辛几何：2025–2026 年度综述

An annual survey of the math.SG arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 108 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Hamiltonian Dynamics and Integrable Systems (22.7%) — Focuses on the dynamics of Hamiltonian systems, integrable structures, and their connections to quantum mechanics and thermodynamics.
• Symplectic and Lagrangian Manifold Geometry (12.7%) — Investigates foundational properties of symplectic and Lagrangian manifolds, including cohomology, reduction, and low-dimensional topology.
• Contact Geometry and Reeb Flows (11.3%) — Explores contact structures, Reeb dynamics, and their applications in dynamical systems and foliation theory.
• Poisson and Shifted Symplectic Structures (11.3%) — Examines Poisson geometry, Dirac structures, and higher categorical generalizations like shifted symplectic geometry.
• Floer Homology and Topological Invariants (9.9%) — Develops Floer-theoretic tools, including Heegaard Floer homology, to compute topological invariants and study symplectic dynamics.
• Fukaya Categories and Homological Mirror Symmetry (9.7%) — Studies Fukaya categories, wrapped Fukaya categories, and their role in homological mirror symmetry and quantum cohomology.
• Mirror Symmetry and Calabi-Yau Geometry (8.3%) — Addresses mirror symmetry, Calabi-Yau manifolds, and enumerative invariants such as Gromov-Witten and Donaldson-Thomas invariants.
• Legendrian Knots and Lagrangian Fillings (7.7%) — Analyzes Legendrian knots, Lagrangian fillings, and augmentation varieties in contact and symplectic topology.
• Convex Geometry and Quantization Methods (6.6%) — Applies convex geometry, uncertainty principles, and quantization techniques to symplectic embeddings and moduli spaces.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.