A new proof of Milnor – Wood theorem

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
$$|\mathcal{E}|\leq 2g-2.$$
We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes $\mathcal{E}$ from the singularities of a quasisection.
(Based on a joint work with Timur Shamazov and Maksim Turevskii)