The μ-constant problem for surface singularities at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The $\mu$-constant problem for surface singularities

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	shapes made of equations	Levels:	1
Category:	Algebraic and complex geometry	Lean version:	not yet
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Topological triviality of μ-constant surface singularities. Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.

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LEVEL 1 Topological triviality of mu-constant families of surface singularities

released 2026-09-24 | 2 theorems · 17 lemmas · 32 proofs · 19,201 words | PLAY LEVEL 1 » (pdf)

We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.

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