Semiconcave functions are a well-known class of nonsmooth
functions that possess deep connections with optimization theory and
nonlinear pde's.
Their singular sets exhibit interesting structures that we will
describe in thistalk. First, by an energy method, we will analyze
the curves along which the singularities of semiconcave solutions
to Hamilton-Jacobi equations propagate, the so-called generalized
characteristics. Then, we will derive the dynamics of propagation for
general semiconcave functions. Finally, we will discuss applications
to Monge-AmpĂ¨re equations and/or weak KAM theory.