Mathematics
The Atiyah–Singer Index Theorem and Its Applications
Quick fact
The Atiyah–Singer index theorem, proved in 1963, equates the analytic index (the difference between the dimensions of the kernel and cokernel) of an elliptic operator to a topological invariant computed from the manifold's geometry. This single formula unifies dozens of classical results, including the Gauss–Bonnet theorem and the Riemann–Roch theorem.
Why this is interesting
Did you know that the number of independent solutions to a differential equation can be determined purely by the 'shape' of the space it lives on, without solving the equation?