The Gauss-Bonnet-Chern theorem: a probabilistic perspective
  We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$. 
Liviu I. Nicolaescu, Nikhil Savale
math.DG math.AP math.PR