In July 2022, Ukrainian-born Maryna Viazovska became the second woman to ever be awarded the prestigious Fields Medal, one of the highest honors in mathematics. This award is commonly known as the Nobel Prize for mathematics. Only a few months after Russia invaded her home of Ukraine, she did something extraordinary. As you can imagine, this backdrop added deep emotional weight to her groundbreaking success. Viazovska’s contributions to mathematics include solving the sphere packing problem in two dimensions and proving that the E8 lattice provides the optimal arrangement for eight-dimensional spheres. She and her collaborators demonstrated that the Leech lattice provides the densest possible packing arrangement in 24 dimensions. This breakthrough fills a gap that has stumped mathematicians for more than 14 decades.
Her award honored her specific accomplishments. Importantly, it brought into sharp focus the essential role of mathematical research in responding to the challenges faced by the world today. The Fields Medal therefore acts as an even brighter beacon of the ideal of excellence. It’s a testament to the essential role that mathematicians play in improving our knowledge in dozens of disciplines.
The Sphere Packing Breakthroughs
In 2016, Viazovska marked a monumental breakthrough by solving the sphere packing problem in two special cases. Her research, especially on the E8 lattice, proved it out to be the most optimal packing configuration in eight-dimensional space. This accomplishment is an incredible mathematical landmark. It further serves as a reminder of what is possible when we work together to reimagine the future.
After her work on E8, she teamed up with other researchers to solve the even more complicated 24-dimensional sphere packing problem. The Leech lattice that turned out to be the best answer, giving mathematicians a wonderful new world to explore and study in higher dimensions. These accomplishments are a testament to the value that sustained, rigorous mathematical inquiry and collaboration can have on solving complex, multifaceted challenges.
“They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved,” – Sidharth Hariharan
Coming from a world of rigid mathematical formalism, Sidharth Hariharan encountered Viazovska while a student in Lausanne, Switzerland. His background with formal verification provided him with a different outlook. Only then would he really be able to gauge the depth of Viazovska’s achievement and its significance for future mathematical pursuits.
The Role of Technology in Formal Proof Verification
Unfortunately, there’s a catch. In March 2024, an exciting new project called Formalising Sphere Packing in Lean started. This initiative aimed to formalize Viazovska’s proof of the Leech lattice using Lean, a popular programming language and proof assistant designed for mathematical verification. At Math, Inc. we’re blazing the trail. Co-founded by Jesse Han, the young company has developed an excitingly original tool to tackle that problem, called Gauss.
Gauss is a next generation reasoning agent that aims to combine classic natural language reasoning with fully formalized reasoning. It was key in autoformalizing Viazovska’s long proof—more than 200,000 lines of code—in two weeks. This pioneering achievement marks a remarkable step forward in the convergence of math and technology.
“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
The implications of such technology are more than economic efficiency. Gauss helps mathematicians confirm complicated proofs at extraordinary speeds and precision. This innovation allows them to focus on the creative and entrepreneurial facets of their work. This symbiotic relationship between technology and mathematics makes it possible to create innovative new approaches using these advances.
“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,” – Jesse Han
Future Prospects and Community Impact
As for-profit advancements such as Gauss gain traction, they open a door to a golden age of mathematical research. The challenge of producing complex proofs in a formal manner not only increases effectiveness of proof validation but promotes collaboration amongst the mathematics and computer science communities. Collaboration was the secret sauce that made Hariharan’s team hum. Backed by Gauss, they were able to quickly detect these errors and optimise their verification process.
“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
These high-dimensional proofs demand a deep understanding of the underlying mathematical theory. You need to have a deep understanding of technological applications. Mathematicians such as Viazovska are always going beyond limits. To push the boundaries of understanding while remaining grounded in rigorous mathematics, they lean heavily on tools such as Gauss.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl
This support from experts like Liam Fowl only serves to reinforce and deepen that transformative potential of harnessing technology with our understanding of mathematics. With the community rallying around these new developments, mathematicians will be better positioned to take on even bolder challenges.