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

격자의 세타함수

[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

 

http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum