Maryna Viazovska is a dynamo who is quickly carving her name into the annals of mathematics. She proved solution of sphere-packing problem in both eight-dimensional and twenty-four dimensional space. This deceptively simple challenge considers packing as many identical spheres or circles as possible in n-dimensional space. She has won many prestigious awards and her work has a profound impact. It further pushes the boundaries of mathematical theory and is a demonstration of the increasing power of artificial intelligence (AI) to verify complex mathematical proofs.
The sphere-packing problem perplexed mathematicians for decades. In three dimensions, the best packing occurs when stacking spheres into pyramids. In two dimensions, the hexagon structure turns out to be the optimal construction. Viazovska’s work was revolutionary on all these counts. Her outstanding research has garnered her the 2022 Fields Medal, an award sometimes thought of as the Nobel Prize for mathematics.
Collaboration and Innovation
Much of Viazovska’s work on sphere packing was possible through her collaboration with Sidharth Hariharan. Collaboratively, they took her sphere-packing proofs through formalization, using a programming language and proof assistant called Lean. This collaboration started in March 2024 with the kick-off of the Formalising Sphere Packing in Lean project. Its purpose was to always have Viazovska’s results hold up under the toughest of checks.
Lean gives mathematicians the power to write proofs that a computer can then check for correctness beyond any doubt. This process is very important in developing trust in rigorous and complicated mathematical statements. According to Hariharan, “They told us that they had finished 30 ‘sorrys,’ which meant that they proved 30 intermediate facts that we wanted proved.” He added, “So it was a pretty fruitful collaboration.”
Beyond this, Jesse Han, CEO and co-founder of Math, Inc., brought further expertise as a logic developer by creating a derived reasoning agent known as Gauss. This tool was meant to focus on combining existing traditional natural-language reasoning capabilities with fully formalized reasoning. Han explained, “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 Role of AI in Proof Verification
The power of Gauss was recently on display when it successfully autoformalized Viazovska’s twenty-four-dimensional sphere-packing proof. Amazingly, this whole process covered more than 200,000 lines of code and took just two weeks to finalize. The speed and accuracy of this remarkable accomplishment illustrates the astonishingly rapid progress AI is making in mathematical research.
Han noted the challenges faced during the formalization process, stating, “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 on line surrounding many of the properties of the Leech lattice, in particular its uniqueness.” Herein lies a profound idea of the depth of knowledge needed to traverse such advanced mathematical terrains.
Liam Fowl, a mathematician and chronicler of these interesting developments, noted just how meaningful these innovations were. “Formal verification of a proof is like a rubber stamp,” he stated. “It’s a kind of bona fide certification that you know your statements of reasoning are correct.” He further added, “These new results seem very, very impressive, and definitely signal some rapid progress in this direction.”
The Future of Mathematics and AI
The confluence of cutting-edge mathematics and artificial intelligence is making possible previously unthinkable breakthroughs in research. The collaboration between Viazovska, Hariharan, and Han serves as a testament to how technology can enhance mathematical understanding and proof verification. As they shape tools, these tools will reshape the way mathematicians work on hard problems.
Illustrating the transformative effect of programming and AI on the field of mathematics, Han went on to expound on the topic. “A programmer used to be someone who punched holes into cards,” he explained. “Then the act of programming became separated from whatever material substrate was used for recording programs.” It’s part of a bigger shift that is happening within our field as mathematicians start using technology more and more in their work.
As you may recall, this unprecedented collaboration has already resulted in extraordinary successes. These achievements represent important individual milestones and signal the growing collective progress of the mathematical community. The power to formalize proofs with these new levels of precision unlocks many exciting opportunities for research and study.