A Revolution in Pure Mathematical Discovery

In spring 2022, something extraordinary happened in a mathematics lab that would have been impossible just a decade earlier. Alexey Pozdnyakov, an undergraduate student, discovered patterns in elliptic curves that had remained hidden for centuries—not through traditional mathematical reasoning, but by training machine learning algorithms to analyze millions of mathematical objects simultaneously. The patterns he found, dubbed "murmurations" for their resemblance to the flowing movements of starling flocks, marked a watershed moment: artificial intelligence had become a true partner in mathematical discovery.

This remarkable story is just one of many chronicled in H. Peter Alesso's groundbreaking new book, The AI Langlands Program, which explores how artificial intelligence is fundamentally transforming the landscape of pure mathematics. Far from the popular narrative of AI replacing human workers, this book reveals something far more profound and exciting: AI as an amplifier of human mathematical intuition, capable of revealing patterns and connections that have existed forever but remained invisible to even the greatest mathematical minds.

The Grand Quest for Mathematical Unity

At the heart of the Langlands Program, often called the "grand unified theory of mathematics." Proposed by Robert Langlands in 1967, this ambitious vision suggests that seemingly unrelated areas of mathematics—number theory, geometry, and algebra—are actually different faces of the same underlying reality. It's as if mathematicians had been studying different parts of an elephant in the dark, and Langlands proposed that these parts all belonged to the same magnificent creature.

The Langlands Program has driven some of the greatest mathematical achievements of the past half-century, including Andrew Wiles' proof of Fermat's Last Theorem. But as these achievements accumulated, so did their complexity. The recent work on the geometric Langlands conjecture spans over 800 pages of mathematics so dense that, as one expert admitted, "almost no one can explain" it fully. This is where AI enters the story—not as a replacement for human insight, but as a necessary partner in navigating complexity beyond any individual's grasp.

Murmurations: The Pattern That Shouldn't Exist

The book's most captivating example of AI-human collaboration is the discovery of murmurations. When Pozdnyakov averaged certain properties of elliptic curves across thousands of examples, the resulting graphs didn't show random scatter or smooth curves. Instead, they traced out graceful, oscillating patterns that no mathematician had thought to look for. As the book explains, "The idea of averaging these values across thousands of curves wasn't mathematically unnatural, but it wasn't motivated by any existing theory."

This discovery exemplifies what makes AI such a powerful mathematical collaborator. Neural networks don't respect mathematical tradition or theoretical prejudice. They simply look for patterns, regardless of whether those patterns align with human intuition. The murmurations were always there, waiting in the data—it just took an alien intelligence to point them out.

From Pattern Recognition to Proof

But discovering patterns is only the beginning. Alesso masterfully guides readers through how these AI discoveries are transformed into rigorous mathematical understanding. After Pozdnyakov's discovery, mathematician Nina Zubrilina derived an explicit formula for the murmuration patterns, converting computational observation into theoretical insight. This exemplifies the book's central theme: the most powerful mathematical advances now come from the interplay between human creativity and artificial intelligence.

The book details numerous other breakthroughs enabled by this partnership. DeepMind's work on Kazhdan-Lusztig polynomials, which had stumped mathematicians for forty years, showcases AI's ability to detect subtle patterns in combinatorial data and suggest specific formulas that humans can then prove. In knot theory, AI discovered unexpected connections between algebraic and geometric properties, providing what one mathematician called "a treasure map" showing exactly where to dig for new theorems.

A New Kind of Mathematical Thinking

Perhaps most fascinating is Alesso's exploration of how AI is developing what can only be called mathematical "taste." Modern language models trained on vast corpora of mathematical text don't just calculate—they recognize elegant approaches, suggest promising strategies, and even express preferences for certain proof techniques. This emergent mathematical intuition, while alien to our own, is proving remarkably effective at guiding research.

The book also addresses the practical challenges of modern mathematics. With major proofs now spanning thousands of pages and requiring expertise across multiple specialized fields, verification has become a critical bottleneck. AI-powered proof assistants are emerging as essential tools, helping to formalize and verify these massive logical structures while maintaining the creative essence of mathematical discovery.

The Future of Mathematical Discovery

What makes The AI Langlands Program essential reading is its vision of mathematics' future. Alesso doesn't present AI as a threat to human mathematicians but as an evolutionary leap in our ability to explore mathematical reality. Young mathematicians are learning to collaborate with AI from the start, treating it as a knowledgeable colleague who can help them explore unfamiliar territory and see connections across vast mathematical landscapes.

The book includes a Python project demonstrating how readers can apply AI techniques to discover patterns in mathematical data, making these cutting-edge methods accessible to students and researchers. Complete code and explanations are available on GitHub, inviting readers to participate in this revolution rather than merely observe it.

lity itself.