ZAP // Dall-E 2
Geometry, number theory and even artificial intelligence. A year of records and important discoveries in mathematics paves the way for a great 2025.
2024 was a year of many — and important — discoveries in the field of mathematics, recalls .
There was developments in famous problems and considered until now intractable, such as the Riemann hypothesis and the abc. The highlights also include the Langlands conjecture, finally proven after several decades, but also the resolution of current problems, regarding, for example, AI.
Here’s what marked mathematics this year, from geometry to more everyday practices, such as packing spheres.
1. Langlands Geometric Conjecture
2024 was the year in which , which dates back to the 80s and allows decompose functions into simpler “waves”.
The conjecture links several fields of mathematics and involves esoteric mathematical objects called beams and is very complex, having taken almost 50 years to be proven.
The problem required overcoming some complex mathematical challenges, e.g. deal with “irreducible representations”a concept from representation theory.
According to Quanta, this discovery proves that mathematicians are excited to spend the next few years exploring its consequences, which they believe will be far-reaching.
2. Prominent role in AI
This year, new models from Google DeepMind turned AI into a serious contender in the International Mathematics Olympiad, the world’s premier math competition for high school students. In January, the company announced the AlphaGeometry, a model capable of solving geometry problems almost as well as a human gold medalist.
It was thanks to mathematics applied in AI that technology reached, this month, a human level of “general intelligence”.
Mathematicians, in order to improve it and make it as intelligent as possible.
When trying, for example, to understand “mathematical murmurs” in important equations called elliptic curvesdiscovered them in many different number theory objects, which led to important new work and knowledge, including the development of a new type of function.
3. Ball packing record
How to organize identical spheres so that they fill as much volume as possible without overlapping each other? This is the problem with sphere packing.
In three dimensions, it is possible to arrange the spheres in a pyramid-shaped pile, in the same way that oranges are stacked in a grocery store. And in larger dimensions?
A new investigation improved the efficiency of previous packaging (first discovered in 2016), while also using a new approach: Instead of packing the spheres neatly, as the first study had done, mathematicians used graph theory to pack the spheres in a very disorderly way.
Two mathematicians — including Thomas Hales, who proved the optimal way to pack spheres in three dimensions in the 1990s — still make a statement about the worst possible ways of packing.
4. Counter-example to the Milnor Conjecture
Three mathematicians found counterexamples to the Milnor conjecture, a 50-year-old geometry problem about the relationship between the general shape of an object and its appearance when made zoom about him.
The work, which involved the development of a new type of structure, revealed that the universe of possible forms is even stranger than mathematicians imagined — although they always imagined it to be, in fact, very strange.
5. Advances in number theory
There were major advances that laid a foundation for the study of number theory this year.
An example is the two mathematicians who proved a new estimate of the number of possible exceptions to the arguably the biggest open problem in mathematics.
The work of these researchers broke the previous record, which had been held for 80 years, and led to new results on the distribution of prime numbers.
Also three postgraduate students proved a better estimate of the largest size that sets can reach before they inevitably have to contain evenly spaced patterns of numbers.
There were new estimates of certain cases of guess abcanother of the most important problems in mathematics — and also one of the most controversial.
