On new structures in the theory of fully nonlinear equations

We describe the current state of the theory of equations with $m$-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of mpositivity of matrices is quite important; we provide a proof of an analog of Sylvester's criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for $m$-Hessian evolution equations. The asymptotic behavior of $m$-Hessian evolutions in a semibounded cylinder is considered as well.