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Shachar Carmeli: The Chromatic Fourier Transform

Time: Tue 2022-10-11 10.15

Location: Albano, house 1, Cramer room

Participating: Shachar Carmeli

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The classical discrete Fourier transform identifies complex-valued functions on a finite group and on its Pontriagin dual. The construction depends on a choice of compatible primitive complex roots of unity. 

In this talk, I will present a work in preparation, joint with Barthel, Schlank, and Yanovski, which generalizes this construction to higher chromatic heights, that is, to the T(n)-local and K(n)-local stable homotopy theories. Like in classical Fourier theory, the inputs for the construction are higher height, "shifted" analogs of roots of unity, previously studied by Schlank, Yanovski, and myself. Correspondingly, the higher Fourier transform identifies functions on a pi-finite p-local connective Z-module spectrum M, with functions on its n-shifted Pontryagin dual M^*[n], valued in T(n)-local rings with enough higher roots of unity.

I will then discuss the relation to previous work of Hopkins and Lurie on Morava E-theories, various applications to the T(n)-local and K(n)-local stable homotopy theories, a categorification of the Fourier transform, and generalizations that allow to define the map for functions over more general spectra.