  A differential operator $D$ commuting with an $S^1$-action is said to be
rigid if the non-constant Fourier coefficients of $\ker D$ and $\coker D$ are
the same. Somewhat surprisingly, the study of rigid differential operators
turns out to be closely related to the problem of defining Chern numbers on
singular varieties. This relationship comes into play when we make use of the
rigidity properties of the complex elliptic genus--essentially an
infinite-dimensional analogue of a Dirac operator. This paper is a survey of
rigidity theorems related to the elliptic genus, and their applications to the
construction "singular" Chern numbers.
