# Shachar Carmeli: The Chromatic Fourier Transform

**Time: **
Tue 2022-10-11 10.15

**Location: **
Albano, house 1, Cramer room

**Participating: **
Shachar Carmeli

**Abstract:**

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.