How AI is Transforming Advanced Mathematics Research

Artificial intelligence is making remarkable strides in the realm of mathematics, moving beyond simple calculations to tackling problems that once seemed exclusive to human mathematicians. Over the past year, large language models (LLMs) and large reasoning models (LRMs) have demonstrated the ability to solve high school level and even International Math Olympiad problems, signaling a new era in AI-assisted mathematical research.

The US Defense Advanced Research Projects Agency (DARPA) recently launched the expMath initiative to accelerate mathematical progress by developing AI coauthors. These AI tools aim to break down complex problems into manageable parts, potentially solving challenges that have stumped humans for decades. Unlike traditional computational aids, these AI systems are designed to collaborate creatively with mathematicians, enhancing both speed and depth of research.

While early AI models struggled with mathematical accuracy, newer LRMs like OpenAI’s o3 and Anthropic’s Claude 4 Thinking have achieved impressive scores on challenging tests such as the American Invitational Mathematics Examination (AIME). Hybrid models that combine language understanding with fact-checking mechanisms, such as Google DeepMind’s AlphaProof and AlphaEvolve, have even outperformed top human competitors in prestigious contests and solved previously unsolved problems.

However, there remains a significant gap between AI solving pattern-based competition problems and addressing open-ended, research-level mathematical challenges. These unsolved problems often require exploring vast sequences of logical steps—sometimes millions long—far beyond the reach of current AI capabilities. Innovative approaches, such as reinforcement learning models that combine multiple steps into 'supermoves,' have shown promise by efficiently navigating these enormous problem spaces.

For example, researchers used this technique to make progress on the Andrews-Curtis conjecture, a 60-year-old unsolved problem, by disproving a long-standing proposed counterexample. This kind of AI-assisted exploration helps mathematicians avoid dead ends and focus their efforts more effectively, saving valuable time and resources.

Beyond step-by-step problem-solving, AI is also enhancing mathematical creativity. Tools like Google DeepMind’s AlphaEvolve and Meta’s PatternBoost enable iterative trial and error, generating new hypotheses and mathematical objects that can inspire human researchers. These AI systems act like advance scouts, uncovering promising avenues and patterns that might otherwise remain hidden.

Yet, despite these advances, AI lacks true intuition and the ability to invent fundamentally new concepts—the kind of breakthrough thinking that has historically driven major mathematical discoveries. As mathematician Geordie Williamson puts it, AI can now play the game of mathematics at a superhuman level but cannot yet invent the game itself.

In summary, AI is rapidly evolving from a computational assistant to a collaborative partner in mathematics. It excels at pattern recognition, managing complex sequences, and generating new ideas, helping mathematicians push the boundaries of knowledge. While the ultimate creative leaps remain human, AI tools are already transforming how math research is conducted, promising faster discoveries and new insights in the years ahead.