A new proof of the Milnor – Wood theorem, and beyond

The Milnor–Wood inequality states that if a (topological) oriented circle bundle over an orientable surface of genus $g$ has a smooth transverse foliation, then the Euler class of the bundle satisfies $|E|\leq 2g-2$. We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes $E$ from the singularities of a quasisection and present some more applications of this approach. (Based on joint works with Ilya Alekseev, Ivan Nasonov, Timur Shamazov and Maksim Turevskii)