In the early 1920s M. Morse discovered that the number of critical
points of a smooth function on a manifold is closely related to the
topology of the manifold. This became a starting point of the Morse
theory which is now one of the basic parts of differential
topology. Circle-valued Morse theory originated from a problem in
hydrodynamics studied by S. P. Novikov in the early 1980s.
Nowadays, it is a constantly growing field of contemporary
mathematics with applications and connections to many geometrical
problems such as Arnold's conjecture in the theory of Lagrangian
intersections, fibrations of manifolds over the circle, dynamical
zeta functions, and the theory of knots and links in the
three-dimensional sphere. The aim of the book is to give a
systematic treatment of geometric foundations of the subject and
recent research results. The book is accessible to first year
graduate students specializing in geometry and topology.
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