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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. Full ellipsoid embeddings and toric mutations

   https://doi.org/10.1007/s00029-022-00765-3  

   Casals, Roger; Vianna, Renato (July 2022, Selecta Mathematica)

   Abstract This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost toric varieties. The construction uniformly recovers the full sequences for the Fibonacci Staircase of McDuff–Schlenk, the Pell Staircase of Frenkel–Müller and the Cristofaro-Gardiner–Kleinman Staircase, and adds new infinite sequences of ellipsoid embeddings. In addition, we initiate the study of symplectic-tropical curves for almost toric fibrations and emphasize the connection to quiver combinatorics. 

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3. PLANAR MINIMAL SURFACES WITH POLYNOMIAL GROWTH IN THE Sp(4,R)-SYMMETRIC SPACE

   Tamburelli, Andrea; Wolf, Michael (January 2025, American journal of mathematics)

   We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the Sp(4,R)-symmetric space. We describe a homeomorphism between the “Hitchin component” of wild Sp(4,R)- Higgs bundles over CP1 with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in H2,2. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of R4. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in H2,2 associated to Sp(4,R)-Hitchin representations along rays of holomorphic quartic differentials. 

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4. Newton–Okounkov bodies and symplectic embeddings into nontoric rational surfaces

   https://doi.org/10.1112/jlms.12831  

   Chaidez, Julian; Wormleighton, Ben (January 2024, Journal of the London Mathematical Society)

   We develop new methods of both constructing and obstructing symplectic embeddings into nontoric rational surfaces using the theory of Newton–Okoukov bodies. Applications include sharp embedding results for concave toric domains into nontoric rational surfaces, and new cases of nonexistence for infinite staircases in the nontoric setting. 

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5. Block perturbation of symplectic matrices in Williamson’s theorem

   Babu, Gajendra; Mishra, Hemant K (August 2023, Canadian Mathematical Bulletin)

   Williamson's theorem states that for any 2n×2n real positive definite matrix A, there exists a 2n×2n real symplectic matrix S such that STAS=D⊕D, where D is an n×n diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of A. Let H be any 2n×2n real symmetric matrix such that the perturbed matrix A+H is also positive definite. In this paper, we show that any symplectic matrix S̃ diagonalizing A+H in Williamson's theorem is of the form S̃ =SQ+(‖H‖), where Q is a 2n×2n real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that S̃ and S can be chosen so that ‖S̃ −S‖=(‖H‖). Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45-58, 2017]. 

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