Explores how dynamic programming underlies diverse algorithms for shortest paths, neural network training, and grammar parsing, tracing its applications from automata and optimal control to reinforcement learning and beyond.

Explores the mathematical capabilities and limitations of large language models following their recent breakthroughs in major open problems, analyzing whether they excel particularly at finding counterexamples versus proofs.

An exploration of how humans solve Coverage Path Planning problems, using an interactive lawn-mowing experiment where tens of thousands of participants found near-optimal paths, compared to the mathematical challenge these routing problems pose for computers. The piece examines why humans are surprisingly efficient at spatial navigation tasks like mowing or vacuuming despite the astronomical number of possible route combinations.

From the article, a good description of algorithms and heuristics.

A slight detour, if we may. There are two main ways to tackle a problem like this. Simplifying things a bit, there are algorithms which guarantee the optimal path, and heuristics which use shortcuts to find a “good enough” path, fast.

Also, this domain is a keeper.

Domain expertise is the most important factor in effectively using LLMs, as demonstrated by Terence Tao's mathematically sophisticated ChatGPT conversation — skilled users can steer models more precisely, recognize flawed outputs, and suggest better approaches because they understand the subject matter deeply. Unlike generic prompting tips, this advantage cannot be replicated without genuine knowledge of the domain.

OpenAI's work on ten notable advances or breakthroughs in mathematics, likely highlighting contributions made by AI systems such as their models in solving or progressing on significant mathematical problems and conjectures.

Core pattern: "Run the inner Skill, record its failures, make a diff to improve it, repeat."

Authors: Tianyi Zhou, Dongrui Liu, Leitao Yuan, Jing Shao, Xia Hu

Architecture: - Multi-agent pipeline with orchestrator, workers, and reviewers operating in a DAG workflow - MCP tool servers: filesystem, git, bash, Lean REPL/LSP, mathlib - Supports multi-node execution via SLURM - Configurable LLM backends (Claude Opus 4.6, GPT, Gemini)

Scale (May 2026): - 26 books - 630,999 total lines of code (483,917 lines of Lean, excluding comments/blanks) - 46,203 declarations, 42,837 proved (92.7%) - 2,855 / 4,007 statements formalized (71.3%) - 183,157M tokens consumed

Key features: - Live presence for agents and humans in the same document - Comments, suggestions, and provenance tracking - Integrates via MCP/skill install with Claude Code, Codex, OpenClaw - Designed for pre-implementation work: scoping, planning, spec review - Free, no login required - Uses X-Agent-Id headers for agent identity in presence

Core idea: Recursive Reasoning Models (RRMs) use repeated computation to refine a persistent latent state rather than append new elements to an output sequence. This decouples reasoning depth from both parameter scale and output length: a compact model can perform many steps of internal computation by repeatedly applying shared transition functions