  This paper concerns generalized differential characterizations of maximal
monotone set-valued mappings. Using advanced tools of variational analysis, we
establish coderivative criteria for maximal monotonicity of set-valued
mappings, which seem to be the first infinitesimal characterizations of maximal
monotonicity outside the single-valued case. We also present second-order
necessary and sufficient conditions for lower-${\mathcal C}^2$ functions to be
convex and strongly convex. Examples are provided to illustrate the obtained
results and the imposed assumptions.
