1) Cauchy Problem for linear hyperbolic operators with multiple
characteristics.
2) Gevrey regularity and propagation of singularities for weakly
hyperbolic psuedo-differential eqautions.
3) Stochastic Hamiltonian Problems.
For the Hyperbolic Cauchy Problem with double characteristics in
recent papers with T.Nishitani we have come to the
complete understanding of the geometric interplay of the linear
algebraic classification of symplectic matrices at multiple points
and the stability of the geometry ot the curves which are soultions
of the Hamiltonian system naturally associated with the principal
symbol of the operator examined.
It has been a problem which only recently can be declared to be
solved the fact that when the fundamental matrix has a Jordan block
of size bigger than 2 in its canonical decomposition new
instabilities tend to appear and can interact with lower order
terms in the classical development of the symbol.
We study numerical methods for systems of Hamiltonian stochastic equations with non Lipschitz coefficients originating from quantitative finance problems.
