Spectral invariants: What can we hear?

 When a stringed instrument is played, the sound we hear consists of a fundamental tone and infinitely many harmonic tones.  Mathematically, what we ``hear'' is the spectrum of the Laplace operator.  In one dimension, the only geometric quantity to be heard is the length of the string.  However, in higher dimensions a natural question is:  What can we hear?  In the 1960s, a popular variation of this question in two dimensions was posed by M. Kac:  Can one hear the shape of a drum?  Any quantity which can be expressed in terms of the spectrum is known as a spectral invariant.  Thus, all spectral invariants can be ``heard.''  Beginning with preliminaries, we'll explore spectral invariants from Weyl's asymptotic formula (1911) to the zeta regularized determinant and its connections with physics (Hawking, 1977), concluding with relationships to dynamics and Smale's (1960s) ``exotic zeta functions.''