The main direction of my research is the construction of the curvature-type differential invariants for a wide class of geometric structures on manifolds with applications to
a) equivalence of nonholonomic vector distributions, sub-Riemannian structures, CR (Cauchy -Riemann) structures, fields of cones, ordinary and partial differential equations;
b) optimality properties of extremals of optimal control problems;
c) state-feedback equivalence of control systems;
d) qualitative study of Hamiltonian systems.
The approach is based on the study of differential geometry of curves in Grassmannians, Lagrange Grassmannians, and spaces of flags.
Among other topics of my research are
1) projective/affine equivalence of sub-Riemannian metrics and inverse optimal control problem ;
2) Gromov's h-principle for vector distribution (global existence of distributions with prescribed differential properties);
3) nonsmooth Morse theory with applications to Mathematical Physics;
4) invariant description of flat control systems;
5) sub-Riemannian Laplace-Beltrami operator.