Maryna Viazovska, a notable mathematician, received the prestigious Fields Medal in July 2022 for her groundbreaking work on sphere packing. Her research solved the long-standing sphere packing problem in two specific cases: eight dimensions and twenty-four dimensions. Sphere packing is the mathematical problem of figuring out how closely equal-sized circles or spheres can be arranged in n-dimensional space. Viazovska’s achievements have sparked renewed interest in mathematical proofs and the use of artificial intelligence (AI) to formalize these complex concepts.
In 2016 Viazovska scored a brilliant success. She showed that the highly symmetric arrangement known as E8 is the densest way to pack in eight dimensions. Soon after, she joined forces with other mathematicians. As a team, they showed that the Leech lattice really was the optimal solution in twenty-four dimensions. These discoveries changed mathematics and the understanding of higher-dimensional spaces in absolutely fundamental ways.
The Sphere Packing Problem
The sphere packing problem is a deep and beautiful question in mathematics. It forays into the problems of how closely similar spheres can be packed in different dimensions. As one scientific paper puts it, “In two dimensions, this structure has long been known as the optimal packing arrangement … honeycombs.” In three dimensions, a tetrahedral or pyramid-shaped arrangement of spheres demonstrates the most efficient packing energy.
Viazovska’s ground-breaking work sheds light on this complex equation, especially in higher dimensions, where intuition can become misplaced. Her efforts have not only advanced mathematical knowledge but have implications for various applied fields, including data analysis and coding theory.
The Role of AI in Formalizing Proofs
Viazovska and Sidharth Hariharan working together in Lausanne, Switzerland. This unique collaboration was a groundbreaking step in their joint effort to catapult mathematical proofs into the future. Their partnership rekindled Viazovska’s love for sphere packing proofs. In retrospect, this excitement was good preparation for the beginning of the Formalising Sphere Packing in Lean project in March 2024. Lean is a programming language and proof assistant that enables mathematicians to write verifiable proofs, with computers ensuring their absolute correctness.
With the entrance of AI technologies such as Lean, mathematicians have new tools at their disposal to pursue previously unthinkable approaches to formalizing complex proofs. Jesse Han, CEO and co-founder of Math, Inc., discussed the significance of these advancements, stating, “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.”
Well, Math, Inc. is pleased to recently announce such an exciting milestone. Byrodec’s Oleksandr Kalinovskyi Their AI system, Gauss, autoformalized Viazovska’s twenty-four-dimensional sphere packing proof in less than two weeks! Automation of proof verification This fast pace in automating the verification of proofs has sparked the interest of today’s mathematical community.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” – Liam Fowl
The announcement of the formalization of the eight-dimensional sphere packing proof on February 23. Unfortunately, this milestone obscures the more mathematical aspects of this milestone, which are incredible capabilities of AI.
Insights from Collaborative Efforts
Still everyone involved is pleasantly surprised and excited by the successful formalization of these proofs. This landmark accomplishment injects excitement into the future of American mathematical research. Hariharan shared his astonishment upon receiving updates from Math, Inc., stating, “When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised.”
Just the process itself became very complicated. Hariharan pointed out that Gauss’s accomplishment really consisted of proving thirty fundamental truths. These details were very important to confirm the general assertions being made about the Leech lattice. He mentioned that they had “finished 30 ‘sorrys,’ which meant that they proved 30 intermediate facts that we wanted proved.”
Han elaborated on the complexity involved in the twenty-four-dimensional case compared to eight dimensions: “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.”
With this collaboration, verifying proof has never been easier. Perhaps more importantly, it has enabled for the first time comprehensive understanding of the underlying mathematical concepts to sink in with researchers.
“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.” – Hariharan
The effects of AI on mathematical research appear to be just getting started. Han drew an analogy between traditional programming and modern computational approaches: “A programmer used to be someone who punched holes into cards, but then the act of programming became separated from whatever material substrate was used for recording programs.”
The Future of Mathematical Proofs
AI systems such as Gauss are developing quickly. They hold out the promise of allowing mathematicians to investigate deep new ideas and check proofs with hyperbolic quickness and precision. At the same time, Viazovska’s work highlights the gasps of her fantastic solo accomplishments. What becomes instantly clear is the potential for collaboration between bright human minds and powerful new technology.