Maryna Viazovska, a mathematician renowned for her groundbreaking work in the sphere packing problem, received the prestigious Fields Medal in July 2022. This recognition came after she made significant advances in understanding how densely identical circles and spheres can be packed in n-dimensional space. In 2016, Viazovska solved the sphere packing problem for two specific cases: 8-dimensional and 24-dimensional spaces. To connect the two, she employed some powerful mathematical functions known as (quasi-)modular forms in her groundbreaking approach. This approach showed the symmetric structure called E8 to be the best possible packing in eight dimensions.
In her collaborative research, Viazovska achieved a remarkable breakthrough. Her work showed that the Leech lattice was the best possible sphere packing arrangement in 24 dimensions. Lack of previous expository background material on the Leech lattice added complexity to this proof. We knew we needed to build a tremendous amount of groundwork to address it deeply. New advances show that artificial intelligence (AI) is revolutionizing mathematical verification to make it faster and better. This advance comes on the heels of work by Sidharth Hariharan and his collaborators to codify Viazovska’s proofs.
The Birth of a New Project
In March 2024, the Formalising Sphere Packing in Lean project began. This new initiative was a huge step ahead at the pivotal crossroad of AI and mathematics. Over the course of this second sequence of events, Hariharan, then a third-year undergraduate, encountered Viazovska in Lausanne, Switzerland. His work with her and other mathematicians has been developed to become a powerful new tool for state-of-the-art verification of mathematical proof.
The project gained attention when Math, Inc. announced that Gauss, an advanced reasoning agent, successfully autoformalized Viazovska’s 24-dimensional sphere packing proof—over 200,000 lines of code—in just two weeks. This achievement highlights the promise AI holds to help mathematicians formalize complex proofs.
“They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved,” – Sidharth Hariharan
Surprisingly, the 24-dimensional case was more complicated than the 8-dimensional case. This complexity emerged due to the necessity of compiling a huge amount of preparatory material about the exceptional structure of the Leech lattice. Progress in each dimension proves an exciting horizon for mathematical proof verification through ongoing collaboration with AI.
The Role of AI in Mathematical Proofs
Gauss aims to combine the best of traditional natural language reasoning with fully formalized reasoning. This ability allows it to perform literature queries and use other tools. It’s the one that produces Lean code, which is super important for actually verifying mathematical proofs. Lean is a fascinating programming language. Mathematicians make use of it as a proof assistant, checking the correctness of their proofs with computer verification.
“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
What we all recognize is the impact this technology potentially could have. It plays a massive role in increasing the efficiency of verifying mathematical proofs. Second, it improves the accuracy of the process by automating many tasks that would have required considerable manual labor.
“So it’s able to conduct literature searches, call up tools, and use a computer to write down Lean code, take notes, spin up verification tooling, run the Lean compiler, etc.,” – Jesse Han
On February 23, the formalization of the 8-dimensional sphere packing proof was completed. This accomplishment is a big leap forward for both autoformalization and AI-human collaboration. Hariharan and his colleagues provided terrific leadership to help make this possible. Beyond that, they provided a largely pre-existing blueprint that proved Gauss’s successful template for autoformalizing his efforts.
“One of them helped us identify a typo in our project, which we then fixed,” – Hariharan
Whether it’s AI’s prowess in detecting mistakes or making proof workflows more efficient, these new developments showcase how AI could revolutionize research in mathematics.
Future Implications and Excitement
Perhaps that’s why the enthusiasm about these advances is electric among these young and old members of the mathematical community. Even experts as far afield as Liam Fowl have recognized the incredible advancements achieved through AI partnership.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl
During the event, Fowl illustrated just how important formal verification is. He likened it to a field as the “rubber stamp” that ensures ultimate infallibility in formalized mathematical proofs. This reliability is of the utmost importance for mathematicians who wish to build upon previous work with certainty.
“Formal verification of a proof is like a rubber stamp,” – Liam Fowl
Hariharan explained the reasons for his optimism about the advances in technology surrounding this work. He noted that this is where AI can help mathematicians to do amazing things.
“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