On July 4, 2022, the mathematician Maryna Viazovska became internationally known. She shared the Fields Medal, an award sometimes colloquially called the Nobel Prize for mathematics. Her innovative research answered the sphere packing problem in both 8-dimensional and 24-dimensional spaces. This accomplishment garnered her much-deserved acclaim and paved the way for exciting new partnerships between humans and AI.
In 2016, Viazovska’s path through mathematics took an extraordinary turn. She mathematically demonstrated that the symmetric structure we now call E8 is indeed the best packing there is in eight dimensions. Following this success, she took her studies to 24 dimensions. Together with her colleagues, she established that the Leech lattice is indeed the optimal packing solution. These achievements have produced an extraordinary moment in mathematics. They invite additional investigation and verification through new technology.
The Role of Technology in Proof Verification
As the field of mathematics is inevitably in a state of change, the impact of technology is more relevant now than ever. Jesse Han, CEO and co-founder of Math, Inc., emphasized the importance of their pioneering reasoning agent called Gauss. This powerful AI unites old school natural language common sense reasoning with cutting edge fully formalized reasoning. Consequently, it greatly increases the efficiency at which complex mathematical proofs can be verified.
Gauss proved its prowess by autoformalizing Viazovska’s 8-dimensional sphere packing proof in five days. Remarkably, it not only accomplished this task in record time but found and fixed a powerful typo in the resulting published paper itself. This instance underscored the potential of AI to assist mathematicians by providing a layer of verification that was previously unattainable.
“One of them helped us identify a typo in our project, which we then fixed.” – Sidharth Hariharan
Gauss demonstrated its incredible power recently by autoformalizing Viazovska’s proof on the sphere packing in 24 dimensions. This monumental work spanned upwards of 200,000 lines of code. The collaboration between AI and researchers like Sidharth Hariharan, a PhD student at Carnegie Mellon University who worked closely with Viazovska to formalize her proofs, has transformed the landscape of mathematical research.
Collaboration Between Humans and AI
The formalization process of Maryna Viazovska’s proofs was a collaborative experience that augmented human ability with AI power. Hariharan said that they were getting reports from Gauss saying that it was done with a bunch of intermediate proofs.
“They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved.” – Sidharth Hariharan
Partnership had really been at the heart of this project. The 24-dimensional case turned out to be much more difficult. We had a much harder time than in the 8-dimensional case. Han found a wealth of existing background material. Orenstein noted that there’s a great deal of information to be included on the different properties of the Leech lattice.
“And it was actually significantly more involved than the 8-dimensional case, because there was a lot of missing background material that had to be brought online surrounding many of the properties of the Leech lattice, in particular its uniqueness.” – Jesse Han
This collaboration has proven to simplify and accelerate the verification process, while improving the accuracy and reliability of mathematical proofs.
Implications for Future Research
The impact of these developments goes far beyond more efficient proof checking. The productive partnership between Viazovska, Hariharan, and Gauss is one step on the path toward leveraging AI to augment mathematical research. Previous infill development and transportation experts such as Liam Fowl have already called these results astounding. These last two, he argues, are signs of dramatic progress made in this direction.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” – Liam Fowl
Having formal verification of proofs functions as a “rubber stamp.” It offers a sense of rigor that was frequently absent from theoretical proofs. This technology offers a thrilling professional opportunity for mathematicians. As for all students, it can better prepare them to tackle issues, assess solutions and iterate faster on answers.
Hariharan expressed optimism about the future of AI in mathematics:
“But at the end of the day, this is technology that we’re very excited about because it has the capability to do great things and to assist mathematicians in remarkable ways.” – Sidharth Hariharan