The AI Langlands Program

AI collaborating with Mathematicians

In spring 2022, undergraduate Alexey Pozdnyakov discovered something that shouldn't exist. Running machine learning algorithms on elliptic curves, he found flowing patterns resembling starling murmuration . . . structures that had escaped mathematical notice for centuries. This moment marks AI's arrival as a true partner in mathematical discovery.

This book reveals how AI is revolutionizing pure mathematics through pattern discovery, automated proof generation, and the management of complexity. Machine learning analyzes millions of mathematical objects simultaneously, uncovering hidden structures, such as Pozdnyakov's murmuration. DeepMind's AlphaProof constructs rigorous proofs at a competition level. AI suggests new conjectures by detecting correlations humans miss. Most crucially, it helps verify and navigate massive proofs, such as the 800-page geometric Langlands conjecture—a cornerstone of mathematics's most ambitious unification project.

The Langlands Program, connecting number theory, geometry, and algebra, provides the book's narrative spine. We follow Andrew Sutherland analyzing a billion elliptic curves with AI, Nina Zubrilina using computational insights to derive mathematical formulas, and teams unlocking forty-year-old problems through machine learning. These aren't just computational feats—AI develops a form of mathematical intuition, recognizing promising strategies and suggesting research directions.

Through mathematician profiles, we explore profound questions: What is mathematical understanding when AI discovers truths humans can verify but not fully grasp? How do we preserve creativity when proofs exceed human comprehension? The book makes complex concepts accessible while offering practical guidance for researchers and educators navigating this transformation.

This isn't AI replacing mathematicians but amplifying their vision. The murmuration was just the beginning—a glimpse of a mathematical universe far richer than imagined, now revealing itself through human creativity and artificial intelligence combined. It's an invitation to witness and participate in mathematics' most significant transformation.

A Python project demonstrates how AI techniques can be applied to discover patterns and correspondences in the spirit of the Langlands program. While simplified for accessibility, it illustrates key concepts from the book.

Complete code and explanation at:
https://github.com/alessoh/AI-Langlands