Mathematics
Elliptic Operators and the Index Theorem
Quick fact
The Atiyah–Singer index theorem, proven in 1963, shows that the analytic index of an elliptic operator—the difference between the dimensions of its kernel and cokernel—equals a purely topological quantity computable from the manifold's geometry. This links the seemingly analytic notion of solvability to global topology, with applications from particle physics to pure mathematics.
Why this is interesting
You know that solving a differential equation often depends on the shape of the space. But what if the number of solutions minus the number of obstructions tells you something about the shape itself?