We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz-Quillen, Cortinas, Ginzburg-Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also review the relation of noncommutative forms to crystalline cohomology (the classical one as well as the noncommutative version due to Cortinas), to operations on Hochschild and cyclic complexes, and with De Rham theory of representation schemes (after Ginsburg-Schedler and Waikato Yeung).
A survey of noncommutative differential forms.
Oct. 20, 2022 11:15am (Math …
Noncomm Geometry
Boris Tsygan (Northwestern University)
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We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz-Quillen, Cortinas, Ginzburg-Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also review the relation of noncommutative forms to crystalline cohomology (the classical one as well as the noncommutative version due to Cortinas), to operations on Hochschild and cyclic complexes, and with De Rham theory of representation schemes (after Ginsburg-Schedler and Waikato Yeung).
We introduced a distance between spectral triples, inspired from our earlier work on the quantum Gromov-Hausdorff distance, and we have established a few first examples of convergence for this distance. In this talk, after reviewing our distance, we will discuss properties which are continuous, in an appropriate sense, for our distance, which includes the spectra of the Dirac operators in spectral triples.
Spectral properties of the convergence of spectral triples Sponsored by the Simons Foundation