One can associate to any ring spectrum \(E\) a non-negative integer called its \emph{chromatic complexity}. Roughly stated, the red-shift conjecture of Ausoni-Rognes says that if a ring spectrum \(E\) has chromatic complexity \(n\), then its algebraic K-theory spectrum \(K(E)\) has chromatic complexity \(n+1\). The red-shift conjecture has been verified for some spectra with low chromatic complexity by Quillen, Hesselholt-Madsen, and Ausoni-Rognes. Work of Bruner-Rognes gives homological evidence for this conjecture at all levels of chromatic complexity.
I will review the red-shift conjecture and the above results, then recall Mahowald's definition of the Thom spectra \(y(n)\), \(n \geq 0\), which have chromatic complexity \(n\). I will then describe how one can apply the work of Bruner-Rognes and Nikolaus-Scholze to prove the red-shift conjecture holds for these homology theories, up to a homotopy limit problem. This is joint work with Gabe Angelini-Knoll.