数学物理：2025–2026 年度综述

An annual survey of the math-ph arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 144 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):

• Lie Algebras and Integrable Hierarchies (15.4%) — Explores the algebraic structures of Lie algebras, higher-spin fields, and integrable systems, focusing on symmetries, contractions, and exact solutions in mathematical physics.
• Nonlinear PDEs and Wave Dynamics (10.7%) — Investigates the stability, scattering, and exact solutions of nonlinear partial differential equations, including wave equations and time-dependent problems in various physical contexts.
• Stochastic Fluids and Differential Equations (8.9%) — Analyzes stochastic differential equations and fluid dynamics, particularly Navier-Stokes systems, focusing on soliton solutions, geometric flows, and entropy maximization in quasi-free states.
• Ising Models and Statistical Mechanics (8.8%) — Studies critical phenomena and thermodynamic limits in Ising models and related spin systems, addressing duality, external fields, and continuum modeling in large-scale networks.
• Non-Hermitian Random Matrix Theory (8.7%) — Examines spectral properties and universality classes in non-Hermitian random matrix ensembles, with applications to Dirac materials, polarized light transport, and graph theory.
• Quantum Information and Entanglement (8.6%) — Focuses on quantum mechanics foundations, entanglement, and quantum information channels, including symmetry breaking, reference systems, and geometric thermodynamics in quantum regimes.
• Gauge and Quantum Field Theories (8.2%) — Develops mathematical frameworks for gauge theories and quantum field theories, covering topological aspects, scalar fields, Kaluza-Klein extensions, and black-hole thermodynamics.
• Integrable Hamiltonian Systems (7.9%) — Explores integrable systems and Hamiltonian formalisms, including spin chains, Calogero-Moser systems, and topological recursion, with emphasis on symmetries and conserved charges.
• Phase Space and Algebraic Structures (7.5%) — Investigates phase-space quantum mechanics, hidden algebras, and phase transitions, utilizing distribution algebras and Hilbert space structures to analyze superintegrable systems and flux quantization.
• Noncommutative and Lorentzian Geometry (5.8%) — Applies spectral geometry and noncommutative methods to Lorentzian manifolds, addressing Ricci flow, black hole uniqueness, and Carrollian geometry in relativistic contexts.
• Schrödinger Operators and Potentials (5.1%) — Analyzes spectral properties and nonlinear dynamics of Schrödinger operators with complex or decaying potentials, including Lieb-Thirring inequalities and rogue wave statistics.
• Classical and Quantum Dynamics (4.5%) — Studies classical and quantum dynamical systems, including chaotic pendulums, relativistic bodies, and optimal transport, focusing on stochastic dissipation and non-equilibrium thermodynamics.

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.