Mathematics Colloquium

Up to continuous deformations, all based continuous maps between two spheres form an abelian group, and is called a homotopy group of the target sphere. It turns out that determination of these groups is a very hard problem in topology. The structures of the homotopy groups of spheres are closely related to many topics in topology, such as the Hopf invariant problem, the Kervaire invariant problem, and the number of smooth structures on a given sphere.

In this talk, I will review some classical methods of computing these groups, and discuss some recent progress using motivic homotopy theory.