rmit.EDU.AU>, rmaig@minyos.xx.rmit.EDU.AU
(Ian Grundy) wrote:

>len@schur.math.nwu.edu (Len Evens) writes:
>
>>In article <1740D1411DS86.RVANRAAM@bcsc02.gov.bc.ca>,
>> <RVANRAAM@bcsc02.gov.bc.ca> wrote:
>>>About 25 years ago I heard an interesting lecture called
>>>"Can you hear the shape of a drum?".  The visiting professor
>>>to the University of Toronto proved that you cannot hear the
>>>shape of a drum, but you could hear how many holes the drum
>>>had.  Does anyone know the proof?
>>> 
>>I believe there have been several expository papers about this
>>subject, including one in the American Math Monthly.    Try looking
>>through some indexes if no one comes forth with an explicit
>>reference.
>
>Try:
>
>M. Kac, "Can one hear the shape of a drum?" Amer. Math. Monthly 73, No. 4,
>        Part 11, (1966), 1-23.
>
>Related references can be found in
>
>E. M. E. Zayed, "Eigenvalues of the Laplacian for the third boundary value
>                problem", J. Austral. Math. Soc. Ser. B 29 (1987), 79-87.
>
>and
>
>H. P. W. Gottlieb, "Eigenvalues of the Laplacian for rectilinear regions",
>                   J. Austral. Math. Soc. Ser. B 29 (1988), 270-281.   
>
>as well as a later paper by Zayed.
>

   The proof that one can hear the number of holes is due, I believe, to
McKean and Singer.  The idea is most easily understood in the context of a
closed surface (rather than a planar domain):  from the heat equation
asymptotics, one can read off the total scalar curvature.  For an
orientable surface, the Gauss-Bonnet theorem says that this is just the
Euler characteristic (times 2 pi), so one can hear the Euler
characteristic and hence the topological type of an orientable closed
surface.  Similar ideas work in the case of surfaces with boundary.
   Kac's original question, Can one hear the shape of a drum (i. e., of a
domain in the Euclidean plane) has a negative answer;  see Inventiones
Mathematicae 110 (1992), 1-22.  A popular article, which I believe is to
appear in the January issue (or shortly thereafter) of American Scientist,
is in preparation.  There have been some short articles explaining the
counterexamples from a very elementary viewpoint, notably one by Pierre
Berard appearing in Afrika Mathematika 3 (1992), 135-146. There is a nice
paper by Peter Buser, John Conway, Peter Doyle and Klaus-Dieter Semmler
which uses the same idea to produce many interesting examples, including a
pair of what Doyle calls "isosonic" drums;  this paper is available on the
web at 

http://www.geom.umn.edu/docs/doyle/drum/cover/cover.html

   For some beautiful images of isospectral drums, as well as movies of
them vibrating, see Toby Dricoll's web page at 

http://cam.cornell.edu/~driscoll/research/drums.html

   Going back a few years, some very nice expository papers discussing
isospectral manifolds include papers by Bob Brooks, Amer. Math. Monthly 95
(1988), 823-839, Pierre Berard, Seminaire Bourbaki 1988-89, no. 705, and
Dennis DeTurck, Rendicotti Sem. Fac. Sci. Univ. Cagliari 58 (suppl. 1988),
1-26.
   The questions of real interest whcih in some sense arise from Kac's
paper are still wide open.
         David Webb

-- 
David.L.Webb@Dartmouth.Edu