The Index Theorem of Atiyah and Singer

The elliptic operator \(D = d/dx\) acting on the circle \(\mathbb {T} = \mathbb {R}/2\pi i \mathbb {Z}\) has a discrete spectrum of the integers. In particular, the dimensions of the kernels of the continuously varying family of operators \(D + \lambda \) , for \(\lambda \in \mathbb {R}\) , jump discontinuously as \(\lambda \) crosses an integer point.