Maryna Viazovska, the esteemed mathematician who received the Fields Medal in July 2022, is at the forefront of a groundbreaking collaboration that merges artificial intelligence with mathematical research. The Fields Medal—considered the Nobel Prize for mathematics—was bestowed upon her. She made influential contributions to sphere packing problems, which are important in theoretical and practical applications. Her most recent creation is a thrilling collaboration with Sidharth Hariharan and the crew at Math, Inc. Combined, they’re showing that AI can play a major role in amplifying the verification of mathematical proofs.
Viazovska’s influence in the field of sphere packing started when she completely solved the problem in two dimensions in 2016. Her proof uses quasi-modular forms to establish that the E8 lattice gives the densest packing in eight dimensions. In addition, she and her collaborators proved that the Leech lattice is optimal – it’s the best possible packing – in 24 dimensions. These accomplishments opened the door to explore further into utilizing AI as a means to verify these long and complex proofs.
The collaboration really came to life when Viazovska first met Sidharth Hariharan in Lausanne, Switzerland. At the time, Hariharan was a third-year undergrad whose original perspective brought Viazovska’s motivation back to life, sparking her interest in proving sphere packing once again. Their shared passion and innovative ideas led to the creation of the Formalising Sphere Packing in Lean project in March 2024, which aims to establish a robust framework for verifying mathematical proofs through AI assistance.
The Role of AI in Mathematical Proof Verification
Math, Inc., co-founded by Jesse Han, has taken a key role in this partnership. The company is working on a more advanced version of Gauss. This new hybrid reasoning agent will bring together classic natural language based reasoning with fully formalized reasoning. This two-fold approach enables Gauss to handle intricate mathematical ideas, while at the same time upholding high standards of proof checking.
Han describes Gauss as “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 new, creative model is definitely one to watch! It has already shown its prowess by autoformalizing proofs that previously took human researchers years of dedicated effort to check.
In one amazing case, Gauss similarly autoformalized Viazovska’s eight-dimensional sphere packing proof in under five days. In the course of doing so, it discovered that the published paper contains a spelling error in the title. Hariharan elaborated, “They came back to us and said — look, we got through 30 ‘apologies.’ This is all in their showing that they just had to prove. This swift correction of errors highlights the promise of AI being an incredibly useful companion tool in math research.
Progress and Challenges in Sphere Packing Proofs
The working relationship that blossomed between Viazovska, Hariharan and their group of researchers has made incredible strides since it began. Not until Gauss formalized the proof of the eight-dimensional sphere packing in 1801. In a mere two weeks, he additionally rendered more than 200,000 lines of code associated with Viazovska’s proof in 24 dimensions. This accomplishment represents an extraordinary advance in what AI can do to help mathematicians check their proofs with rigor.
This macroeconomic task was not without its challenges. As Han explained, “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.” The intricacies of such an issue show just how advanced the dance between human gut instinct and AI technology remains.
Liam Fowl, a noted expert in the latest area of mathematic research touted the development. He said in utter shock, “These new findings are amazing beyond belief and obviously show phenomenal gains overnight! These endorsements are a concrete indication of the increasing awareness about the potential of AI to transform mathematics.
Future Prospects for AI in Mathematics
The sky’s the limit for future cooperation between mathematicians and AI technology. So far, the Formalising Sphere Packing in Lean project has already provided a tangible foundation for future research initiatives. Hariharan shared the massive time investment it took the project to develop its own repository. Next, in June of 2025, they made it available and public to everyone. This collaborative spirit fosters a deep commitment to sharing resources and best practices that advance knowledge across the mathematical community.
The productivity efficiencies gained from the improvements made to Gauss have resulted in greater efficiency associated research outputs. Han noted a significant breakthrough achieved around mid-January that resulted in a more robust version of Gauss: “This new version reproduced our three-week PNT result in 2–3 days.” These developments serve as a reminder of how AI can accelerate the process of mathematical discovery without compromising on rigor.