Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.

In May, a team of nine mathematicians announced a major breakthrough. They had proved what is called the geometric Langlands conjecture — a central component of a broader research program to build a “grand unified theory” of mathematics. The proof, which totaled more than 800 pages and marked the culmination of 30 years of work, was, in the words of one mathematician, a “crowning achievement.”

“It is beautiful mathematics,” said another. “The best of its kind.”

It is the best of its kind not just because it’s a seminal piece of mathematics — one that resolved a huge open problem and is now poised to influence decades of research to come — but because it involved forging deep and unexpected connections. Often, the greatest results come about when mathematicians find ways to put seemingly unrelated ideas in dialogue with each other, breaking down the barriers between different areas of study. The proof of the geometric Langlands conjecture is just such a result.

This wasn’t the only major advance of 2024. In fact, there were several landmark proofs in the field of geometry alone. Some, as in the Langlands case, finally put decades-old conjectures to bed. Others offered surprising counterexamples instead.

But such breakthroughs don’t usually come out of nowhere. They’re made possible by decades of effort, by an accumulation of incremental steps. This year, there were many exciting results of this flavor, too, especially in number theory. Among them were developments on famously intractable problems such as the Riemann hypothesis and the abc conjecture.

That’s how mathematical progress works, for the most part: a new idea here, another there, until what once seemed completely impossible becomes a little bit less so.

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