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In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation…
We present a general procedure for constructing triangulated categories, linear over a field, with distinct enhancements. Some of our examples can be equipped with a (non-degenerate) t-structure, thereby showing that the existence of a…
We introduce the wrapped Donaldson-Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical…
We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in $(\mathbb{C}^*)^n$ (B-model) and their mirror toric Landau-Ginzburg A-models. The main technical ingredient of our construction is a…
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 a complex torus. Hypersurfaces with every Newton…
Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(\Sigma_A, \Lambda_A; \eta_A)$, where $\Sigma_A$ is an oriented smooth surface with…
Inspired by the homological mirror symmetry conjecture of Kontsevich, we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth Calabi-Yau variety.
A sequel to arXiv:1111.1460, this paper elaborates on some of the themes in the above paper. Connections to Symplectic Field Theory (SFT) and mirror symmetry are explored.
We construct the cyclic open--closed map for the big (i.e., bulk-deformed) relative Fukaya category, in the semipositive case, and show that it is a morphism of `polarized variations of semi-infinite Hodge structures'. We also give a…
We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface…
We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In…
We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any…
We prove that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial is equivalent to the triangulated category of singularities associated with the same polynomial together with a grading by an…
We construct the Fukaya category of a surface with genus greater than one and compute its Grothendieck group. We consider here a topological variant, in which we disregard the area form and use instead an admissibility condition borrowed…
Here we carefully construct an equivalence between the derived category of coherent sheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the most accessible case of homological mirror symmetry. We also…
This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of…
We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus. We show that the derived Fukaya category of this mirror coincides with the derived…
We construct a sheaf-theoretic analogue of the wrapped Fukaya category in Lagrangian Floer theory, by localizing a category of sheaves microsupported away from some given $\Lambda \subset S^*M$ along continuation maps constructed using the…
It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of…
This is an expository article on the A-side of Kontsevich's Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_\infty$-categories and their homological algebra, and later restrict to Fukaya categories, with…