The zeros of random sections of real vector bundles

We define integral geometric analogues of the Chern classes for real vector bundle on a smooth real variety. More precisely, we define the Chern densities of a real bundle. These densities are analogues of the Chern forms of a complex vector bundle and inherit some of their properties. \noindent (The text is a summary of a report on the conference PCA’2024 in Euler International Mathematical Institute, St. Petersburg)