Course 1 · Path stage 1
Calculus & Linear Algebra
Fluency in linear algebra and calculus, taught as the tools they will become: derivatives as linear maps, eigenvalues, quadratic forms.
© 2026 NeckPinch · www.neckpinch.com · All rights reserved.
The Guidebook
From the axioms
to Perelman.
Twelve courses, one for each stage of the Path, and for each course a book written here: one continuous route from the natural numbers to the proof of the Poincaré conjecture. Every idea is built once, in full, and every chapter says where it came from and where it is going.
- Courses
- 12
- Books
- 17
- Chapters planned
- 139
- Chapters written
- 11
Course 2 · Path stage 2
Proofs & Real Analysis
Analysis rebuilt from the axioms: numbers, limits, continuity, derivatives, integrals, then distance, compactness, function spaces and differential equations.
Course 3 · Path stage 3
Measure & Integration
An integral that behaves under limits, complete function spaces, and the Gaussian integrals that run all the way to Perelman.
Course 4 · Path stage 4
Functional Analysis
Analysis in infinite dimensions: Banach and Hilbert spaces, weak compactness, spectra, and the Sobolev spaces of geometric PDE.
Course 5 · optional · Path stage 5
Complex Analysis
Optional. Holomorphic functions, conformal maps and uniformization: the two-dimensional ancestor of geometrization.
Course 6 · Path stage 6
PDE & the Heat Equation
The heat equation in depth: maximum principles, regularity, nonlinear existence and blow-up, the first geometric flows, and entropy.
Course 7 · Path stage 7
Topology & the Fundamental Group
Spaces up to deformation: the fundamental group, covering spaces, and the precise statement of the Poincaré conjecture.
Course 8 · Path stage 8
Smooth Manifolds
Manifolds, tangent bundles, flows, tensors and forms: the machinery Ricci flow is written in.
Course 9 · Path stage 9
Riemannian Geometry
Metrics, connections and curvature, then comparison, convergence and heat on manifolds: everything Ricci flow needs from geometry.
Course 10 · Path stage 10
3-Manifolds & Geometrization
What the answer should look like: decompositions, Thurston’s geometries and hyperbolic manifolds.
Course 11 · Path stage 11
Hamilton's Ricci Flow
Hamilton’s Ricci flow: existence, maximum principles, his 1982 theorem, solitons, compactness and singularities.
Course 12 · Path stage 12
Perelman & the Proof
Perelman’s entropy, noncollapsing, canonical neighbourhoods, surgery and finite extinction: the proof.
How to read the guidebook
Each book is written to be read beside a classic text. The book page names its companions, and each chapter says which part of them it accompanies. The guidebook doesn't replace those books. It joins them: it says what to read, what to skip because you've already met it, and what none of them covers.
Every idea has one home. It is built in full once, and later chapters recall it in a paragraph and link back. Every chapter opens where the previous one stopped and closes by naming the exact places the ideas return, often years later in the route.
Examples are real. They come from physics, engineering, computing and data. Each is labelled as a model (the mathematics is the governing law), in use (a technology relies on it), data (numbers you can check) or an analogy, and an analogy always says where it breaks.
Exercises are part of the text. Hints and solutions are folded away until you ask for them. Try first.
Your progress stays with you. Reading settings, where you stopped and which chapters you've finished are saved in your own browser, not on a server.
About the guidebook. Written with AI assistance for this site, at my request, and published as I study. Facts about the world (dates, figures, events) are checked against primary sources before a chapter is published. The mathematics follows the standard texts named in each chapter, but it has not yet been reviewed by a specialist. If you find a mistake, please write: corrections are always welcome.
© 2026 NeckPinch (www.neckpinch.com). All content in the guidebook (text, mathematics, figures and exercises) is protected by copyright. All rights reserved. No part may be copied, republished or redistributed without written permission.