Maria Yakerson — How motivic homotopy theory helps to classify vector bundles

Classically, topological vector bundles are classified by homotopy classes of maps into infinite Grassmannians. This allows us to study topological vector bundles using obstruction theory: we can detect whether a vector bundle has a trivial subbundle by means of cohomological invariants.

In the context of algebraic geometry, one can ask whether algebraic vector bundles over smooth affine varieties can be classified in a similar way. Recent advances in motivic homotopy theory give (partial) positive answers, which will be discussed in the talk. In particular, we will see how in the algebro-geometric version of the story, quadratic forms enter the scene.