数理逻辑：2025–2026 年度综述

An annual survey of the math.LO arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

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

• Descriptive Set Theory and Borel Graph Complexity (20.9%) — Investigates the complexity and structural properties of Borel graphs, equivalence relations, and hyperfiniteness within descriptive set theory.
• Model Theory of Valued and Ordered Fields (13.4%) — Focuses on the model-theoretic properties of valued fields, henselian fields, and definable groups, including existential fragments and geometric applications.
• Foundations of Set and Dependent Type Theory (10.4%) — Explores foundational issues in intuitionistic set theory, dependent type theory, and the interplay between model theory and type-theoretic structures.
• Definable Groups and Topological Dynamics (10.1%) — Analyzes definable groups, particularly abelian and topological groups, and their connections to dynamics, small cancellation, and mereogeometry.
• Minimal Structures and Lattice Extensions (9.7%) — Studies minimal structures, weakly sparse structures, and extensions in varieties of hoops and lattices, focusing on automorphism groups and topologies.
• Semantics of Modal and Quantum Logics (9.2%) — Develops semantic frameworks for modal, intuitionistic, and quantum logics, including bisimulation invariance and relational semantics.
• Algebraic Semantics of Heyting and Boolean Algebras (8.0%) — Examines duality theories, free algebras, and categorical structures related to Heyting, Boolean, and MV-algebras.
• Proof Theory and Combinatorial Set Theory (7.8%) — Addresses proof complexity, Ramsey theory variants, and the strength of combinatorial theorems in the context of choice principles and computability.
• Completeness and Interpolation in Modal Logics (7.0%) — Investigates completeness, interpolation, and constructive properties of various modal logics, including causal and quantum modalities.
• Set Theory of Borel Sets and Entanglement (3.6%) — Explores properties of Borel sets, independent families, and entangled sets within specific set-theoretic models and ideal topologies.

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.