Maryna Viazovska’s groundbreaking work on the sphere-packing problem should inspire further discoveries in the world of mathematics. Then in 2016, she made mincemeat of the sphere-packing problem, getting extraordinary results in two different cases. Her research ultimately proved the best configuration of spheres in any number of dimensions. The sphere-packing problem asks how tightly we can pack identical circles and spheres in n-dimensional Euclidean space. This question is as basic to geometry as it is to number theory.
In three dimensions, Viazovska discovered that spheres organized in a pyramid shape gives the most compact packing density. In the case of two dimensions, Ingham’s research found that the honeycomb structure is the best packing solution. Her work did not stop there. Specifically, Viazovska showed that a special symmetric configuration called E8 is in fact the best packing possible in eight dimensions. She worked with fellow mathematicians to think outside the box. Collectively, they proved that the Leech lattice is indeed the optimal packing arrangement in 24 dimensions.
Advancements in Proof Verification
Sidharth Hariharan helping to lead proof verification team on sphere-packing proof He is deeply committed to proving these intimidating mathematical claims. He and his collaborators have labored wonderfully to make sure the correctness and trustworthiness of those proofs.
The partnership between Hariharan’s team and Math, Inc., founded and operated by CEO Jesse Han, has returned very positive results. In February 2023, Gauss, a new, advanced reasoning agent developed by Math, Inc., was able to autoformalize Viazovska’s eight-dimensional sphere-packing proof. This intersectional milestone represents an amazing step forward not just in the field of autoformalization but for math-AI collaboration more broadly.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” – Sidharth Hariharan
The importance of this accomplishment can’t be overstated. Gauss’s powers go far beyond just checking answers. It combines natural-language reasoning with fully formalized reasoning. This two-pronged strategy raises the whole stack’s efficiency and by taking advantage of provably accurate math, verification accuracy too.
“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
The Role of Lean Programming Language
The interactive theorem proving programming language Lean is at the heart of this effort. As a widely used proof assistant, Lean lets mathematicians write proofs that computers can check for perfect correctness. This process reshapes conventional mathematical practices through the integration of computational support to promote exactness.
In March 2024, a nascent project called Formalising Sphere Packing in Lean would come out of this collaboration. This project will help deepen our understanding and verification of sphere-packing problems through formalization in Lean.
“Formal verification of a proof is like a rubber stamp,” – Liam Fowl
This increased emphasis on formal verification is a fundamental change for mathematicians. It indicates the growing reliance on computational tools to validate complex mathematical concepts that were previously susceptible to human error.
Collaboration and Future Directions
The partnership between mathematicians and AI is still paying off in spades. With this in mind, T4America’s Jesse Han pointed to what we think is the biggest achievement during this partnership. He clustered thousands of artificial neurons together and then announced that discovery in mid-January strengthened the earlier Gauss.
“We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” – Jesse Han
The journey did not have its challenges. Even the verification of Viazovska’s 24-dimensional proof was intensely complicated. It was a process that included working through reams of complicated underlying policy. It took Gauss just two weeks to autoformalize all 200,000+ lines of code in this proof.
“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,” – Jesse Han
As this progress moves forward, mathematicians and computer scientists alike look hopefully to the future, to what new possibilities mathematical proofs — and their verification — may hold. The ramifications extend well beyond sphere-packing conundrums. They herald a new era, in which AI will enable mathematicians to take on even greater challenges.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl