In July 2022, Maryna Viazovska – a highly respected mathematician who makes up our panel – came into the spotlight when she was awarded the coveted Fields Medal. This award is widely considered the Nobel Prize for mathematics. She received this honor after her trailblazing solutions to the sphere packing problems in 8-dimensional and 24-dimensional space. Viazovska’s work has advanced the boundaries of mathematical knowledge. Alongside this now-standard result, it has sparked a resurgence in using contemporary technology to formalize these proofs.
The sphere packing problem, or how to pack spheres as tightly as possible in a given space, became much more complex in higher dimensions. Viazovska tackled the problem successfully, demonstrating that the symmetric arrangement known as E8 is the optimal solution in 8 dimensions. In the 24-dimensional case, among other things, she showed that the Leech lattice is the optimal packing configuration. Her achievements have set a new standard in mathematical research, paving the way for further exploration in this complex field.
A chance meeting between Sidharth Hariharan and Maryna Viazovska in Lausanne, Switzerland, reignited Hariharan’s enthusiasm for formalizing proofs related to sphere packing. Hariharan is a first-year PhD student at Carnegie Mellon University. He had helped previously to get the proof formalized for packing in an 8-dimensional sphere. Their animated conversation led him to explore the topic more extensively.
The Role of Technology in Mathematical Research
Technology is a powerful tool Technology has been an invaluable tool in other disciplines for years. Jesse Han is the co-founder of Math, Inc. As a result, the company has been building an innovative reasoning agent, Gauss, which assists researchers in formalizing proofs. Gauss uses state-of-the-art algorithms to analyze and check deep mathematical theory.
Gauss would have achieved the autoformalization of Viazovska’s proof of the optimal packing of spheres in 24 dimensions. He did it in just two weeks’ time—amazing! This included solving 200,000+ lines of code—demonstrating the power of AI to manage big math workloads. Han emphasized the collaborative nature of this project, stating, “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 use of AI for the first time in enriching mathematical research is indeed a radical change. Today, researchers can use AI tools to catch mistakes and help them understand more complicated proofs, which has made their work much more efficient. “One of them helped us identify a typo in our project, which we then fixed,” noted Hariharan, showcasing the practical benefits of incorporating AI into traditional research methodologies.
Progress and Future Prospects
While the partnership between AI and mathematicians is still growing, this new technology has produced groundbreaking results. In March 2024, the Formalising Sphere Packing in Lean project was initiated to further explore and validate sphere packing proofs. Lean is a widely-used functional programming language and interactive theorem prover. Its ability to formalize all mathematical proofs is what makes it ideal for such an ambitious project.
Over the next two years, Hariharan and his team compiled the project’s repository. In response to immense public pressure, they finally allowed public use in June 2025. This innovative strategy was designed to promote civic involvement and inspire teams of thinkers to work together to develop more formalized proofs of their mathematics.
That project had a major breakthrough in mid-January that resulted in plans to develop an improved version of Gauss. Han reported that this upgraded model demonstrated remarkable efficiency: “This new version reproduced our three-week PNT result in 2–3 days.” This blinding speed demonstrates just how powerful AI can be to advancing mathematical research. More importantly, it stokes greater excitement about its potential to transform how mathematicians approach and solve big, hairy problems.
Liam Fowl, a noted mathematician, commented on the significance of these developments: “These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” From the unique relationship between mathematical research and AI, more discoveries and new inventions await.