08-12-2026, 12:12 PM
The Neurosurgery Resident Who Proved Crouzeix’s Conjecture
Summary
The article describes the apparent 2026 solution of Crouzeix’s conjecture, a major problem in matrix analysis that had remained open for more than 20 years. The conjecture states that for any complex matrix (A) and polynomial (p), the spectral norm of (p(A)) is at most twice the maximum value of (|p(z)|) over the matrix’s numerical range. The breakthrough came from Shanmu Jin, a neurosurgery resident and largely self-taught mathematician, whose proof emerged during an approximately 16-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode.
His approach reduced the difficult problem to a surprisingly simple positivity condition, and the proof was subsequently checked by experts including Michel Crouzeix; an independent proof by Emiel Lorist and Felix Schwenninger appeared shortly afterward. The authors view the episode as an important demonstration of AI's emerging ability to contribute genuinely new mathematical ideas and suggest that AI-assisted breakthroughs by researchers outside traditional mathematics may become increasingly common, potentially transforming the practice of applied mathematics.
ARTICLE
Summary
The article describes the apparent 2026 solution of Crouzeix’s conjecture, a major problem in matrix analysis that had remained open for more than 20 years. The conjecture states that for any complex matrix (A) and polynomial (p), the spectral norm of (p(A)) is at most twice the maximum value of (|p(z)|) over the matrix’s numerical range. The breakthrough came from Shanmu Jin, a neurosurgery resident and largely self-taught mathematician, whose proof emerged during an approximately 16-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode.
His approach reduced the difficult problem to a surprisingly simple positivity condition, and the proof was subsequently checked by experts including Michel Crouzeix; an independent proof by Emiel Lorist and Felix Schwenninger appeared shortly afterward. The authors view the episode as an important demonstration of AI's emerging ability to contribute genuinely new mathematical ideas and suggest that AI-assisted breakthroughs by researchers outside traditional mathematics may become increasingly common, potentially transforming the practice of applied mathematics.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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│ KONSTANTINOS MICHAILIDIS │
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