Collaborative Research: AIMing: Advancing Combinatorics and Number Theory with AI

This project will develop a rigorous framework for collaboration between mathematicians and artificial intelligence (AI) systems to accelerate mathematical discovery, with a focus on fundamental problems in number theory and combinatorics. Mathematics relies increasingly on large and complex datasets, where subtle patterns can be difficult for researchers to recognize using traditional methods. By transforming patterns detected by AI into understandable mathematical statements and rigorously verified results, the project will advance both mathematics and the responsible use of AI in scientific research. Beyond pure mathematics, the reproducible methodologies established in this project will provide other data-intensive scientific fields with reliable tools for pattern discovery. Thereby, it supports national priorities in AI by emphasizing interpretable, trustworthy, and verifiable methods. Furthermore, the project will link and strengthen the mathematics and AI communities through a biweekly virtual seminar, summer schools and conferences, and research opportunities for students and other early-career researchers. Datasets, software, and educational resources developed in this project will be made publicly available to bolster future research endeavors. Toward its goals, the project will establish and test a four-phase, iterative workflow for mathematical discovery. First, large datasets of complex number-theoretic and combinatorial objects will be generated and curated. Second, machine-learning methods will be applied to detect non-obvious patterns within these datasets. Third, specialized tools will be employed to interpret the way in which the trained AI models make decisions, allowing the team to translate the this internal logic into precise mathematical conjectures and algorithms. Finally, the resulting claims will be validated through rigorous mathematical proof, automated theorem provers, or AI-guided searches for counterexamples. Initial results have already demonstrated the success of this approach by extracting new mathematical patterns and algorithms from the internal representations of AI models, and subsequently proving them as theorems. Specific objectives include building a public database of noncongruence modular forms to study their arithmetic properties, using reinforcement learning (RL) to optimize equations for algebraic curves, and developing RL agents to discover explicit mathematical mappings for unsolved conjectures. Through these activities, the project will produce new mathematical knowledge while establishing a reproducible and broadly-applicable methodology for interpretable, verifiable, and human-guided AI-assisted discovery. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.