The Year of Math | Quanta Magazine

In May, a group of nine mathematicians announced a major breakthrough. They have proves the so-called geometric Langlands conjecture — a central part of a broader research program to build a “grand unified theory” of mathematics. The proof, which ran to more than 800 pages and marked the culmination of 30 years of work, was, in the words of one mathematician, a “supreme achievement.”

“This is beautiful math,” said another. “The best of its kind.”

It is the best of its kind not only because it is a significant piece of mathematics – one that solves a huge open problem and is now poised to influence decades of research to come – but because it involves making deep and unexpected connections. Often, the greatest results occur when mathematicians find ways to put seemingly unrelated ideas into dialogue with each other, breaking down barriers between different areas of study. The proof of Langlands geometric conjecture is the same result.

This is not the only major development in 2024. In fact, there are many signs of evidence in the field of geometry alone. Others, like the Langlands case, are finally putting decades-old assumptions to rest. Others, however, offer surprising counterexamples.

But such developments do not usually come from nowhere. It was made possible through decades of effort, by accumulating incremental steps. This year, there are many exciting results of this flavor, too, especially in number theory. These include developments in famous intractable problems such as the Riemann hypothesis and the abc speculation.

That’s how mathematics progresses, for the most part: a new idea here, another there, until what once seemed completely impossible becomes trivial.