From first-year calculus to Perelman

The Path

This is the plan I'm actually following: 12 stages from calculus to the three preprints that proved the Poincaré conjecture. It will take years, and I'd rather say that out loud than pretend otherwise. If you're walking it too, tick the milestones as you go — they're saved in your browser, not on my server, so your progress is yours alone.

1 stage behind me · working through Stage 2: Proofs & Real Analysis now

This is my own reading plan, with a few changes I made after thinking it through: measure theory added before the PDE books, topology moved ahead of manifolds, complex analysis marked optional (it's needed for the Kähler branch, not for Poincaré), Taylor and Lieberman kept as references rather than cover-to-cover reading, and Hamilton's 1982 paper moved into the Ricci flow stage where I'll finally have the PDE to follow it.

0%0 of 54 milestones

Stage 1 · 8–16 weeks Finished

Calculus & Linear Algebra

The language everything else is written in. Derivatives as linear maps, eigenvalues, inner products and quadratic forms come up on nearly every page later.

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My notes from this stage

Stage 2 · 20–32 weeks I'm here now

Proofs & Real Analysis

Tao's two volumes, built from the ground up and written for self-study. Volume II's chapters on metric spaces, uniform convergence and several-variable calculus are the ones you'll lean on later.

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My notes from this stage

Stage 3 · 6–10 weeks

Measure & Integration

I added this stage myself: Evans and Brezis both assume Lebesgue measure and the convergence theorems from page one. A short stage, but skipping it makes the PDE books much harder than they need to be.

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Stage 4 · 10–16 weeks

Functional Analysis

Banach and Hilbert spaces, weak convergence, and above all Sobolev spaces — the setting in which every existence theorem for geometric PDE is stated.

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Stage 5 · optional · 8–12 weeks

Complex Analysis

Beautiful, and genuinely optional for the Poincaré route. You need it only for the Kähler–Ricci flow, which is a parallel branch of the subject. Treat it as a side track you can take any time — or skip until the flow leads you there.

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Stage 6 · 14–22 weeks

PDE & the Heat Equation

Ricci flow is a nonlinear heat equation, so this is the analytic engine of the whole subject. Read Evans properly; keep Taylor and Jost as references rather than cover-to-cover reading, or this stage will swallow a year.

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Stage 7 · 12–18 weeks

Topology & the Fundamental Group

I moved this earlier than I first planned: the Poincaré conjecture is a statement about π₁, so you want this before manifolds rather than after. Read Lee and the first chapters of Hatcher; Milnor's two short books are the best-written mathematics on this list.

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Stage 8 · 14–20 weeks

Smooth Manifolds

Charts, tangent bundles, vector fields, flows, tensors and forms. Lee is the standard, and the exercises are where the subject actually sinks in.

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Stage 9 · 16–24 weeks

Riemannian Geometry

Metrics, connections, geodesics, curvature — and the comparison theorems that let you turn curvature bounds into control on volume and distance. This is where the Ricci tensor finally means something.

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Stage 10 · 8–14 weeks

3-Manifolds & Geometrization

What the answer is supposed to look like: prime and JSJ decompositions, Seifert fibrations, hyperbolic geometry, and Thurston's eight geometries.

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Stage 11 · 16–28 weeks

Hamilton's Ricci Flow

Everything so far was preparation for this. Short-time existence, evolution of curvature, maximum principles for tensors, solitons, and Hamilton's 1982 theorem. His 1982 paper sits here rather than in the Riemannian stage: it needs the PDE theory you now have.

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Stage 12 · the target · open-ended weeks

Perelman & the Proof

The three preprints, read with an exposition open beside them. Nobody reads Perelman cold — the expositions exist precisely because the preprints compress years of work into 70 pages.

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