Title: Vanishing results in Chromatic homotopy theory at prime 2
Abstract: Chromatic homotopy theory is a powerful tool to study periodic phenomena in the stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories $E_h$. The homotopy groups of these fixed points are periodic and computed via homotopy fixed points spectral sequences. In this talk, we prove that at the prime 2, for all heights $h$ and all finite subgroups $G$ of the Morava stabilizer group, the $G$-homotopy fixed point spectral sequence of $E_h$ collapses after the $N(h,G)$-page and admits a horizontal vanishing line of filtration $N(h,G)$.
Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem. As an application, we give a computation $E_2^{hG_{48}}$ based on this vanishing result. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.