Maryna Viazovska, a prominent mathematician, has made significant strides in solving the sphere packing problem, particularly in eight-dimensional and twenty-four-dimensional spaces. This mathematical conundrum centers around figuring out the maximum packing density of identical spheres within an n-dimensional space. Experts have conventionally located the ideal configurations in lower dimensions. Honeycomb structures are best in two dimensions, and pyramid stacking — tetrahedral coordination — is ideal in three dimensions. There’s more to Viazovska’s groundbreaking work than these dimensions, using otherworldly mathematical functions to extend optimal solutions years later.
In 2016, Viazovska showed that the E8 lattice yields optimal packing in eight-dimensional space. Then, together with her coauthors, she went on to show that the Leech lattice is the densest known packing in twenty-four dimensions. The proof was difficult and fraught with nuance, which led to creative breakthroughs and ideas. Sidharth Hariharan and his colleagues took it upon themselves to formalize Viazovska’s discoveries.
Formalizing Complex Proofs
On February 23, 2024, Hariharan, along with co-author Max D. Klementiev, formally announced the discovery of Viazovska’s eight-dimensional sphere packing proof. This formalization process seeks to guarantee that mathematical proofs are not only true, but provably true by a computer system. This kind of formal verification serves as a “rubber stamp” for mathematicians, providing assurance of their work’s accuracy.
This intense process underlines all the complexities that go into ensuring a stamp of truth for something as advanced as mathematical algorithms.
“They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved.” – Sidharth Hariharan
Together this unique collaboration has made great progress. It’s an important reminder of the value of old ways, even as it demonstrates the apparently limitless ways that modern technologies can improve the mathematical verification process.
“One of them helped us identify a typo in our project, which we then fixed.” – Sidharth Hariharan
AI is already upending things, as with mathematical proof verification. This increase comes on the heels of a big announcement by Math, Inc. on the release of their AI system Gauss. Our advanced reasoning agent is able to interleave traditional natural language reasoning with fully formalized reasoning. In a truly impressive achievement, Gauss autoformalized Viazovska’s twenty-four-dimensional sphere packing proof—over 200,000 lines of code—in two weeks flat.
AI and Proof Verification
According to research team member Jesse Han, Gauss’s most distinguishing capabilities were special.
The speed and efficiency of this process illustrates AI’s potential to truly transform practices mathematically. Fortunately, Gauss was soon able to spot an error in the published advance paper. He made that discovery only five days after the new eight-dimensional proof was created.
“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
This simultaneous convergence of AI and mathematics has created a vibrant new space for researchers. As of March 2024, our most recent project is called Formalising Sphere Packing in Lean. Our aim is to produce a strong topological skeleton suitable for formalising proofs concerning the sphere packing.
The Future of Mathematical Research
It’s really, really, really incredible to see these new results come in,” stated Liam Fowl, a leading advocate for these types of programs.
Researchers are currently studying the relationship between AI and math. We love to see them succeed in their goals of advancing knowledge and improving efficiency in intricate disciplines. This union of human creative power and AI will transform the way mathematicians solve problems, providing unprecedented breakthroughs.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” – Liam Fowl
The excitement surrounding these developments is palpable among 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.” – Sidharth Hariharan
As researchers continue to explore the intersection of AI and mathematics, they aim to enhance understanding and streamline processes within complex fields. The collaboration between human intellect and artificial intelligence may revolutionize how mathematicians approach problems, leading to unprecedented discoveries.