Rabinowitz Floer Homology

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The Rabinowitz action functional is a Lagrange multiplier functional whose critical points are periodic orbits of fixed energy. A Floer homology for this action functional was first constructed by the au-thors of this proposal. Meanwhile Rabinowitz Floer homology has found numerous applications, see e.g. the ICM 2022 talk of one of the authors of this proposal. The Lagrange multiplier at a critical point corresponds to the period, where a negative value means that the periodic orbit is traversed back-wards in time. This feature distinguishes Rabinowitz Floer homology from symplectic homology and symplectic field theory where periodic orbits can only be traversed in forward time. It leads to deep relations of Rabinowitz Floer homology with Tate homology and Poincare duality, and it is the reason that Rabinowitz Floer homology has the structure of a graded Topological Quantum Field Theory. The goal of this proposal is to understand this structure in more depth, to study how it extends to Hamil-tonian delay equations, and to explore applications of this structure to quantum mechanics in the semiclassical limit.