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MIT math researcher David Roe ’06 and Andrew Sutherland ’90, PhD ’07 were the first AI to receive math grants for Renaissance philanthropy and the XTX market.
Four other MIT alumni – Anshula Gandhi ’19, Viktor KunčakSM ’01, PhD’07; Graaja Ranade ’07; and Dr. Damiano Testa ’05- were also honed by the settlement program.
The first 29 award-winning projects will support mathematicians and researchers from universities and organizations who work to develop artificial intelligence systems to help advance mathematical discovery and research across multiple critical tasks.
Roe and Sutherland and Chris Birkbeck of the University of East Anglia will use their grants to enhance the automatic theorem, demonstrated by establishing a connection between the L-functional and modular form database (LMFDB) and the Lean4 Mathematics library (Mathlib).
“Automatic theorem plunders are technically involved, but they have insufficient resources for development,” Sutherland said. With AI technologies such as the Big Language Model (LLMS), barriers to entry to these formal tools are rapidly declining, allowing working mathematicians to access formal verification frameworks.
Mathlib is a large, community-driven math library for lean theorems for you, a formal system that verifies the correctness of each step in the proof. Mathlib currently contains 10 orders5 Mathematical results (e.g. lemma, propositions, and theorems). LMFDB is a huge collaborative online resource, an “encyclopedia” of modern digital theory, containing more than 10 kinds of9 Specific statements. Sutherland and Roe are executive editors of LMFDB.
The Roe and Sutherland grants will be used for a project aimed at enhancing both systems, making the results of LMFDB available in Mathlib as an assertion that has not yet been formally proven and provide an exact formal definition of the numerical data stored in LMFDB. This bridge will benefit both human mathematicians and AI agents and provide a framework for connecting other mathematical databases to formal theorem systems.
The main obstacles to automated mathematical discovery and proof are the limited number of formal mathematical knowledge, the high cost of formalizing complex results, and computationally accessible gaps, and the feasibility of formalization.
To address these barriers, researchers will use funds to build tools for accessing LMFDB from Mathlib, thus making a large database of non-forming mathematical knowledge accessible to the formal proof system. This approach enables the proof assistant to determine specific goals for formalization without the need to formalize the entire LMFDB corpus in advance.
“Building a large database of digital theory facts will provide powerful techniques for mathematical discovery, because an agent may want to consider a set of facts that may be considered when searching for the theorem or evidence, with an exponential fact greater than the fact that ultimately needs to formally prove theorem,” ROE.
Researchers point out that proofing the new theorem at mathematical knowledge boundaries often involves steps that rely on non-trivial calculations. For example, Andrew Wiles proved Fermat’s final theorem, using what is called “3-5 tips” at the key points in the proof.
According to Sutherland, “The fact that this technique depends on the fact that the modular curve X_0 (15) has only limited rational points, none of which correspond to semi-stable elliptic curves.” “This fact was well known before Wiles’ work and was easily verified using computational tools available in modern computer algebra systems, but it is not something that people can actually prove using pencils and paper, nor is it necessarily easy to formalize.”
While formal theorem plunderers are connecting to computer algebra systems for more efficient verification, leveraging computational output in existing mathematical databases also offers some other benefits.
The results of using storage take advantage of thousands of CPU-year computing time has been spent on creating LMFDBs, saving money needed to redo these computing. Having pre-calculated information can also make searches for examples or counterexamples without knowing in advance how widespread the search is. Furthermore, mathematical databases are curated repositories, not just random sets of facts.
“Number theorists emphasize the role of conductors in elliptic curve databases, which is crucial for a noteworthy mathematical discovery using machine learning tools,” Sutherland said.
“Our next step is to build a team to interact with the LMFDB and Mathlib communities to start formalizing the definition of LMFDB’s elliptic curves, numerical and modular table segments and to perform LMFDB searches from within Mathlib,” ROE. “If you are a MIT student interested in participating, feel free to contact us!”