How AI is Revolutionizing Mathematics: From High School Problems to Unsolved Mysteries

DARPA's push to accelerate mathematics with AI

The US Defense Advanced Research Projects Agency (DARPA) launched a new initiative called expMath (Exponentiating Mathematics) aimed at speeding up the progress in mathematics, a field fundamental to many real-world applications such as computer science, medicine, and national security. DARPA envisions AI as a coauthor helping mathematicians by breaking down complex problems into simpler parts to solve them faster.

Current AI capabilities in mathematics

While mathematicians have long used computers for calculations and verification, recent advances in AI, particularly large reasoning models (LRMs) like OpenAI's o3 and Anthropic’s Claude 4 Thinking, have shown promising results by achieving high scores on challenging math exams like the American Invitational Mathematics Examination (AIME). Hybrid models combining language models with fact-checking systems, such as Google DeepMind’s AlphaProof and AlphaEvolve, have made breakthroughs by solving unsolved problems and matching top human performances in prestigious competitions.

However, there remains a significant gap between AI solving high school level problems and tackling complex research challenges. Experts note that while AI can master problems with recognizable patterns, many research problems require explorative approaches beyond current AI capabilities.

The challenge of managing long sequences in problem solving

Mathematician Sergei Gukov highlights the difficulty in solving complex problems lies in finding very long sequences of logical steps, sometimes involving millions of moves, which is far beyond current AI's brute-force capabilities. To address this, researchers developed methods combining multiple steps into "supermoves" using reinforcement learning, dramatically shortening the path needed to tackle such problems. This approach has already yielded progress on longstanding puzzles like the Andrews-Curtis conjecture by ruling out previously proposed counterexamples.

AI and mathematical insight

Advanced mathematics often requires creative insight and experimentation rather than just step-by-step procedures. Tools like Google DeepMind’s AlphaEvolve use iterative trial and error, generating and refining solutions through multiple rounds, sometimes incorporating human input to guide the process.

Similarly, AI tools like PatternBoost, developed in collaboration with Meta, assist mathematicians by generating variations of mathematical ideas, supporting exploratory brainstorming essential for breakthroughs. For example, the icosahedron, a 20-sided 3D shape, was a mathematical object discovered by ancient Greek mathematicians purely through mental insight, illustrating the importance of creative thinking in math.

The future of AI in mathematics

Though AI is becoming a powerful assistant by identifying promising directions and eliminating dead ends, true mathematical breakthroughs still depend on human intuition and creativity. Machines excel at playing within known rules but have yet to invent entirely new concepts or approaches. As AI continues to improve, it may become an indispensable collaborator, but for now, the deepest insights remain a human domain.