Vanishing lines in chromatic homotopy theory

Chromatic homotopy theory studies periodic phenomena in stable homotopy theory via fixed points of Lubin–Tate theories. The homotopy groups of these homotopy fixed points are periodic and computed via homotopy fixed points spectral sequences. In this talk, we present a result of an upper bound of the complexity of these computations. In particular, at the prime $2$, for any given height, and a finite subgroup of the Morava stabilizer group, we find a number $N$ such that the homotopy fixed point spectral sequence of collapses after page $N$ and admits a horizontal vanishing line of a certain filtration $N$. The proof uses new equivariant techniques developed by Hill–Hopkins–Ravenel in their solution of the Kervaire invariant one problem and has applications to computations. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.