Algebra/Topology Seminar
Speaker: Lior Yanovski
Title: The chromatic discrete Fourier transform
Abstract: The classical DFT can be thought of as an isomorphism of rings ![\mathbb{C}[A] \overset{\sim}{\to} \mathbb{C}^{A^*}](https://ci6.googleusercontent.com/proxy/fNlhbzxZL-Nsd13j2kga9bUbXELnEyGwtegRRBddg86vnutx6FSxiInsZy4RBcHiIIg2f5lac3mgYWamyGSU79L5u4PWkWgfckxgnqpPci9f9nTaV9NtxikWgOZQ8pvDqBS42uJMUJ46DhIEmmxx6zPip7jAZD9c_yYPLZXsh4KZPTcbSwc3Sup23WoWlDrk4oDTL8AWcx568t1vRXdAVq4hErqbZoxzYzT0T7s=s0-d-e1-ft#https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Cmathbb%7BC%7D[A]%09%5Coverset%7B%5Csim%7D%7B%5Cto%7D%09%5Cmathbb%7BC%7D%5E%7BA%5E*%7D "\mathbb{C}[A] \overset{\sim}{\to} \mathbb{C}^{A^*}")
, between the complex group algebra of a finite abelian group 
and the algebra of functions on its Pontyagin dual 
. In their work on ambidexterity, Hopkins and Lurie proved an analogous result in the chromatic world, where the field of complex numbers is replaced by the Lubin-Tate spectrum 
, the finite abelian group is replaced by a suitably finite 
-power torsion 
-module spectrum, and the Pontryagin dual is modified by an 
-fold suspension. From this, they deduce a number of structural properties of the 
-category of  "K(n)")
-local spectra, such as affineness and Eilenberg-Moore type formulas for -finite spaces. In this talk, I will present a joint work with Barthel, Carmeli, and Schlank, in which we develop the notion of a "higher Discrete Fourier transform" for general higher semiadditive -categories. This allows us, among other things, to extend the above results of Hopkins and Lurie to the  "T(n)")
-local setting. Furthermore, by replacing Pontryagin duality with Brown-Comenetz duality, we extend the Fourier transform over the Lubin-Tate spectrum beyond -modules. Finally, we study the interaction of Fourier transforms with categorification and redshift phenomena.
