Curios

A collection of links to things I find interesting — math, geometry, computation, and whatever else catches my eye.

• Resting Bodies

  "This paper explores the analysis and design of the resting configurations of a rigid body, without the use of physical simulation. In particular, given a rigid body in R³, we identify all possible stationary points, as well as the probability that the body will stop at these points, assuming a random initial orientation and negligible momentum. The forward version of our method can hence be used to automatically orient models, to provide feedback about object stability during the design process, and to furnish plausible distributions of shape orientation for natural scene modeling. Moreover, a differentiable inverse version of our method lets us design shapes with target resting behavior, such as dice with target, nonuniform probabilities. Here we find solutions that would be nearly impossible to find using classical techniques, such as dice with additional unstable faces that provide more natural overall geometry."

• Remarks on the Disproof of the Erdős Unit Distance Conjecture — Alon, Bloom, Gowers, et al.

  "Erdős's unit distance conjecture, posed in 1946, asked whether n points in the plane can span at most n^{1+o(1)} unit distances — a bound that resisted proof or disproof for nearly eighty years. In 2026, an internal model at OpenAI found a counterexample, constructing point sets with at least n^{1+ε} unit distances for some fixed ε > 0. The construction is an unexpected application of algebraic number theory: by embedding rings of integers from CM number fields of increasing degree — drawn from infinite class field towers of Golod-Shafarevich type, which have bounded root discriminant — into the plane via the Minkowski embedding, one obtains point sets where primes that split completely contribute exponentially many unit-norm differences, overwhelming the class number. This paper is a human-digested and verified account of the AI proof, accompanied by reflections from nine prominent mathematicians. Gowers writes: “The breakthrough of Larry Guth and Nets Katz that solved the closely related Erdős distance problem introduced new and unexpected tools into combinatorial geometry that led to the solutions of many further problems. However, the unit distance conjecture continued to resist (for reasons we now understand!), so appeared to require a further major idea.”"

• Ricci Flow, Entropy and Optimal Transportation — McCann & Topping

  "McCann and Topping prove that heat-diffused probability measures on a compact Riemannian manifold have non-increasing Wasserstein-2 distance precisely when the metric's time derivative is bounded above by twice the Ricci tensor — the defining condition of reverse Ricci flow. Put another way: if a manifold is expanding yet optimal transport distances between heat-evolved measures are still shrinking, that expansion is exactly constrained by Ricci flow. What makes this result so appealing is the clarity with which it exposes the coupling between the two flows — a condition posed entirely in the language of the heat equation, measured by optimal transport, turns out to characterize one of the central objects in geometric analysis."