- Professor Yang-Hui He discusses the rapidly evolving role of AI in pure mathematics and theoretical physics, focusing on a new paradigm where AI assists in genuine research-level discovery. The conversation centers on his “murmuration conjecture,” a pattern in number theory found through machine learning, and how AI is transforming the way conjectures are formulated and proofs are constructed.
The Changing Landscape of AI in Research
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AI is now solving problems at the research level, not just undergraduate exercises.
- DeepMind’s AlphaGeometry2 and AlphaProof are benchmarked on increasingly difficult problems, with AlphaGeometry2 reaching silver-medal Olympiad level and AlphaProof approaching college-level theorem proving.
- Epoch AI’s FrontierMath project has defined Tier 1–4 problems, where Tier 1–3 are graduate-level and Tier 4 are active research-level problems. As of December 2024, AI could solve about 2% of Tier 1–3 problems, but by March 2025, performance reached 10–25% on those tiers, with Tier 4 benchmarking underway.
- He is flying to Berkeley to help benchmark Tier 4 problems, which are so difficult that even humans would struggle to solve them.
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Three broad categories of AI-assisted mathematical research are identified:
- Top-down (intuition-guided): Pattern recognition in mathematical data, often visualized as image processing. This is how conjectures like the murmuration phenomenon are discovered.
- Bottom-up (formal): Automated proof assistants like Lean, which formalize and verify proofs. This requires large corpora of formalized mathematics, which is still limited.
- Meta-mathematics (LLM-assisted): Using large language models as research copilots—summarizing literature, suggesting proof strategies, and connecting disparate fields. Examples include Terence Tao using LLMs to assist in proofs and Madhu Das using ChatGPT to help complete a technical lemma.
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The role of the future academic may shift from “doer” to “director” or “curator.”
- Just as professional mathematicians no longer compute tedious integrals by hand (outsourced to Mathematica or SageMath), in 5–10 years, routine parts of proofs or derivations may be delegated to AI.
- The human’s role will be to decide what problems to tackle, interpret results, and guide the overall research direction.
The Murmuration Conjecture: A Case Study in AI-Guided Discovery
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The murmuration conjecture emerged from a collaboration between Yang-Hui He, Kyuhwan Lee, Thomas Oliver, and Alexey Pozdnyakov, later joined by Andrew Sutherland.
- The name “murmuration” was inspired by the visual resemblance of the discovered patterns to the flocking behavior of starlings.
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The problem began with the Birch and Swinnerton-Dyer (BSD) conjecture, one of the Millennium Prize Problems.
- The BSD conjecture relates the rank of an elliptic curve (the number of independent infinite families of rational solutions) to the behavior of its associated L-function at a specific point.
- Computing the rank directly from the equation of an elliptic curve is extremely difficult; the BSD conjecture provides an indirect analytic method via the L-function.
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Initial machine learning experiments using the Weierstrass form of elliptic curves (parameterized by coefficients g2 and g4) failed to predict rank.
- The Weierstrass form is natural for algebraic geometers but does not capture the arithmetic structure relevant to rank.
- A null result was obtained despite using millions of curves and advanced techniques.
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The breakthrough came when number theory experts suggested using Euler coefficients (ap coefficients) instead.
- For each prime p, the Euler coefficient ap measures the deviation between the number of solutions to the elliptic curve modulo p and the expected value p + 1.
- These coefficients encode deep arithmetic information and are directly related to the L-function in the BSD conjecture.
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Using the first 100 Euler coefficients as input features, a simple neural network or PCA classifier predicted rank with nearly 100% accuracy.
- This was a dramatic improvement over the Weierstrass-based approach, which had failed completely.
- The high accuracy suggested that the neural network was capturing something fundamental about the BSD conjecture.
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To interpret what the AI was doing, the team performed principal component analysis (PCA) and examined the projection matrices.
- They found that the PCA was effectively computing a “vertical average”: for a fixed prime p, averaging the Euler coefficients across many elliptic curves ordered by conductor.
- This averaging procedure was unusual—traditional methods average over primes for a fixed curve, not over curves for a fixed prime.
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Plotting these vertical averages against primes revealed striking oscillatory patterns, dubbed “murmurations.”
- Elliptic curves of rank 0 showed one type of oscillation; rank 1 showed a different pattern; higher ranks followed accordingly.
- Even and odd parity ranks could be distinguished by their oscillation patterns.
- The patterns were so clear that they could be used to predict rank visually.
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The murmuration phenomenon surprised experts like Peter Sarnak, who initially expected the result to be trivial but found it novel and significant.
- Sarnak noted that while the conjecture could have been formulated by someone like Bertrand Swinton-Dyer, it was never considered because the vertical averaging procedure is counterintuitive.
- AI, lacking preconceptions, simply spotted the pattern in the data.
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The conjecture has since been proven for Dirichlet characters and weight-2 modular forms.
- Nina Zubrilina and Alex Cohen provided proofs in 2023–2024.
- The phenomenon is now understood as a generalization of Chebyshev’s bias (a known imbalance in the distribution of primes modulo 4) to all L-functions in the Langlands program.
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The murmuration conjecture passes two key criteria of the “Birch test” (a proposed benchmark for AI-guided discovery):
- Interpretability (I): The pattern is described by a precise mathematical formula.
- Novelty (N): It galvanized a new field of study, with conferences and workshops dedicated to it.
- It fails the autonomy (A) criterion because human expertise was essential in choosing the right representation and interpreting results.
The Broader Impact on Mathematics and Science
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The murmuration conjecture exemplifies a new mode of scientific discovery: AI identifies patterns that humans then interpret and formalize.
- This complements traditional top-down intuition (as in the work of Gauss and Riemann) and bottom-up formalization (as in Lean-based proof assistants).
- The interplay between human and machine is crucial—AI excels at pattern recognition, while humans provide context, interpretation, and proof.
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The future of mathematical research may involve a closed loop:
- Literature is processed by LLMs to identify open problems and connections.
- AI formulates conjectures from data (top-down).
- Conjectures are auto-formalized into systems like Lean.
- Proofs are generated or assisted by AI (bottom-up).
- Humans interpret results and feed them back into the literature.
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This loop is not yet fully automated, but each component is advancing rapidly.
- Auto-formalization is limited by the lack of large-scale formalized mathematical corpora (only millions of lines of Lean exist, billions are needed).
- Proof generation is improving, with LLMs like ChatGPT already capable of producing proof strategies that humans can verify and refine.
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The pace of progress is exponential, with major breakthroughs occurring every few months.
- He emphasizes that we are entering a “brave new world” of discovery, where AI enables humans to tackle problems that were previously intractable.
- The combination of human intuition and machine pattern recognition is transforming not just mathematics but all of theoretical science.