Closed 1-forms in topology and dynamics

This article surveys recent progress of results in topology and dynamics based on techniques of closed 1-forms. Our approach lets us draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically, a Lusternik–Schnirelmann type theory for closed 1-forms is described, along with the focusing effect for flows and the theory of Lyapunov 1-forms. Also discussed are recent results about cohomological treatment of the invariants $\operatorname{cat}(X,\xi)$ and $\operatorname{cat}^1(X,\xi)$ and their explicit computation in certain examples.