On the motivic Adams conjecture

In the motivic homotopy theory one associates to a scheme X the socalled motivic stable homotopy category SH(X) that is a tensor triangulated category and is a universal source of cohomological invariants that are A^1-invariant and satisfy descent in the Nisnevich topology. Motivic Adams conjecture provides a partial information on the Picard group of SH(X). Every vector bundle over X gives rise to an element in the Picard group of SH(X) via the Thom spectrum construction and this yields a homomorphism from the Grothendieck group of vector bundles over X to the Picard group of SH(X). Motivic Adams conjecture claims that the difference between the Thom space of a vector bundle and the Thom space of the value of the k-th Adams operation on the vector bundle in Pic(SH(X)) is k-torsion. In my talk I will introduce all this notions and give an outline of the proof for this conjecture. The talk is based on the work in progress joint with Elden Elmanto, Oliver Roendigs and Maria Yakerson.