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Description
An annual survey of the math.MG 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):
- Ricci Curvature Bounds and Manifold Regularity (21.1%) — Investigates quantitative lower bounds on Ricci curvature, their implications for manifold regularity, completeness, and discrete-to-continuous approximations.
- Coarse Geometry and Quasi-Isometric Rigidity (12.2%) — Explores large-scale geometric properties of groups and graphs, focusing on hyperbolicity, quasi-isometric rigidity, and boundary structures.
- Metric Measure Spaces and Convergence Theory (11.8%) — Studies the geometry of metric measure spaces, including convergence notions, magnitude, and applications to Lorentzian and voting geometries.
- Self-Similar Sets and Fractal Geometry (11.3%) — Analyzes geometric properties of self-similar sets, fractals, and convex sets, with a focus on perimeter, area, and Kakeya-type problems.
- Brunn-Minkowski and Isoperimetric Inequalities (8.8%) — Develops sharp inequalities related to Brunn-Minkowski, isoperimetry, and log-concave measures, including stability and dual formulations.
- Convex Bodies and Geometric Inequalities (8.5%) — Examines sections, centroids, and inequalities for convex bodies, including spectral geometry and non-Euclidean equilibria.
- Minkowski Problem and Covering Conjectures (8.0%) — Addresses uniqueness and existence in the Minkowski problem, alongside classical covering and minimal area conjectures like Rado's and Erdős's.
- Gromov-Hausdorff Distance and Metric Structures (6.7%) — Focuses on distance metrics between spaces, including Gromov-Hausdorff and Wasserstein distances, and their applications in reconstruction and geometry.
- Discrete Geometry and Distinct Distances (5.9%) — Investigates combinatorial geometry problems such as distinct distances on curves, equilateral sets, and counterexamples to geometric conjectures.
- Synthetic Geometry and Stability Theorems (5.8%) — Establishes synthetic geometric theorems, stability results, and quantitative bounds in spherical and non-positively curved 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.
