The study of the zeros, critical points,
etc., of random polynomials goes back many years to Bloch-Polya,
Littlewood-Offord, Kac and others. Since complex holomorphic
homogeneous polynomials are sections of the hyperplane bundle
over projective space, it is natural to develop the theory of
random polynomials in the setting of positive line bundles over
algebraic manifolds. We will present results on the probabilities
that holomorphic sections do various things as the degree tends to infinity.
The results also apply to "almost-holomorphic" sections of ample
line bundles over integral symplectic manifolds.