Maryna Viazovska and AI: A Mathematical Revolution in Progress

Maryna Viazovska: An Iconic Figure in Modern Mathematics

In July 2022, Maryna Viazovska, a Ukrainian mathematician, was honored with the prestigious Fields Medal, often compared to the Nobel Prize in mathematics. This event was particularly significant as she was only the second woman to receive this distinction in 86 years. Moreover, her recognition came shortly after Russia's invasion of Ukraine, adding a poignant dimension to her success. Today, Viazovska continues to make headlines through an innovative collaboration between artificial intelligence and mathematicians, which has enabled the formal verification of her proofs. Liam Fowl, an AI reasoning expert at Princeton University, emphasizes the importance of these advancements, even though he is not directly involved in the project.

The Sphere Packing Challenge

Viazovska's research, which earned her the Fields Medal, focuses on the complex problem of sphere packing. This mathematical problem seeks to determine how to optimally arrange identical spheres in multi-dimensional space. In two dimensions, the solution is the honeycomb pattern, while in three dimensions, a pyramidal arrangement is the most efficient. However, beyond these dimensions, the complexity increases significantly. In 2016, Viazovska solved this problem for eight and twenty-four dimensional spaces. She demonstrated that the E8 arrangement is optimal in eight dimensions, while the Leech lattice is optimal in twenty-four dimensions, using (quasi-)modular forms. These discoveries, although theoretical, have practical applications, particularly in the field of error-correcting codes used in communication technologies.

Formal Verification: A New Challenge

Although Viazovska's proofs have been validated by the mathematical community, formal verification by computer represents a distinct challenge. Since 2022, significant progress has been made in this area, notably with the assistance of artificial intelligence. Liam Fowl describes formal verification as a "rubber stamp," a certification that ensures the reasoning is correct.

A Meeting That Changed Everything

An unexpected meeting between Sidharth Hariharan, then a student, and Viazovska rekindled interest in formalizing sphere packing proofs. Hariharan, although still early in his career, had already distinguished himself in the art of formalizing proofs. He explained to Viazovska how formalization helped him better understand mathematical concepts. Intrigued, Viazovska expressed her interest in formalizing her own work. This interaction led to the creation of the "Formalising Sphere Packing in Lean" project in March 2024. Lean is a programming language that allows for the computer verification of mathematical proofs.

A team was formed to create a detailed plan of the elements of the proof in eight dimensions, identifying those that still required formalization. The project was developed for about 15 months before being opened to the public in June 2025. It was then that Math, Inc. emerged.

The Role of AI with Math, Inc.

Math, Inc., an innovative startup, is developing Gauss, an AI designed to automate the formalization of proofs. The startup first made headlines when it announced that Gauss had completed a Lean formalization of the strong prime number theorem in just three weeks. Math, Inc. then reached out to Hariharan and his team to inform them that Gauss had proven several elements of their sphere packing project. "They solved 30 'sorrys,' intermediate facts that we needed to prove," explains Hariharan. This collaboration helped correct a typo in the project, illustrating the effectiveness of AI in this field.

Towards New Dimensions

The impact of AI on the verification of mathematical proofs is undeniable, opening new perspectives for scientific research. Collaborations between mathematicians and AI, such as the one between Viazovska and Math, Inc., illustrate the potential of these technologies to transform scientific inquiry. By automating certain complex tasks, AI frees researchers to focus on more creative and innovative aspects of their work.

A Promising Future for AI-Human Collaboration

The future of collaboration between AI and mathematicians looks promising. The advancements made by startups like Math, Inc. demonstrate that AI can play a crucial role in solving complex mathematical problems. By combining the computational power of machines with human intuition, it is possible to reach new heights in scientific research. Future projects could benefit from these synergies, paving the way for even more revolutionary discoveries in mathematics and beyond.