What does the Atiyah-Singer index theorem connect in geometry and differential equations?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

The Atiyah-Singer index theorem is a landmark result in mathematics that establishes a profound connection between topology, geometry, and analysis.1

Specifically, the theorem relates the analytical properties of elliptic differential operators on a compact manifold to the topological invariants of that manifold. By equating the analytical index—the difference between the dimensions of the kernel and the cokernel of an operator—with a topological index derived from characteristic classes, the theorem provides a bridge between the local behavior of differential equations and the global structure of the underlying geometric space. This discovery, for which Isadore Singer and Sir Michael Francis Atiyah were awarded the Abel Prize, has also played a critical role in developing connections between mathematics and theoretical physics.1