The \$K(n)\$-local sphere is a fundamental object of interest in chromatic homotopy theory. One may access this object by Galois descent from its separable closure, the Lubin--Tate spectrum of height n attached to the Honda formal group law. However, as the Galois group \$\mathbb{G}_n\$ is a large \$p\$-adic Lie group, this is not very practical. An easier task is to approximate the \$K(n)\$-local sphere by descent along finite subgroups of \$\mathbb{G}_n\$. The resulting homotopy fixed points are known as higher real \$K\$-theories, and they are among the most important and well-studied spectra we have at our disposal.
In this series of talks I will discuss the role that higher real \$K\$-theories play in chromatic homotopy theory. In talk 1, I will introduce the main characters and landscape and mention tools for computation. In talk 2, I will introduce redshift in algebraic \$K\$-theory and blueshift in Tate fixed points for higher real \$K\$-theories.