Maryna Viazovska, a renowned mathematician, achieved a significant milestone in the field of mathematics by solving the sphere-packing problem in two complex cases: 8-dimensional and 24-dimensional space. This challenge asks you to consider how closely filled identical circles, spheres, etc., can pack into n-dimensional space. Mathematicians have been stumped by this question for decades. Viazovska’s groundbreaking work was recognized with the highly coveted Fields Medal in July of 2022. With this achievement, she became the second woman in the 86-year history of the award to receive this prestigious honor.
The sphere-packing problem has a distinct dimensionality: its optimal solutions vary in form across different dimensions. In higher dimensions, such as three dimensional space for example, spheres in a pyramid shape offer the densest packing. In two dimensions, the honeycomb lattice provides the most efficient arrangement. Viazovska’s work, along with subsequent results, has taken the study of these configurations to higher dimensions.
The Role of AI in Mathematical Proofs
AI and human mathematicians are working together to an unprecedented degree. They have taken a rather transformative turn with Gauss, a reasoning agent created by Math, Inc. Gauss is engineered to interleave traditional natural-language reasoning with fully formalized reasoning, enabling it to assist mathematicians in proving complex theorems.
Jesse Han, CEO and cofounder of Math, Inc., describes Gauss’s capabilities:
“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 algorithmic approach extends the other researchers’ work to encode Viazovska’s proofs with extraordinary efficiency. On February 23 www.researchgate.net announced their formalization of the 8-dimensional sphere-packing proof. Just two weeks later, Gauss would go on to accomplish the unbelievable feat of autoformalizing the entirety of Viazovska’s 24-dimensional sphere-packing proof—over 200,000 lines of code—within the time dimension.
During the collaborative efforts of mathematicians and AI, significant breakthroughs were made in the field. According to Liam Fowl, a respected figure in mathematics:
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction.”
This synergy beautifully illustrates how AI can complement mathematicians, offering the best of both worlds—speed and creativity combined with rigor and intuition.
Formalizing Sphere Packing Proofs
This led to the creation of the Formalising Sphere Packing in Lean project in March 2024, a watershed moment for the formal verification of mathematical proofs. Lean is a functional programming language. As a proof assistant, a mathematician pairs with a proof assistant like Lean to write a proof that the computer is able to verify is correct. This project seemed to catch fire. Along the way, Sidharth Hariharan and Jesse Han collaborated to find a more formal proof of their findings about sphere packing.
The partnership paid powerful dividends, with Gauss helping to prove many intermediate lemmas required for the complete verification pipeline. Hariharan mentioned an interesting anecdote from their journey:
“We had been building the project’s repository for about 15 months when we enabled public access in June 2025.”
Especially with the integration of AI into mathematical research, the process has only made it easier to find mistakes and correct pre-established proofs. Hariharan reflected on the collaborative efforts:
“They told us that they had finished 30 ‘sorrys,’ which meant that they proved 30 intermediate facts that we wanted proved.”
The promise of these advances, ushered in by Viazovska’s findings and the power of AI, mark the beginning of a new era in mathematical research. A partnership between human ingenuity and artificial intelligence has created new avenues of possibility that were once considered impossible. As Jesse Han pointed out:
“So it was a pretty fruitful collaboration.”
The Future of Mathematical Research
This evolution in programming is indicative of larger shifts within academia and research. The combination of traditional mathematical approaches with innovative technology like Gauss represents a promising future where complex problems can be tackled more efficiently.
“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.”
This evolution in programming reflects broader changes within academia and research. The combination of traditional mathematical approaches with innovative technology like Gauss represents a promising future where complex problems can be tackled more efficiently.