On July 4, 2022, Maryna Viazovska broke new ground. In doing so, she became the second woman in history to receive the highly esteemed Fields Medal, an award widely considered to be the Nobel Prize for mathematics. This award came only months after Russia initiated a full-scale war on her native Ukraine. It made a grim counterpoint to her extraordinary successes. Viazovska rose to prominence throughout the mathematical community for her resolution of the sphere packing problem in two impressive instances. In doing so, she has established that this elegant, symmetric arrangement known as E8 provides the densest packing in eight dimensions. Moreover, she’s shown that the Leech lattice is indeed optimal in 24 dimensions.
Her groundbreaking contributions did not stop there. In March 2024, Viazovska and fellow mathematician Sidharth Hariharan pictured above right, reunited to celebrate the award in Lausanne, Switzerland. Their project discussions inspired the new Formalising Sphere Packing in Lean project. This new initiative seeks to make formal the proofs of her high-dimensional sphere packing solutions using state of the art technology. The teamwork of human mathematicians and artificial intelligence has achieved a significant milestone in formal proof verification.
The Historical Impact of Maryna Viazovska
Maryna Viazovska’s historic accomplishment, becoming the second woman and first female mathematician to receive the Fields Medal is certainly one such event in the field of mathematics. Since it was created in 1936, only one other woman has ever received this award. Her singular creativity has opened new avenues for mathematicians and scholars to follow and remind us of her brilliance and determination in the face of world discord.
The way Viazovska resolved the sphere packing problem was indeed groundbreaking. This challenge is about packing spheres into a space in the most efficient way. Her solutions have far-reaching applications in mathematics fields like coding theory and even the field of crystallography. She’s already shown us that E8 is the best possible configuration in eight dimensions. Furthermore, her research demonstrates that the Leech lattice has this property in 24 dimensions, unlocking a wealth of further study and application.
Ultimately, Viazovska’s contributions aren’t just some impressive mathematical ideas—they’re turning into real breakthroughs in our understanding of other rich, complicated mathematical structures. Her grit and brilliance in the face of adversity has established her as one of the foremost leaders in math.
The Role of AI in Formal Proof Verification
The joint work of Maryna Viazovska and Sidharth Hariharan launched a process, using EiAi, to use artificial intelligence to prove new mathematical proofs. Their discussions laid the groundwork for the Formalising Sphere Packing in Lean project, which aims to use advanced computational techniques to verify mathematical claims.
One particularly exciting thing about this pilot project is the participation of Gauss, a newly minted reasoning agent conceived and created by Math, Inc. Gauss autoformalized Viazovska’s 24-dimensional sphere packing proof in record time, generating more than 200,000 lines of code in under two weeks. This remarkable achievement illustrates how AI can speed up complex mathematical research and open up new realms of exploration.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” – Sidharth Hariharan
Gauss thus autoformalized the 8 dimensional proof in five days. Specifically, it pointed out a typo in the Appendix of the originally published article. This correction is a testament to its power to increase math precision.
“One of them helped us identify a typo in our project, which we then fixed,” – Hariharan
This partnership between AI and human mathematicians is an exciting step towards incorporating democratizing technology into the long established and traditional research process. Jesse Han, CEO of Math, Inc., called Gauss “revolutionary,” noting its ability to support more mature, large-scale formalizations.
“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,” – Jesse Han
Future Prospects and Implications
The improvement achieved by the Formalising Sphere Packing in Lean project has deep consequences for the future of mathematics. Experts are optimistic that these new innovations will speed up verification timelines. They believe it will embolden mathematicians to venture into new areas of investigation.
The Nobel winner and member of the mathematical establishment, Liam Fowl, said these innovations were like nothing in the bird kingdom. It is a crucial step of the formal verification process that adds an additional layer of security, providing a corrected mathematical proof with greater confidence.
“Formal verification of a proof is like a rubber stamp,” – Liam Fowl
Realizing the successful potential for AI’s integration into this multifaceted field presents thrilling opportunities. Now mathematicians are freed to focus on truly creative problem-solving rather than being clogged up by the need for verification. Jesse Han was excited about that future, envisioning how technology like this could make such an impact.
“I think the end result of technology like this will be to free mathematicians to do what they do best, which is to dream of new mathematical worlds,” – Han
AI is changing software development and more specifically, its imperative role in mathematics. Moreover, it has a tremendous potential to reinvent the ways researchers approach proof verification and even problem solving in general.