This text features a careful treatment of flow lines and algebraic
invariants in contact form geometry, a vast area of research connected
to symplectic field theory, pseudo-holomorphic curves, and Gromov-Witten
invariants (contact homology). In particular, this work develops a novel
algebraic tool in this field: rooted in the concept of critical points
at infinity, the new algebraic invariants defined here are useful in the
investigation of contact structures and Reeb vector fields. The book
opens with a review of prior results and then proceeds through an
examination of variational problems, non-Fredholm behavior, true and
false critical points at infinity, and topological implications. An
increasing convergence with regular and singular Yamabe-type problems is
discussed, and the intersection between contact form and Riemannian
geometry is emphasized. Rich in open problems and full, detailed proofs,
this work lays the foundation for new avenues of study in contact form
geometry and will benefit graduate students and researchers.