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, with a specific focus on a unified approach to non-compactness in both disciplines.  Fully detailed, explicit proofs and a number of suggestions for further research are provided throughout. Rich in open problems and written with a global view of several branches of mathematics, this text lays the foundation for new avenues of study in contact form geometry.  Graduate students and researchers in geometry, partial differential equations, and related fields will benefit from the book's breadth and unique perspective.

Table of Contents:
Introduction, Statement of Results, and Discussion of Related Hypotheses.- 1 Topological results.- 2 Intermediate hypotheses (A4), (A4)’ (A5), (A6).- 3 The non-Fredholm character of this variational problem, the associated cones, condition (A5) (discussion and removal).- 4.a Hypothesis $$\overline {(A4)}$$ and statement of the most general results, discussion of $$\overline {(A4)}$$.- 4.b Discussion of (A2), (A3), and $$\overline {(A4)}$$.- Outline of the Book.- I Review of the Previous Results, Some Open Questions.- I.A Setup of the Variational Problem.- I.B The Flow Z0 of [2]: Critical Points at Infinity, False and True.- II Intermediate Section: Recalling the Results Described in the Introduction, Outlining the Content of the Next Sections and How These Results are Derived.- III Technical Study of the Critical Points at Infinity: Variational Theory without the Fredholm Hypothesis.- III.A True Critical Points at Infinity.- III.B False Critical Points at Infinity of the Second Kind.- IV Removal of (A5).- IV.1 The Difference of Topology Due to a False Critical Point at Infinity of the Third Kind.- IV.2 Completion of the Removal of (A5).- IV.3 Critical Points at Infinity of Mixed Type.- IV.4 (A5) and the Critical Points at Infinity of the Third Kind or of Mixed Type.- V Conditions (A2)—(A3)—(A4)—(A6).- V.1 An Outline for the Removal of (A2).- V.2 Discussion of (A3).- V.3 Weakening Condition (A4).- V.4 Removing Condition (A6).- References.