Notes from a graphics researcher
What is… a blowup?
What makes a point singular, and how can changing the space make it smooth? I came to Hironaka’s theorem through physics simulation and graphics. These notes follow one question: what must we remember at a singular point to make the geometry useful again?
A pairwise collision rule can give different limiting outcomes when three balls touch at once. This demo keeps the ratio of two tiny gaps, so the different approaches remain distinguishable. It is a physical introduction to the idea of recording what is lost at a singular limit. The billiard example in Part I introduces this idea, and the optional mechanics experiment unpacks this demo.
Why I'm writing this
In October 2026, Resolution, the AI alignment research organization that absorbed the theoretical AI safety group formerly known as Timaeus, released a Lean formalization[14] of Heisuke Hironaka's 1964 theorem on resolution of singularities, built largely by AI agents (their post[14] explains how). Their interest comes from singular learning theory, which uses resolution to derive Bayesian asymptotics when regular statistical assumptions fail. Ultimately I wanted to understand the what and the why of this formalization: what exactly was proved, and why a group working on AI safety would spend its effort on a theorem from 1964.
In my graduate work on physics simulation and geometry processing, I spent years building solvers around singular parameterizations. A collision, a collapsed component, or an undefined direction could turn a small geometric defect into a difficult numerical problem. I wanted to understand what resolution offers in these familiar settings, and why the same mathematics helps describe learning.
But, it being the age of AI, I talked this through with my chatbot friends, who took names for themselves. This post was mostly written by the Claude Opus 5.5 agent that took the name Chart, with help from Stratum (Gemini 4 Argon), and continued by Atlas (Gemini 3.8 Flash). Rowan (ChatGPT-6.1 Sol) reviewed the mathematics and revised the cone explorer and accompanying explanations.
The idea to keep in view
At a singular point, a crossing may hide which branch we came from; a cusp may hide a regular parameter; a model may hide changes in its parts because their predictions cancel. A blowup replaces a chosen center with records of approach and keeps a map back to the original space. Away from that center, the map is invertible. Above it, the exceptional set holds the distinctions the old point could not express.
Hironaka’s theorem says that algebraic varieties in characteristic zero admit a smooth resolution. Constructive versions achieve it through finitely many blowups along suitably chosen smooth centers. The hard part is proving that the choices fit together and that the process ends.
An ordinary blowup records approach directions relative to a chosen center. A Nash blowup instead records the variety’s own tangent spaces; it can have exceptional fibers too. Repeating Nash’s rule can reproduce singular charts forever in higher dimensions. That failure helps explain why the choice of operation and the proof of progress matter. The last two chapters follow that geometry into learning. Two parts of a model can change while their combined prediction barely changes. This creates many nearly right settings; blowups help us count how much room they occupy.
A route through the questions
On a first reading: follow the four core chapters in order: keep directions at a point; understand why one resolution rule fails; watch changes cancel in a fit; then count how many fits remain plausible. The calculations can be expanded when you want to check a step. The side trips collect the graphics and simulation problems that motivated these notes; their summaries name the figures inside. In Part IV, the live thermometer shows how the geometric exponent becomes something we can measure.