Geometric spectral invariants (I)

We will begin with the discovery of the first geometric spectral invariants by Hermann Weyl in 1912. 
In the first lecture, we will recall Weyl's original proof using Dirichlet-Neumann bracketing, to demonstrate
"das asymptotische verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,'' which is
now known as Weyl's law.  In the second lecture, we will turn to the spectral invariant which lead to the
discovery of new geometric spectral invariants in the 1960s and 1980s:  the heat trace.  We will recall the
now famous paper of M. Kac, as well as his proof that one can hear the number of holes in a smoothly
bounded domain, based on the short time asymptotic expansion of the heat trace.  This will naturally lead
us to the spectral zeta function and the zeta-regularized determinant of the Laplacian.  Finally, in the third
lecture, we will turn to a more elusive spectral invariant:  the wave trace.  We will discuss geometric spectral
invariants which can be extracted from the wave trace, and we shall conclude the mini-course with a discussion
of open problems.