Hearing the shape of a drum
리만기하학의 문제 Is a Riemannianmanifold (possibly with boundary) determined by its spectrum?
1964, John Milnor found two distinct 16-dimensional manifolds with the same spectrum.
1991년, CarolynGordon, David Webb, and Scott Wolpert found examples of distinct plane "drums"which "sound" the same. See the illustrations below.
Milnor, John (1964), "Eigenvalues of the Laplace operator on certain manifolds", Proceedings of the National Academy of Sciences of the United States of America 51: 542ff
Kac, Mark (1966), "Can one hear the shape of a drum?", American Mathematical Monthly 73 (4, part 2): 1–23
expositions
http://www.ams.org/samplings/feature-column/fcarc-199706
YouCan't Always Hear the Shape of a Drum by Barry Cipra, which appeared inVolume 1 of What's Happening in the MathematicalSciences.
16 dimensioanl lattices
[1]http://www.facstaff.bucknell.edu/ed012/bucknell.pdf
http://www.facstaff.bucknell.edu/ed012/Altoona.pdf
http://math.berkeley.edu/~alanw/240papers03/vitocruz.pdf