In and around stable homotopy groups of spheres

My thesis focuses on computations of stable homotopy groups of spheres, with applications and connections to differential geometry and motivic homotopy theory. The Adams spec- tral sequences and Toda brackets play a major role in my work. We have introduced two methods to compute Adams differentials and solve extension problems: one is very technical but inductive, using the algebraic Kahn-Priddy theorem; the other one is more systematic, using a new connection between motivic homotopy theory and chromatic homotopy theory. Combining both methods, we have computed stable stems into a larger range. As a consequence, I solved the strong Kervaire invariant problem in dimension 62 and showed that the 61-sphere has a unique smooth structure, which is the last odd dimensional case.