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1. An elementary alternative to ECH capacities

   https://doi.org/10.1073/pnas.2203090119  

   Hutchings, Michael (August 2022, Proceedings of the National Academy of Sciences)

   The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg–Witten theory. In this paper we define a sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff and Siegel [D. McDuff, K. Siegel, arXiv [Preprint] (2021)], giving a similar elementary alternative to symplectic capacities from rational symplectic field theory (SFT). 

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2. Symplectic capacities, unperturbed curves, and convex toric domains

   Dusa McDuff and Kyler Siegel (January 2023, Geometry topology)

   We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute these capacities for all four-dimensional convex toric domains. This gives various new obstructions to stabilized symplectic embedding problems which are sometimes sharp. 

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3. Mirror Symmetry for Truncated Cluster Varieties

   https://doi.org/10.3842/SIGMA.2022.055  

   Gammage, Benjamin; Le, Ian (July 2022, Symmetry, Integrability and Geometry: Methods and Applications)

   In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory which are known to symplectic geometers. 

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4. 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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5. Floer cohomology and flips

   https://doi.org/10.1090/memo/1372  

   Charest, François; Woodward, Chris (September 2022, Memoirs of the American Mathematical Society)

   We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program, analogous to the description of Bondal-Orlov ( Derived categories of coherent sheaves , 2002) and Kawamata ( Derived categories of toric varieties , 2006) of the bounded derived category of coherent sheaves on a compact complex manifold. 

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