K-Theory
Harmonic forms are differential forms that are both closed and co-closed, meaning they satisfy the equations $d\omega = 0$ and $\delta\omega = 0$, where $d$ is the exterior derivative and $\delta$ is the codifferential. These forms arise naturally in the study of differential geometry and play a crucial role in connecting analysis, topology, and geometry, particularly in the context of the Atiyah-Singer index theorem and its proof.
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