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, it 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.
This monograph is devoted to Krull domains and its invariants. The book shows how a serious study of invariants of Krull domains necessitates input from various fields of mathematics, including rings and module theory, commutative algebra, K-theory, cohomology theory, localization theory and algebraic geometry. About half of the book is dedicated to so-called involutive invariants, such as the involutive Brauer group, and is essentially the first to cover these topics. In a structured and methodical way, the work presents a large quantity of results previously scattered throughout the literature. Audience: This volume is recommended as a first introduction to this rapidly developing subject, but will also be useful as a state-of-the-art reference work, both to students at graduate and postgraduate levels and to researchers in commutative rings and algebra, algebraic K-theory, algebraic geometry, and associative rings.
In this paper the author establishes some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. He then uses these ideas to prove the Hanna Neumann Conjecture of the 1950s; in fact, he proves a strengthened form of the conjecture.