AI-Assisted Exploration in Algebra and Number Theory
Mentor: Dr. Deependra Singh
Overview
Inspired by the recent work Mathematical exploration and discovery at scale by Georgiev, Gomez-Serrano, Tao, Wagner, and Google DeepMind, this project explores how LLM-based agents can be systematically integrated into research in algebra and number theory. The central idea is to combine three complementary modes of mathematical work into a single pipeline:
Computational exploration: Using evolutionary and code-based agents (in the spirit of AlphaEvolve) to perform large-scale numerical and symbolic experiments, with the aim of identifying patterns and generating plausible conjectures
Informal mathematical reasoning: Developing insights from experimentation into coherent, human-readable arguments, in the style of extended reasoning systems such as DeepThink
Formal verification: Translating successful arguments into fully rigorous, machine-checked proofs using proof assistants such as Lean, guided by tools similar to AlphaProof
Approach
As a first stage, we will apply this pipeline to a set of classical problems in number theory, including results on sums of squares, quadratic forms, and the $u$-invariant of local and global fields. These benchmarks serve both as a training ground for deploying the methodology and as a means of evaluating its strengths and limitations, since the underlying results are well understood.
Having calibrated the approach on these known cases, we will then turn to carefully selected open problems in less explored settings. In particular, we aim to investigate whether this framework can yield new insights for arithmetic questions over fields beyond the local and global fields, such as semi-global fields (for example, $\mathbb{C}((t))(x)$ and $\mathbb{Q}_p(x)$), where similar questions are less well-understood and systematic exploration may be especially informative.
Prerequisites
Fields, Galois Theory.