Parings of \(K\)-theory with Fredholm modules given by Dirac operators can be computed numerically as the half-signature of a suitable finite dimensional-matrix called the spectral localizer. This allows to access integer-valued pairings, \(\mathbb{Z}_2\)-valued pairing arising in Real $K$-theory as well as semifinite index pairings. Topological insulators constitute a rich field of applications within which the above can be illustrated. (Joint work with Nora Doll, Terry Loring, Tom Stoiber).

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