In this paper, we explore loops of non-autonomous Hamiltonian diffeomorphisms with degenerate fixed maxima. We show that such loops can not have totally degenerate fixed global maxima. This has applications for the Hofer geometry of the group of Hamiltonians for certain symplectic four manifolds and also gives criteria for certain four manifolds to be uniruled.

We construct smooth actions of arbitrary compact Lie groups on complex projective spaces, such that the corresponding transformations arising from the group action do not preserve any symplectic structure on the complex projective space.

Given an exact relatively Pin Lagrangian embedding $Q \subset M$, we construct an $A^∞$ restriction functor from the wrapped Fukaya category of $M$ to the category of modules on the differential graded algebra of chains over the based loop space of $Q$. If $M$ is the cotangent bundle of $Q$, this functor induces an $A^∞$ equivalence between the wrapped Floer cohomology of a cotangent fibre and the chains over the based loop space of $Q$, extending a result proved by Abbondandolo and Schwarz at the level of homology.

Various Seiberg–Witten–Floer cohomologies are defined for a closed, oriented three-manifold; and if it is the mapping torus of an areapreserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg–Witten–Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.