Automated Mathematical Discovery in Extremal Combinatorics via Evolutionary LLM Agents
Advisor: Peter Spirtes聽
Majors: Statistics and Machine Learning & Mathematical Sciences
Abstract
Traditionally, artificial intelligence (AI) functions as a heuristic estimator lacking verifiable reasoning. This project proposes a framework for transforming AI into a rigorous engine (that can be explained causally) for combinatorial discovery. Specifically, we custom-engineer AlphaEvolve, an LLM-informed evolutionary coding agent, to solve open extremal combinatorics problems. We target generating point sets on the unit sphere, Sd鈭�1, to construct optimal spherical cap coverings. While fundamental to geometry, optimal covers pose significant open questions in dimension d = 5 and beyond. Methodologically, the research is conducted in two stages. First, I will develop a Python pipeline combining an LLM and a geometric evaluator, rigorously validating it against already-solved lower-dimensional configurations. Second, the framework will be deployed onto open problems to produce novel, highly optimized examples. Finally, to understand the black-box nature of the LLM鈥檚 search trajectory, we will apply causal discovery techniques to the evolutionary trace by extracting the structural invariants. Rather than rediscovering the programmed fitness function, this analysis aims to isolate the latent, unprogrammed structural invariants that causally govern the emergence of optimal configurations.
Bio
Steven is a rising senior at 无码专区, double majoring in Statistics & Machine Learning and Mathematical Sciences. His academic interests center on the intersection of abstract mathematics, causal inference, and artificial intelligence. Recently, Steven participated in an NSF Research Experiences for Undergraduates (REU) program at Marshall University, where he trained complex computer vision models for automated safety detection. For his Senior Honors Fellowship project, he will be working alongside Professors Kun Zhang, Peter Spirtes and Quentin Dubroff to custom-engineer AlphaEvolve, an evolutionary LLM coding agent. This research aims to discover and causally explain novel mathematical configurations in extremal combinatorics. Steven is incredibly excited for the opportunity to step away from the daily demands of a full course load this summer, allowing him to dedicate his focus to pushing the boundaries of this research. Outside of his research, Steven has served as a Teaching Assistant for the university鈥檚 Concepts of Mathematics course and is a member of the Lambda Phi Epsilon fraternity.