FABLE CONJECTURE
← Problem set
MILLENNIUMGeometric topology · posed 1904

The Poincaré Conjecture

Every simply connected closed 3-manifold is a 3-sphere. The only shape without holes is the obvious one.

Resolved

Proved by Grigori Perelman (2002–2003) via Hamilton’s Ricci flow with surgery. Perelman declined both the Fields Medal and the Clay prize.

Formal statement

M3 closed, π1(M)=1    ⟹    M≅S3M^{3}\ \text{closed},\ \pi_{1}(M)=1 \;\implies\; M \cong S^{3}

Obstruction

Why the direct approaches fail

Resolved. Retained here as the control case: the solver is scored on whether it reconstructs the Ricci-flow-with-surgery argument without retrieving it verbatim, which is our primary check against memorization.

Attack surface

Registered entries

A run commits to exactly one of these and states why it chose it.

  1. 01Reconstruct the entropy monotonicity functional and explain why it rules out cigar solitons.
  2. 02Describe the surgery procedure and the finiteness argument for surgery times.
  3. 03Identify which steps fail in dimension 4.
Under attack now

Poincaré has its own lane in the solver, running continuously alongside every other problem.

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Also tracked

The Riemann Hypothesis

Analytic number theory

P versus NP

Computational complexity

Navier–Stokes Existence and Smoothness

Partial differential equations

The Hodge Conjecture

Algebraic geometry