Since there are several meanings of spectrum there are accordingly several things one may mean by “spectral geometry”. Most of those belong either to the geometry as seen either by point spectra of spectral theory (of operators, families of operators, operator algebras, rings, associative algebras, abelian categories etc.), or by spectra in the sense of stable homotopy theory like symmetric spectra, -spectra, ring spectra…
For the notion of spectrum of an operator spectral geometry is geometry as seen by the spectra of operators (“hearing the shape of a drum”), which is geometry as seen by spectral triples (often inaccurately referred to as “noncommutative geometry”).
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For the notion of spectrum as in stabilization in stable homotopy theory, spectral geometry is geometry over formal duals of E-∞ rings (the full version of derived algebraic geometry).
See at E-∞ geometry.
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Last revised on March 5, 2015 at 21:36:04. See the history of this page for a list of all contributions to it.