results: 1) Gauge gravitation theory on natural bundles
where gravity is a classical Higgs field providing a world manifold
with a Lorentz structure. 2) Comprehensive geometric
formulation of classical field theory where classical fields are
represented by sections of fibre bundles, including: (i)
the differential calculus on graded fibred manifolds and
cohomology of a variational bicomplex, (ii) Lagrangian formalism
on fibre bundles and graded manifolds in terms of infinite-order
jets, (iii) generalized Noether theorems for reducible degenerate
Lagrangian systems in homology terms, (iv) prequantum BRST-extended
Lagrangian theory of fields, higher-order antifields and ghosts.
3) Covariant Hamiltonian field theory on
polysymplectic manifolds. 4) Classical
Higgs field theory on composite bundles.
5) Geometric formulation of non-relativistic time-dependent mechanics
in terms of fibre bundles over R. 6) Its quantization in a form
of geometric quantization of symplectic foliations. 7) Geometric
formulation of relativistic mechanics in terms of jets of one-dimensional
submanifolds. 8) Extension of the Liouville-Arnold, Nekhoroshev
and Mishchenko-Fomenko theorems on integrable Hamiltonian systems
to a general case of not necessary compact invariant submanifolds.