In brief
- AlphaEvolve, an AI that “evolves” code solutions, rediscovered and improved proofs for the finite-field Kakeya conjecture.
- Gemini Deep Think verified the logic, and AlphaProof formalized the result—creating a closed AI-research loop.
- The project shows how AI can now generate, test, and prove mathematical ideas, not just predict or summarize them.
In a striking example of how artificial intelligence is reshaping scientific research, Google DeepMind has teamed up with renowned mathematicians to harness AI tools for tackling some of mathematics’ toughest riddles.
The collaboration, announced this week, highlights a new AI system called AlphaEvolve that not only rediscovers known solutions, but also uncovers fresh insights into longstanding problems.
“Google DeepMind has been collaborating with Terence Tao and Javier Gómez-Serrano to use our AI agents (AlphaEvolve, AlphaProof, & Gemini Deep Think) for advancing math research,” Pushmeet Kohli, a computer scientist leading science and strategic initiatives at Google DeepMind, tweeted on Thursday. “They find that AlphaEvolve can help discover new results across a range of problems.”
Kohli cited a recent paper that outlined the breakthroughs, and pointed to a standout achievement: “As a compelling example, they used AlphaEvolve to discover a new construction for the finite field Kakeya conjecture; Gemini Deep Think then proved it correct and AlphaProof formalized that proof in Lean.”
He described it as “AI-powered math research in action!” Tao also detailed the findings in a blog post.
The Kakeya conjecture
The finite field Kakeya conjecture, first proven in 2008 by mathematician Zeev Dvir, deals with a deceptively simple question in abstract spaces known as finite fields—think of them as grids where numbers wrap around, like in modular arithmetic. The puzzle asks for the smallest set of points that can contain a full “line” in every possible direction without unnecessary overlaps. It’s like finding the most efficient way to draw arrows in all directions on a chessboard, without wasting squares.
In layman’s terms, it’s about packing and efficiency in mathematical spaces, with implications for fields like coding theory and signal processing. The new work doesn’t overturn the proof, but refines it with better constructions—essentially, smarter ways to build these sets that are smaller or more precise in certain dimensions.
The paper details how the AI system was tested on 67 diverse math problems from areas like geometry, combinatorics, and number theory.
“AlphaEvolve is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems,” the authors said in the abstract.
A Darwinian approach to AI-assisted math
At its heart, AlphaEvolve mimics biological evolution. It starts with basic computer programs generated by large language models and evaluates them against a problem’s criteria. Successful programs are “mutated” or tweaked to create variations, which are tested again in a loop. This allows the system to explore vast possibilities quickly, often spotting patterns humans might miss due to time constraints.
“The evolutionary process consists of two main components: (1) A Generator (LLM): This component is responsible for introducing variation… (2) An Evaluator (typically provided by the user): This is the ‘fitness function’,” the paper states.
For math problems, the evaluator might score how well a proposed set of points satisfies the Kakeya rules, favoring compact and efficient designs.
The results are impressive. The system “rediscovered the best known solutions in most of the cases and discovered improved solutions in several,” according to the abstract. In some cases, it even generalized findings from specific numbers to formulas that work universally.
These tweaks refine earlier bounds by tiny but meaningful amounts, like shaving off extra points in higher-dimensional grids.
Supercharging mathematicians
Tao, a Fields Medal-winning mathematician at UCLA, and Gómez-Serrano of Brown University, brought human expertise to guide and verify the AI’s outputs. The integration with other DeepMind tools—Gemini Deep Think for reasoning and AlphaProof for formal proofs in the Lean programming language—turned those raw discoveries into rigorous math.
The collaboration underscores a broader shift: AI is supercharging mathematicians.
“These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems,” the paper reads.
That could mean faster innovations in tech areas reliant on math, like cryptography or data compression. But it also raises questions about AI’s role in pure science—can machines truly “invent” or just optimize?
This latest effort suggests the field is just getting started.
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