This year's lecture covered various aspects of viscosity solution theory and its applications to the study of completely nonlinear, second-order, elliptic and possibly degenerate equations, and in particular1st-order Hamilton-Jacobi equations. The study of edge singularities caused by the simultaneous presence of nonlinearity cancellation at the edge and superlinear Hamiltonians was addressed, as was a new approach to the uniqueness of solutions by using auxiliary functions that generalize the geodesic distances associated with a Hamiltonian in the special case where it is homogeneous of degree 2.
The first theme was motivated by a mathematical economy model introduced for the mining industries. And the second theme has as its natural application the theory of stochastic viscosity solutions, introduced in collaboration with P. E. Souganidis.
We present here only a brief Abstract of the main results on the second theme.