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1. On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

   https://doi.org/10.1112/S0010437X22007886  

   Bai, Shaoyun; Côté, Laurent (March 2023, Compositio Mathematica)

   We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $$3$$ . As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $$3$$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy. 

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2. Homological mirror symmetry without correction

   https://doi.org/10.1090/jams/973  

   Abouzaid, Mohammed (October 2021, Journal of the American Mathematical Society)

   Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) \pi _2(Q) 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 topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras. 

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3. Tate’s thesis in the de Rham setting

   https://doi.org/10.1090/jams/1010  

   Hilburn, Justin; Raskin, Sam (July 2023, Journal of the American Mathematical Society)

   We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures. 

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4. Coherent Springer theory and the categorical Deligne-Langlands correspondence

   https://doi.org/10.1007/s00222-023-01224-2  

   Ben-Zvi, David; Chen, Harrison; Helm, David; Nadler, David (February 2024, Inventiones mathematicae)

   Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant$$K$$ $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$$F$$ $F$with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$$K$$ $K$-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, thecoherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of$$\mathrm{GL}_{n}(F)$$ ${\mathrm{GL}}_n\left(F\right)$into coherent sheaves on the stack of Langlands parameters. 

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5. Fukaya categories of surfaces, spherical objects and mapping class groups

   https://doi.org/10.1017/fms.2021.21  

   Auroux, Denis; Smith, Ivan (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $$2$$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $$1$$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included. 

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