Atiyah–Singer index theorem

The Atiyah–Singer index theorem equates the analytic index of an elliptic differential operator on a compact manifold with a topological quantity determined by its symbol.

Elliptic differential operator: A differential operator whose principal symbol is invertible away from the zero section of the cotangent bundle. Ellipticity makes the operator's index finite and permits the theorem's analytic side.
Fredholm operator: A bounded linear operator with finite-dimensional kernel and cokernel and closed range. Elliptic operators on compact manifolds yield Fredholm maps, making their indices well-defined.
Atiyah–Singer index theorem for the Dirac operator: An index formula for Dirac operators expressed through characteristic classes of spin or spin-c structures. This celebrated specialization links spin geometry directly to topological invariants.
Michael Atiyah: A British mathematician known for work in geometry, topology, and mathematical physics. He co-formulated the theorem and developed its K-theoretic framework.
Index theory: The study of numerical invariants associated with operators and their relation to geometry and topology. The theorem is the foundational bridge organizing this broad subject.
Analytic index: The dimension of an operator's kernel minus the dimension of its cokernel. This is the analytic quantity the theorem identifies with topology.
Kernel of a linear operator: The vector space of inputs that a linear operator maps to zero. Its dimension contributes positively to the analytic index.
Gauss–Bonnet theorem: A theorem equating the integral of curvature on a compact oriented even-dimensional manifold with its Euler characteristic. It is recovered as an index formula for the de Rham operator.
Isadore Singer: An American mathematician whose work spans differential geometry, topology, and mathematical physics. He co-formulated the theorem and helped connect it to spectral geometry.
Anomaly cancellation: The removal of quantum anomalies through constraints or compensating contributions in a physical theory. Index formulas help explain when fermionic anomalies vanish or are quantized.

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