Title: Some Things You Can't Hear The Shape Of
Speaker: John Conway
The problem popularized by Mark Kac's famous lecture "Can you hear
the shape of a drum?" was finally solved about ten years ago, by
Gordon, Wolpert and Webb. However, this is not the end of the problem,
just as Kac's question wasn't the beginning.
On the way to being solved, the problem led to several related
notions of "isospectrality", each of which gives new problems of its
own. I shall discuss these problems and their solutions, when known,
for open and closed manifolds, lattices, codes, and groups.
One most interesting particular result is the recent theorem of
Juan-Pablo Rossetti that there is a unique non-trivial pair of
isospectral "platycosms" (closed flat 3-manifolds). So I shall
also discuss the 10 types of platycosm and their geometry.