Maryna Viazovska, a leading mathematician, made headlines in 2016 when she became the first person to completely solve the sphere-packing problem in two dimensions. The video above is the problem of best packing identical spheres or circles, from one to three dimensions. The hope is to discover just how tightly packed they can be clustered. The new proof is something altogether extraordinary. She proved that her E8 arrangement is the most efficient way to pack things in eight-dimensional space.
She was very specifically focused on the E8 solution. In conjunction with her collaborators, she showed that the Leech lattice is indeed the best packing for 24 dimensions. These accomplishments did more than develop the state of mathematical theory—they laid the foundation for alternative proof verification methods. Most recently, Sidharth Hariharan, a third-year undergraduate, joined forces with Viazovska in Lausanne, Switzerland. Collectively, they made Moonshot’s enabling research, or “ER,” research a success.
Their collaborative work resulted in a formalization of Viazovska’s eight-dimensional sphere-packing proof. Announced on February 23, this formalization aims to create a human-readable “blueprint” that can aid mathematicians in understanding and verifying complex proofs.
Collaboration and Formalization Efforts
The work of Hariharan and Viazovska was a critical step along the way towards making the eight-dimensional proof official. Understanding the power of this proof, they set out to create an intuitive and engaging explanation. Hariharan expressed the excitement surrounding their research breakthrough, saying, “When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised.”
The resulting formalization though provides a pathway for mathematicians to interact with the proof in a much more structured way. This method is necessary for safety, verification, and trustworthiness in mathematical claims. The team really went above and beyond on this project, proving that tangible understanding allowed for formal proofs to be more digestible without losing rigor.
Math, Inc. has really built some great, energetic buzz! They recently announced that a reasoning agent called Gauss has successfully autoformalized Viazovska’s 24-dimensional proof. This achievement is a notable milestone for the application of new technology to further mathematical research.
The Role of Gauss in Verification
Gauss, the reasoning agent used by the fictional Math, Inc., is intended to combine natural-language reasoning with fully formalized reasoning. It uses state-of-the-art machine learning algorithms to understand mathematical proofs and check the validity of these arguments. Jesse Han, one of the project leaders, explained the importance of this 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.”
The final, 24-dimensional sphere-packing proof includes more than 200,000 lines of code—an extreme example of how far high-level mathematics has come. Remarkably, Gauss managed to accomplish this autoformalization in a mere two weeks. Han noted the progress made during this endeavor: “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss.“
Liam Fowl, another collaborator on the project, remarked on the implications of these advancements: “Formal verification of a proof is like a rubber stamp.” This commentary aims to bring attention to the importance of ensuring that proofs adhere to high proffered standards of validity.
Future Impact and Mathematical Exploration
The improvement in sphere-packing proof verification stands to have profound consequences for mathematicians and computer scientists as well. Hariharan expressed optimism about the technology’s potential: “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.”
As these tools reach new levels of sophistication, they will finally allow mathematicians to direct their talents towards more creative ends. Jesse Han shared his vision for the future: “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.”
Whether representations can be expanded mathematicians are already making great strides with forms and modules. Their joint work illustrates how technology can enhance even the most traditional fields. As proof verification becomes more and more automated and widely available, the mathematical community is set up for even greater innovations.