The Year in Math | Quanta Magazine

IIn May, a team of nine mathematicians announced a major breakthrough. they have Proved the so-called geometric Langlands conjecture ——A core component of a broader research program to construct a “grand unified theory” of mathematics. The proof, which totaled more than 800 pages, marked the culmination of 30 years of work and was, in the words of one mathematician, “a crowning achievement.”

“This is beautiful mathematics,” said another. “Best in class.”

It is the best of its kind, not only because it is a groundbreaking mathematical result—it solves a huge open problem and is now expected to influence research for decades to come—but also because it involves building deep, meaningful Unexpected connections. Often the best results occur when mathematicians find ways to bring seemingly unrelated ideas into conversation with each other, breaking down barriers between different fields of research. The proof of the geometric Langlands conjecture is such a result.

This isn’t the only big development coming in 2024. Some, like the Langlands case, finally overturned decades-old speculation. Others offer surprising counterexamples.

But such breakthroughs don’t usually come out of nowhere. They were made possible through decades of hard work, through the accumulation of incremental steps. This year has also seen many exciting results in this style, especially in number theory. These include advances in famously difficult problems such as the Riemann Hypothesis and ABC Speculation.

This is how mathematical progress works, for the most part: a new idea here, another idea there, until things that once seemed completely impossible become less impossible.