The vector field problem is a classical problem in differential topology. It asks for the maximum number of vector fields \(X_1, \ldots, X_k\) on a smooth manifold \(M\) such that they are everywhere linearly independent. It is not an exaggeration to say that vector field problem (for spheres) had been a major driving force in the development of algebraic topology up to the middle of the last century. In a celebrated landmark paper, J F Adams settled this completely for the spheres. For a general smooth manifold, it is still a hard problem to solve.
This talk will be largely of expository nature and should be accessible to a wide audience.
