Introduction to AI for Math Grants
MIT Department of Mathematics researchers David Roe ’06 and Andrew Sutherland ’90, PhD ’07 are among the inaugural recipients of the Renaissance Philanthropy and XTX Markets’ AI for Math grants. Four additional MIT alumni — Anshula Gandhi ’19, Viktor Kunčak SM ’01, PhD ’07; Gireeja Ranade ’07; and Damiano Testa PhD ’05 — were also honored for separate projects. The first 29 winning projects will support mathematicians and researchers at universities and organizations working to develop artificial intelligence systems that help advance mathematical discovery and research across several key tasks.
The Project: Connecting LMFDB and Mathlib
Roe and Sutherland, along with Chris Birkbeck of the University of East Anglia, will use their grant to boost automated theorem proving by building connections between the L-Functions and Modular Forms Database (LMFDB) and the Lean4 mathematics library (mathlib). Mathlib is a large, community-driven mathematical library for the Lean theorem prover, a formal system that verifies the correctness of every step in a proof. Mathlib currently contains on the order of $10^5$ mathematical results. The LMFDB, a massive, collaborative online resource that serves as a kind of “encyclopedia” of modern number theory, contains more than $10^9$ concrete statements.
Benefits of the Project
The main obstacles to automating mathematical discovery and proof are the limited amount of formalized math knowledge, the high cost of formalizing complex results, and the gap between what is computationally accessible and what is feasible to formalize. To address these obstacles, the researchers will use the funding to build tools for accessing the LMFDB from mathlib, making a large database of unformalized mathematical knowledge accessible to a formal proof system. This approach enables proof assistants to identify specific targets for formalization without the need to formalize the entire LMFDB corpus in advance.
Applications of the Project
Proving new theorems at the frontier of mathematical knowledge often involves steps that rely on a nontrivial computation. For example, Andrew Wiles’ proof of Fermat’s Last Theorem uses what is known as the “3-5 trick” at a crucial point in the proof. Using stored results leverages the thousands of CPU-years of computation time already spent in creating the LMFDB, saving money that would be needed to redo these computations. Having precomputed information available also makes it feasible to search for examples or counterexamples without knowing ahead of time how broad the search can be.
Next Steps
The researchers’ next steps are to build a team, engage with both the LMFDB and mathlib communities, start to formalize the definitions that underpin the elliptic curve, number field, and modular form sections of the LMFDB, and make it possible to run LMFDB searches from within mathlib. If you are an MIT student interested in getting involved, feel free to reach out!
Conclusion
The project aims to augment both the LMFDB and mathlib systems, making the LMFDB’s results available within mathlib as assertions that have not yet been formally proved, and providing precise formal definitions of the numerical data stored within the LMFDB. This bridge will benefit both human mathematicians and AI agents, and provide a framework for connecting other mathematical databases to formal theorem-proving systems.
FAQs
Q: What is the goal of the AI for Math grants?
A: The goal of the AI for Math grants is to support mathematicians and researchers at universities and organizations working to develop artificial intelligence systems that help advance mathematical discovery and research.
Q: What is the LMFDB?
A: The LMFDB is a massive, collaborative online resource that serves as a kind of “encyclopedia” of modern number theory, containing more than $10^9$ concrete statements.
Q: What is mathlib?
A: Mathlib is a large, community-driven mathematical library for the Lean theorem prover, a formal system that verifies the correctness of every step in a proof.
Q: How will the project benefit mathematicians and AI agents?
A: The project will provide a bridge between the LMFDB and mathlib, making it possible to access a large database of unformalized mathematical knowledge and providing precise formal definitions of numerical data.
Q: How can I get involved in the project?
A: If you are an MIT student interested in getting involved, feel free to reach out to the researchers.
