Abstract: In the first part, we briefly introduce chromatic homotopy theory. This is a powerful tool to study periodic phenomena in stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories. The homotopy groups of these fixed points are periodic and computed via homotopy fixed points spectral sequences. In the second part, 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.
