Maryna Viazovska is a Ukrainian mathematician whose achievements are nothing short of extraordinary. Coxeter thought she was wonderful because she had solved the sphere-packing problem in both 8-dimensional and 24-dimensional spaces. This beautiful and deep problem asks how closely the same circles or spheres can be packed in n dimensional space. In 2022, Viazovska was awarded a Fields Medal for her pioneering work. This success made her the second woman in the 86-year history of this highly competitive and prestigious award to win!
The sphere-packing problem has enthralled mathematicians for centuries. In 3D, the most efficient packing of spheres is pyramidal stacking. When looking at two-dimensional packing, the best packing is provided by a honeycomb structure. Viazovska still took the challenge a step further. To prove that the most symmetric arrangement—an exceptional lattice called E8—gives the densest known packing of space in eight dimensions, she invoked powerful mathematical tools called (quasi-) modular forms. Along the way she proved that the Leech lattice is the optimal packing arrangement in 24 dimensions.
A Collaborative Effort
Viazovska’s groundbreaking research did not go unnoticed. Working with Sidharth Hariharan, she produced a much more formalized version of her sphere-packing proofs. Collaboratively, and using the programming language/proof assistant Lean, they were able to guarantee the validity of their results.
“Formal verification of a proof is essentially a rubber stamp,” said collaborator Liam Fowl, a researcher formerly with the U.S. This underscores the dire need to formalize mathematical proofs, to ensure their correctness and reliability.
Hariharan characterized their partnership as productive, calling it “a pretty fruitful collaboration. Together their contributions and collaboration led to a clear and powerful blueprint. This commentary describes the state of the art of the 8-d proof, demystifying it for other researchers and practitioners within the field.
Once they set out to get it right, the obstacles started piling up. Hariharan noted that they were on the 30th “sorry” when he made the statement. This achievement was as a result of them proving 30 additional intermediate facts that we had asked for. While this process is demanding and ultimately triumphant, it underscores the often daunting and unfriendly atmosphere towards rigorous proof verification.
The Role of AI in Mathematics
The evolution of artificial intelligence tools, specifically Chat GPT, has provided expanded opportunities for research and fact-checking. Math, Inc.’s AI system, Gauss, autoformalized Viazovska’s 24-dimensional sphere-packing proof in an impressive two weeks. This automated process generated over 200,000 lines of code, showcasing the potential of AI to assist mathematicians in their work.
Jesse Han, a key player in enhancing Gauss’s capabilities, remarked, “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss.” This step was crucial in allowing the AI system to go beyond formalizing proofs and actually fix errors in them. As an example, Gauss himself discovered an error in the printed paper on the 8-dimensional case and made the necessary corrections.
Han referred to Gauss as an unusual breed of language model. He provided the wonderful high-level description of it, as a reasoning agent, intended to gracefully blend traditional natural-language reasoning with fully formalized reasoning. This dual functionality allows Gauss to continue to explore intricate mathematical ideas while still maintaining accuracy.
Liam Fowl expressed optimism about the future impact of AI on mathematics: “These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” When mathematicians adopt AI more extensively into their workflows, they expect to see greater productivity and efficiency at working on difficult problems.
Implications for Future Research
The breakthroughs achieved by Viazovska and her collaborators represent a major, paradigm-shifting change in the way that mathematical research is conducted. AI is already assisting mathematicians in formalizing proofs and checking their work. This frees them up to focus on innovative solutions to problems rather than bogging down in the weeds.
According to Han, this kind of technology will be a boon for mathematicians in the long run. Most importantly though, it will free them up to focus on what they do best—dreaming up new mathematical universes. This view is telling of a much more positive outlook—one in line with the increasing consensus that AI will enhance human skills, instead of supplanting them.
The importance of Viazovska’s accomplishments reach well past the sphere-packing problem. Her work is a fantastic example of how collaboration—both human-human and human-AI—can help mathematicians make and understand deeper breakthroughs.
