Top Math Breakthroughs of 2024: Langlands, AI, Sphere Packing and More

Hope you haven’t started getting bored yet this week. Let’s get straight to the point — well, to last year’s point. 2024 was a genuinely remarkable year in mathematics: long-awaited proofs, unexpected counterexamples, and problems solved with the help of AI. See for yourself.

The Geometric Langlands Program: A Math Breakthrough 30 Years in the Making

One of the biggest achievements of the year was the proof of the geometric Langlands conjecture. This conjecture is part of a sweeping program that unifies different areas of mathematics into a single theoretical framework. The resulting proof runs over 800 pages and represents the culmination of three decades of work. Mathematicians now expect a wave of new discoveries built on top of this result.

AI Starts Doing Math — and Doing It Well

Not long ago, AI systems made embarrassing errors on the simplest calculations. In 2024, Google DeepMind’s AlphaProof learned to prove theorems at the level of an International Mathematical Olympiad winner. AI is no longer just solving problems — it’s uncovering new patterns in mathematics, such as strange symmetries in elliptic curves that resemble the murmuration of a flock of birds.

Sphere Packing and the First Major Math Breakthrough in 75 Years

How do you pack spheres in space to fill as much volume as possible? In three dimensions, the answer is well known: stack them in a pyramid, like oranges at a grocery store. In 8- and 24-dimensional spaces, answers were found about a decade ago. But what about all other dimensions in the general case? In 2024, mathematicians improved on known methods by exploiting chaotic configurations — marking the first major advance in this area in 75 years.

The Milnor Conjecture Falls: A Surprising Counterexample

Sometimes mathematicians don’t prove conjectures — they disprove them. That’s exactly what happened with the Milnor conjecture, which for 50 years had seemed almost self-evident. It turned out that the space of possible shapes of geometric objects is even stranger than anyone had imagined.

New Progress on the Riemann Hypothesis and the abc Conjecture

Number theory had its share of math breakthroughs in 2024 as well. Researchers managed to tighten the bounds on possible exceptions to the Riemann hypothesis — one of the most important unsolved problems in all of mathematics. On top of that, new proofs emerged for special cases of the abc conjecture, which links deep properties of prime numbers in a surprising way.

What’s Next for Mathematics in 2025?

Mathematics advances not only through landmark breakthroughs, but through the gradual accumulation of ideas. Progress in number theory, geometry, and AI-driven research suggests that 2025 has even more surprises in store.

Which of these breakthroughs surprised you the most?