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We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flows such as coupled mean curvature/Kaehler-Ricci flow in the sense of Smoczyk as a pair of self-intersection points is born or dies at a self-tangency, using results of Ekholm-Etnyre-Sullivan. This proves part of a conjecture of Joyce. We give a lower bound on the time for which the Floer cohomology is invariant under the (forward or backwards) flow, if it exists.
We show that cellular Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain
We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding $S^1$-equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicit
We use quilted Floer theory to construct functor-valued invariants of tangles arising from moduli spaces of flat bundles on punctured surfaces. As an application, we show the non-triviality of certain elements in the symplectic mapping class groups o
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow
In this paper, we write down a special Heegaard diagram for a given product three manifold $Sigma_gtimes S^1$. We use the diagram to compute its perturbed Heegaard Floer homology.