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-Ampère equations and/or weak KAM theory.