Terence Tao

Terence Tao Uses GPT-5 Pro to Solve a 3-Year Unsolved Problem in 11 Minutes

Honghao Wang

— 5 min read
Terence Tao Uses GPT-5 Pro to Solve a 3-Year Unsolved Problem in 11 Minutes

Terence Tao and GPT‑5 Pro Tackle a 3‑Year‑Old Problem

Terence Tao and GPT‑5 Pro recently teamed up to tackle a mathematics problem that had remained unsolved for three years — in an area outside Tao’s primary expertise: differential geometry.

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While Tao’s strengths lie in analysis, number theory, and combinatorics, differential geometry focuses on the properties of manifolds and relies on very different tools. Tao began with a rough intuitive approach, while GPT‑5 Pro performed complex calculations and provided rigorous proofs, allowing Tao to capture key logical steps and explore unconventional methods.

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Interestingly, AI helped Tao realize that his initial intuition was wrong — but the process still expanded his understanding of the problem.

> Tao’s summary on AI’s role:

> “AI is very useful at small scales, somewhat unhelpful at medium scales, and helpful again at large scales.”

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From Calculation to Proof — With AI

The Problem (from mathoverflow, posed 3 years ago):

> Given a smooth topological sphere in 3D space enclosing a region with the absolute value of its principal curvatures not exceeding 1 — is its enclosed volume at least that of the unit sphere?

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Tao’s Initial Plan

  • Restrict the problem to star‑shaped regions.
  • Push forward using integral inequalities.
  • Use AI for calculations due to his rusty differential geometry skills.
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GPT‑5 Pro’s Performance:

  • Time taken: 11 minutes 18 seconds.
  • Completed all calculations.
  • Provided a full proof for the star‑shaped case.

Tools and Methods Used:

  • Known results like Stokes’ theorem, Willmore inequality.
  • New to Tao: Minkowski’s first integral formula.
  • Combined with the arithmetic–geometric mean inequality, yielding a streamlined proof.
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Verifying the Proof

Tao struggled to find references online and asked AI again.

  • Two satisfactory proofs resulted:
  • One via the divergence theorem (Tao’s suggestion).
  • Another based on a flow method (new to Tao).

AI also worked under a perturbed elliptic PDE approach, though with a minor estimation error.

  • Small data case: Solved — when the surface is close to a unit sphere.
  • Large data case: Still unsolved — vastly different shapes remain challenging.
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Outcome of the Attempt

Tao concluded:

  • Numerical methods may solve many special cases in finite time.
  • General case remains out of reach.
  • AI followed instructions literally, even if the approach wasn’t ideal.
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Lessons Learned — AI Helps Understanding Even Without Full Solutions

While the problem wasn’t completely solved:

  • AI accelerated exploration.
  • Helped test and discard unsuitable methods early.
  • Expanded Tao’s perspective on AI’s role in research.
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Tao’s Multi‑Scale Evaluation of AI

AI’s usefulness varies by scale:

  • Small scale: Minor errors, frequent useful ideas.
  • Medium scale: Often reinforced incorrect intuition.
  • Large scale: Helpful in broader strategic sense.

Four Scales of Formal Proof Work (Tao’s framework):

  • Single line of a formal proof.
  • Individual lemma.
  • Full theorem.
  • Entire textbook.
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> Optimal automation: Between 0% and 100%. Enough to cut repetitive work, but with humans in the loop to keep awareness at all scales.

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Tao’s AI Collaboration History

Early ChatGPT Era (≈ 3 years ago)

  • First tested AI on the Riemann Hypothesis — results looked convincing but lacked depth.
  • Concluded early AI produced “surface‑level” outputs needing line‑by‑line human verification.
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GPT‑4 Arrives — First Major Benefit

  • Used GPT‑4 for statistical data processing, completing in minutes what would take a day.
  • Began using it for LaTeX compilation fixes.

> “Today was the first day GPT‑4 saved me a huge amount of tedious work.”

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GPT‑4 in Mathematical Proofs

  • Suggested eight approaches for a recursion problem — generating functions led to a quick solution.
  • Wrote Python code for sequence length computations — creative but required human correction.

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Copilot for Paper Writing

  • Worked for repetitive text, but proof outlines wandered off‑topic.
  • Conclusion: AI writing in mathematics is still more interesting than useful.
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GPT‑o1 — Less Hallucination, More Formalism

  • Compared to predecessors, much improved at formal tasks.
  • Still mediocre at conceptual problem‑solving.
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AI as Numerical Search Assistant

A recent example:

  • Problem: Need numerical parameters to test an inequality.
  • Approach: Step‑by‑step interactive calculations with AI.
  • Verified outputs independently in Python.

Results:

  • Clear time savings.
  • Explored solutions he wouldn’t have considered without AI.
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> Tao’s belief: AI’s best role is as a copilot, not a replacement — aiding experiments, exploration, and scaling of research.

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Broader Implications

The Tao–AI collaboration reflects patterns applicable beyond mathematics:

  • AI accelerates exploration and repetitive work.
  • Human oversight maintains problem‑scale awareness.
  • Open‑source platforms like AiToEarn官网 integrate content generation, multi‑platform publishing, analytics, and monetization, enabling researchers to share and scale their AI‑assisted work efficiently.

For deeper integration of AI into mathematical or creative workflows, such platforms provide:

  • Cross‑disciplinary reach (Douyin, Bilibili, YouTube, X, etc.).
  • Seamless links between generation → review → distribution.

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Sources:

  • https://chatgpt.com/share/68e85cba-7228-800e-8804-a0f41aa64e14
  • https://mathoverflow.net/questions/425509/sphere-with-bounded-curvature
  • https://terrytao.wordpress.com/mastodon-posts/

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If you’d like, I can also create a clear timeline chart of Tao’s AI experiments so the progression from ChatGPT to GPT‑5 Pro is instantly visual. Would you like me to add that?

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