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.
Nan Cao for Quanta Magazine
The Geometric Langlands Conjecture Is Proved
Arguably the biggest result of 2024 comes from the Langlands program, an ambitious, 50-year-old vision that, if realized, will connect very different realms of mathematical study. Its goal is essentially to redraw the map of mathematics — to bring its separate continents together into one united Pangaea.
But proving actual results in the Langlands program is often prohibitively difficult. The statements themselves are incredibly complicated and technical, to say nothing of the techniques needed to prove them.
In the 1980s, a mathematician posed a geometric version of one of the program’s key components — the geometric Langlands conjecture, which involves esoteric mathematical objects called sheaves. The conjecture was seen as a central part of the Langlands program, but for decades, no one could solve it — until this year. The proof is a huge boon to the rest of the program, and mathematicians are excited to spend the next several years exploring its consequences, which they believe will be far-reaching. As one mathematician put it, “It’s going to seep through all the barriers between subjects.”
A Growing Role for Artificial Intelligence
When new artificial intelligence systems such as ChatGPT (a type of “large language model”) were first introduced, their mathematical abilities were the stuff of memes, and not in a good way. The chatbots failed to add numbers correctly, much less solve more complicated word problems. As for producing full-fledged proofs — forget it. When it came to math, it seemed that AI would only continue to struggle.
But this year, new models from Google DeepMind turned AI into a serious competitor in the International Mathematical Olympiad, the world’s premier math competition for high school students. In January, the company announced AlphaGeometry, a model capable of proving geometry problems nearly as well as a human gold medalist. Within half a year, AlphaGeometry 2 could achieve gold easily — and, when integrated with Google’s large language model Gemini, could prove more general problems well enough to score a silver medal on the full Olympiad exam. DeepMind called this newest model AlphaProof.
AlphaProof was a massive achievement. It showcased AI’s fast-growing mathematical prowess and offered hope for how the technology might one day act as a mathematical “copilot” in original research.
In March, Quanta reported on how this is already happening. In 2022, mathematicians used machine learning to uncover strange patterns in important equations called elliptic curves. The patterns were beautiful, startling: If you looked at certain numerical properties of elliptic curves from the right perspective, they resembled the way birds flock, a phenomenon called a murmuration. In the couple of years since, researchers have been trying to understand these mathematical murmurations. In doing so, they’ve found them in lots of different objects in number theory, leading to important new work and insights, including the development of a novel kind of function.
As AI methods get more sophisticated, this story will only become more common. We’ve seen it before — in how computers gradually entered the world of mathematics and opened up new vistas for mathematicians to explore. Now mathematicians are trying to predict what that will look like with AI.
Sphere-Packing Records Get Broken
Unlike the geometric Langlands conjecture, the sphere-packing problem is simple to state: How do you arrange identical spheres to fill as much volume as possible without overlapping? In three dimensions, you can arrange spheres in a pyramid-shaped pile, the way oranges get stacked in a grocery store. But what about in higher dimensions?
No one knew the answer for any dimension higher than three until 2016, when the Ukrainian mathematician Maryna Viazovska proved that particular lattices are optimal for packing spheres in eight- and 24-dimensional space. In all other dimensions, the exact answer remains unknown.
Mathematicians also want to find a general solution — a formula that gives a way to densely pack spheres in arbitrarily high dimensions, even if the packing isn’t completely optimal. In April, Quanta reported on the first significant advance on this version of the sphere-packing problem in 75 years. The result improved on the efficiency of previous packings, while making use of a novel approach: Rather than packing spheres in a nice, organized way, as Viazovska had done, the mathematicians used graph theory to pack spheres in a very disorderly fashion.
This wasn’t the only packing result of 2024. Two mathematicians — including Thomas Hales, who proved the optimal way to pack spheres in three dimensions back in the 1990s — proved a statement about the worst possible shapes to pack, too.
Alex Eben Meyer for Quanta Magazine
Mathematicians Find Counterexample to 50-Year-Old Milnor Conjecture
Proving old conjectures is important — but so is disproving them. As one mathematician told Quanta, “We have to be suspicious, even about things that intuitively look very likely to be true.” That’s the approach that resulted in yet another major geometry proof: Three mathematicians found counterexamples to the Milnor conjecture, a 50-year-old problem about the relationship between an object’s overall shape and what it looks like when you zoom in on it. The work, which involved the development of a new kind of structure, revealed that the universe of possible shapes is even stranger than mathematicians imagined — even though they’d always imagined it to be pretty strange indeed.
Nico Roper for Quanta Magazine
Important Steps Taken in Number Theory
Solving these major geometry problems was akin to erecting towering monuments in the landscape of mathematics. But it’s also crucial to lay better foundations for the monuments to come. This is exactly what happened in number theory in 2024: Mathematicians made crucial, albeit incremental, progress toward a better understanding of some of the most important questions in the field.
Two mathematicians, for instance, proved a new estimate of the number of possible exceptions to the Riemann hypothesis, arguably the biggest open problem in math. The work not only broke the previous record, which had stood for 80 years, but led to new results about the distribution of prime numbers.
Similarly, three graduate students proved a better estimate of the largest size that sets can reach before they must inevitably contain evenly spaced patterns of numbers. The work, which deals with how order inevitably emerges from disorder in mathematics, marks the first progress on “Szemeredi’s problem” in decades.
Meanwhile, the mathematician Hector Pasten recently made progress toward understanding properties of the sequence 2, 5, 10, 17, 26 and so on — that is, all numbers of the form n2 + 1. His proof allowed mathematicians to probe the complicated relationship between the operations of addition and multiplication. It also allowed him to prove new estimates of certain cases of the abc conjecture, another of the most important problems in math — and also one of the most controversial.
All of these number theory questions are still a long way off from being solved. But by inching closer, mathematicians have developed powerful new tool kits and illuminating new perspectives. Who’s to say what might happen in 2025 and beyond?
