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 $e_h$.  these fixed points are 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 $n(h,g)$.

this vanishing result has proven to be computationally powerful, as demonstrated by hill--shi--wang--xu’s recent computation of $e_4^{hc_4}$.  our proof uses new equivariant techniques developed by hill--hopkins--ravenel in their solution of the kervaire invariant one problem.  as an application, we extend kitchloo--wilson’s $e_n^{hc_2}$-orientation results to all $e_n^{hg}$-orientations at the prime 2. this is joint work with zhipeng duan and xiaolin danny shi.