Robert McLachlan

Structure in the algebra of rooted trees

When the vector field f is canonical Hamiltonian, the algebra of
elementary differentials generated by f has special structure. For
example, f''(f,f) - 2 f'(f'(f)) is Hamiltonian and f'(f'(f)) is
energy-preserving.  There are subalgebras of Hamiltonian
(resp. energy-preserving) B-series corresponding to symplectic
(resp. energy-preserving) integrators. In this talk I will introduce
and describe these subalgebras and their relationships and also the
more complicated sets of B-series that are conjugate to Hamiltonian or
energy-preserving.