  One interpretation of Torelli's Theorem, which asserts that a compact Riemann
Surface $X$ of genus $g > 1$ is determined by the $g(g+1)/2$ entries of the
period matrix, is that the period matrix is a message about $X$. Since this
message depends on only $3g-3$ moduli, it is sparse, or at least approximately
so, in the sense of information theory. Thus, methods from information theory
may be useful in reconstructing the period matrix, and hence the Riemann
surface, from a small subset of the periods. The results here show that, with
high probability, any set of $3g-3$ periods form moduli for the surface.
