  We prove pointwise variational Lp bounds for a bilinear Fourier integral
operator in a large but not necessarily sharp range of exponents. This result
is a joint strengthening of the corresponding bounds for the classical Carleson
operator, the bilinear Hilbert transform, the variation norm Carleson operator,
and the bi-Carleson operator. Terry Lyon's rough path theory allows for
extension of our result to multilinear estimates. We consider our result a
proof of concept for a wider array of similar estimates with possible
applications to ordinary differential equations.
