Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that include both toric manifolds and the K3 surface. We classify closed almost toric four-manifolds up to diffeomorphism and indicate precisely the structure of all almost toric fibrations of closed symplectic four-manifolds. A key step in the proof is a geometric classification of the singular integral affine structures that can occur on the base of an almost toric fibration of a closed four-manifold. As a byproduct we provide a geometric explanation for why a generic Lagrangian fibration over the two-sphere must have 24 singular fibers.

We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.

This paper studies a notion of enumerative invariants for stable A-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable A-branes and their counts is provided by the string theoretic origin of the topological A-model. This is the Witten index of the supersymmetric quantum mechanics of a single D3 brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the A-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the A-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of B-branes related by homological mirror symmetry.

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

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