We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in (C × ) n . Hypersurfaces with every Newton polytope can be obtained.Our proof has the following ingredients. Using Mikhalkin-Viro patchworking, we compute the skeleton of the hypersurface. The result matches the [FLTZ] skeleton and is naturally realized as a Legendrian in the cosphere bundle of a torus. By [GPS1, GPS2, GPS3], we trade wrapped Fukaya categories for microlocal sheaf theory. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties. loc 4. Microlocalizing Bondal's correspondence 4.1. Bondal's coherent-constructible correspondence 4.2. Restriction is mirror to microlocalization 4.3. Microlocalizing sheaves of categories of microlocal sheaves 4.4. At infinity 5. A glimpse in the mirror of birational toric geometry References

We prove the conjecture of Yankı Lekili and Kazushi Ueda on homological mirror symmetry for Milnor fibers of Berglund-Hübsch invertible polynomials. The proof proceeds as usual by calculating the "very affine" Fukaya category and then deforming it, relating the local categorical deformations to a calculation of David Nadler. The proof may be understood as a basic calculation in the deformation theory of perverse schobers.

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The proposed physical duality known as 3d mirror symmetry relates the geometries of dual pairs of holomorphic symplectic stacks. It has served in recent years as a guiding principle for developments in representation theory. However, due to the lack of definitions, thus far only small pieces of the subject have been mathematically accessible. In this paper, we formulate abelian 3d mirror symmetry as an equivalence between a pair of 2-categories constructed from the algebraic and symplectic geometry, respectively, of Gale dual toric cotangent stacks.In the simplest case, our theorem provides a spectral description of the 2-category of spherical functors -i.e., perverse schobers on the affine line with singularities at the origin.We expect that our results can be extended from toric cotangent stacks to hypertoric varieties, which would provide a categorification of previous results on Koszul duality for hypertoric categories O. Our methods also suggest approaches to 2-categorical 3d mirror symmetry for more general classes of spaces of interest in geometric representation theory.Along the way, we establish two results that may be of independent interest: (1) a version of the theory of Smith ideals in the setting of stable ∞-categories; and (2) an ambidexterity result for co/limits of presentable enriched ∞-categories over ∞-groupoids.

We prove a homological mirror symmetry equivalence between the A-brane category of the pair of pants, computed as a wrapped microlocal sheaf category, and the B-brane category of its mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the A-model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces.