we begin this lecture by reviewing the concept of the trace of a matrix. then we move on to study the vibrations of a guitar string of length $\pi$. the noise that the string produces contains a base tone and higher overtones. in the idealized mathematical model, the list of frequencies of all these tones are $$ 1,\ 2^2,\ 3^2,\ 4^2,\ \dots, $$ and so the wavelengths are $$ 1,\ \frac1{2^2},\ \frac1{3^2},\dots\dots $$ we explain why the {\it total wavelength}, $1+\frac1{2^2}+\frac1{3^2}+\dots$ can be computed by integrating the {\it green's function} $x(1-x/\pi)$ along the guitar string, giving an answer of $\pi^2/6$. the trouble starts when we try to replace the guitar string by a sphere and carry out the same process. the resulting formula $$ \sum_{n=1}^\infty \frac{2n+1}{n(n+1)}\ =\ \int_{s^2} g(x,x)\,ds(x) $$ simply boils down to $\infty=\infty$. however, by subtracting infinity very carefully from each side, we obtain an interesting formula which works for any surface, not just a sphere. although this mathematics may never be used to help us design musical instruments, it is related to the vortex theory of fluids, and einstein's relativity.