Abstract. We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY 3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.

We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspace Sm of the Lie algebra sl2m(C) which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submanifold L℘ ± of a fibre Ym,t 0 of this fibre bundle; we regard the braid β as a symplectic automorphism of the fibre, and apply Lagrangian Floer cohomology to L℘ ± and β(L℘ ± ) inside Ym,t 0 . The main theorem asserts that this group is invariant under the Markov moves, hence defines an oriented link invariant. We conjecture that this invariant co-incides with Khovanov's combinatorially defined link homology theory, after collapsing the bigrading of the latter to a single grading.

Let X be a closed symplectic manifold equipped a Lagrangian torus fibration. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y , which can be considered as a variant of the T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of X embeds fully faithfully in the derived category of coherent sheaves on Y , under the technical assumption that π 2 (X) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topologised infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.

Abstract. We build the wrapped Fukaya category W(E) for any monotone symplectic manifold E, convex at infinity. We define the open-closed and closed open-string maps, OC : HH * (W(E)) → SH * (E) and CO : SH * (E) → HH * (W(E)). We study their algebraic properties and prove that the string maps are compatible with the c 1 (T E)-eigenvalue splitting of W(E). We extend Abouzaid's generation criterion from the exact to the monotone setting. We construct an acceleration functor AF : F (E) → W(E) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map c * : QH * (E) → SH * (E). We define the SH * (E)-module structure on the Hochschild (co)homology of W(E) which is compatible with the string maps (this was proved independently for exact convex symplectic manifolds by Ganatra). The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category F (E), and also hold for closed monotone symplectic manifolds.As an application, we show that the wrapped category of O(−k) → CP m is proper (cohomologically finite) for 1 ≤ k ≤ m. For any monotone negative line bundle E over a closed monotone toric manifold B, we show that SH * (E) = 0, W(E) is non-trivial, and E contains a non-displaceable monotone Lagrangian torus L on which OC is non-zero.

Abstract. We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M . Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.

We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more Calabi-Yau manifolds (e.g. rigid ones) than previously thought or there exist "symplectic Calabi-Yaus" -non-Kähler symplectic 6-folds with c1 = 0. The analogous surgery in four dimensions, with a generalisation to ADE-trees of Lagrangians, implies that the canonical class of a minimal complex surface contains symplectic forms if and only if it has positive square.

Abstract. We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach.

Abstract. We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology, over fields of characteristic zero. The key ingredient is the construction of a degree one Hochschild cohomology class on a Floer A∞-algebra associated to the (k, k)-nilpotent slice Y k , obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactificationȲ k . The spaceȲ k is obtained as the Hilbert scheme of a partial compactification of the A 2k−1 -Milnor fibre. A sequel to this paper will prove formality of the symplectic cup and cap bimodules, and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.