In July 2022, Maryna Viazovska became the first woman to receive the prestigious Fields Medal. This recognition is frequently described as the Nobel Prize for mathematics, underscoring just how remarkable her accomplishments have been in this arena. This particular accolade is contemporaneously and especially poignant. She has made history by becoming only the second woman to receive this award in its 86 years. Viazovska’s pioneering achievement demonstrated that E8 lattice is simply the best way to pack spheres in eight dimensions. She collaborated with other mathematicians to demonstrate the existence of a major new principle. Collectively, they proved that the Leech lattice is the optimal packing in 24 dimensions.
The collaboration between Viazovska and her peers has not only advanced the field of mathematics but has opened new avenues for artificial intelligence (AI) to assist in mathematical research. Sidharth Hariharan represents this synergy beautifully with his pioneering research. He first encountered Viazovska at work formalizing sphere-packing proofs as a third-year undergraduate in Lausanne, Switzerland. Now, he’s a first-year Ph.D. student at Carnegie Mellon University. He’s digging just as deeply in his research, going laser-focused on the role of AI in mathematics.
The Role of AI in Mathematical Proof Verification
AI is already having a transformative impact on formal verification processes. One, particularly successful AI system known as Gauss has been especially effective in helping researchers to solve difficult, complex mathematical problems. Gauss, as reasoning agent, is meant to bridge the gap between conventional natural-language reasoning and completely formalized reasoning.
According to Jesse Han, one of the developers involved with Gauss, “It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural-language reasoning and fully formalized reasoning.” This unique methodology gives Gauss the ability to perform in-depth literature searches. Moreover, it allows him to write Lean code, document his code, and help out with using other verification tools.
Gauss’s remarkable efficiency is evident in its ability to autoformalize Viazovska’s 24-dimensional sphere-packing proof within just two weeks. This proof is over 200,000 lines of code deep, demonstrating the complexity and depth of these mathematical concepts. This productive collaboration between Gauss and human mathematicians is a sign of a major breakthrough in AI’s potential to support cutting-edge math research.
Collaboration Between Humans and AI
The collaboration that developed between researchers and AI systems such as Gauss has already yielded strides in computing formal verification. Hariharan pointed out that while their eventual project repository was developed over 15 months, they didn’t open things up to the public until June 2025. After this milestone, they received communication from Math, Inc., which indicated that they had successfully proven 30 intermediate facts necessary for their work.
Now, doing 30 ‘sorrys’ is not enough. This meant they won definitively on 30 disputed facts that we needed to have shown to be true, as Hariharan further explained. The featured collaborative effort is a testament to the necessity of interdisciplinary teamwork in today’s mathematical research. One of these interactions even resulted in us catching a typo in the project, which we were then able to quickly address and correct.
This verification process is a great example of how AI can augment rather than replace humans. The team hit a key research breakthrough right around mid-January that led to a much more enhanced version of Gauss. This remarkable achievement is evidence of the rapid evolution possible through collaboration between human mathematicians and AI systems.
The Future of Mathematics and AI Integration
If mathematicians are willing to engage with the opportunities and challenges of this new technology, the future is bright. Given how quickly these AI technologies have come about, we can only expect AI’s presence in mathematical research to expand. These discoveries in turn foster the development of more efficient solutions to practical problems and creative approaches to age-old mathematical conundrums.
Jesse Han remarked on the evolving nature of programming in relation to AI: “A programmer used to be someone who punched holes into cards, but then the act of programming became separated from whatever material substrate was used for recording programs.” The purpose of this analogy is to illustrate how AI is revolutionizing the craft of mathematics. It’s not just impacting how we learn the material itself.