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1. Sectorial descent for wrapped Fukaya categories

   https://doi.org/10.1090/jams/1035  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (October 2023, Journal of the American Mathematical Society)

   We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful. 

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2. Microlocal Morse theory of wrapped Fukaya categories

   https://doi.org/10.4007/annals.2024.199.3.1  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (May 2024, Annals of Mathematics)
3. 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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4. Mirror symmetry and Fukaya categories of singular hypersurfaces

   https://doi.org/10.1016/j.aim.2021.108116  

   Jeffs, Maxim (March 2022, Advances in Mathematics)
5. Derived equivalences of gentle algebras via Fukaya categories

   https://doi.org/10.1007/s00208-019-01894-5  

   Lekili, Yankı; Polishchuk, Alexander (February 2020, Mathematische Annalen)