Neckpinch
The shape of space,
one day at a time.
I'm teaching myself the mathematics behind Perelman's proof, in public, one day at a time. Every new paper as it lands, the ideas I'm walking toward, and a journal of what clicked and what didn't.
500papers indexed
9new this week
35topics
2upcoming events
93days learning
Papers
Everything new, as it lands.
Most of it is over my head today. I collect it anyway.
- Sep 23 Cohomogeneity one solutions of the IIB system
- Sep 23 On the singularity formation of gauge fields coupled to Ricci flow
- Sep 22 A continuous Positive Mass Theorem for perturbations of Euclidean space
Journal
What I learned today.
Back to basics: Tao and the Peano axioms
Sep 23, 2026 · 1 min
Read the latestEvents
Where they all get together.
Geometric Partial Differential Equations
Apr 25–30, 2027 · Banff, Canada
See calendarThe Path
12 stages. I'm on Stage 2.
Concepts
The ideas I'm walking toward.
Perelman's 𝓕-functional. Ricci flow is its gradient flow.
New on arXiv
Latest papers
Fresh off the press, whether or not I can read them yet.
A century-long story
From Poincaré to Perelman
A question asked in 1904, answered with a heat equation for geometry. This is the story that got me into all this.
Poincaré asks the question
In the fifth supplement to Analysis Situs, Henri Poincaré asks whether a closed 3-manifold with trivial fundamental group must be the 3-sphere.
Higher dimensions fall first
Stephen Smale proves the analogue in dimensions ≥ 5 using the h-cobordism theorem. Michael Freedman settles dimension 4 in 1982.
Thurston and Hamilton
Thurston announces the geometrization conjecture. Hamilton introduces Ricci flow and proves that 3-manifolds with positive Ricci curvature are spherical space forms.
Hamilton's program
Harnack inequalities, singularity analysis and the idea of surgery. Hamilton lays out a plan to prove geometrization with Ricci flow, but the singularities remain an obstacle.
A Millennium Problem
The Clay Mathematics Institute lists the Poincaré conjecture among its seven Millennium Prize Problems.
Perelman's three preprints
Grisha Perelman posts three papers to arXiv: the entropy formula, Ricci flow with surgery, and finite extinction time. They complete Hamilton's program.
Verification and the Fields Medal
Kleiner–Lott, Morgan–Tian and Cao–Zhu publish detailed expositions. Perelman is awarded the Fields Medal at ICM Madrid and declines it.
Beyond dimension three
Brendle and Schoen use Ricci flow to prove the differentiable sphere theorem, showing the method reaches well beyond 3-manifolds.
The Millennium Prize
Clay awards Perelman the Millennium Prize. He declines it as well.
Flows through singularities
Kleiner–Lott construct canonical singular Ricci flows. Bamler–Kleiner prove uniqueness and stability, Brendle classifies 3D ancient solutions, and Bamler develops a structure theory in all dimensions.
Richard Hamilton (1943–2024)
The inventor of the Ricci flow dies. His flow is now a standard tool of geometric analysis.
Your turn
Hundreds of new papers each year push the flow into higher dimensions, Kähler geometry, physics and beyond. The newest ones are on the Papers page.