Smooth structures, stable homotopy groups of spheres and motivic homotopy theory

Abstract: Following Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, Guozhen Wang and I showed that the 61-sphere has a unique smooth structure and is the last odd dimensional case: $S^1, S^3, S^5$ and $S^{61}$ are the only odd dimensional spheres with a unique smooth structure. The proof is a computation of stable homotopy groups of spheres. We introduce a method that computes differentials in the Adams spectral sequence by comparing with differentials in the Atiyah-Hirzebruch spectral sequence for real projective spectra through Kahn-Priddy theorem. I will also discuss recent progress of computing stable stems using motivic homotopy theory with Dan Isaksen and Guozhen Wang.