Surprisingly given its significance to the human condition and our daily lives, seldom will you hear about mathematics in our news. It would be understandable for you to assume that because of this mathematics is static and unchanging. However, as ever in 2024, some of the smartest people around the world have been working on both old and new problems, theories and programmes with the potential to change our world. Whether it’s bringing us a step closer to creating a Grand Unified Theory of mathematics, understanding randomness in our universe, the astonishing advances of AI in maths, or even just discovering a new prime number, there have been many great achievements in the world of maths over the past year.
This is a summary of the past year in mathematics.
A step towards unifying mathematics? Geometry’s Langlands Conjecture finally proved
It might have taken three decades, five academic studies and one thousand pages, but this year a team of nine mathematicians led by Sam Raskin have solved one third of what has been called mathematics’ ‘Rosetta Stone’. This achievement has been hailed as having far-reaching implications for mathematics, physics and quantum field theory.
Originated by Robert Langlands in the late 1960s, the Langlands program is a series of yet-to-be-proved ideas establishing deep connections between seemingly unrelated areas of mathematics, translating complex waves into smoothly oscillating sine waves. These conjectures have been fundamental to modern mathematics and unlock new ways of thinking about mathematical ideas. Much like the Rosetta Stone, the program covers three disparate branches of mathematics: number theory, harmonic analysis and geometry.
A new set of papers have now settled the Langlands conjecture in the geometric column of this Rosetta Stone, with a proof involving more than 800 pages spread across five papers. Raskin’s research specifically focused on algebraic geometry: using shapes and geometry to imagine algebraic equations. The team spoke of finding many routes into the heart of the problem, having “encircled the problem from every direction” so “it had no way to escape,” and creating a “rising sea” of ideas to tackle the problem.
“It is extremely beautiful, beautiful mathematics, which is connected very much with other mathematics and with mathematical physics.”
– Alexander Beilinson, one of the main progenitors of the geometric version of the Langlands program
“We always knew that there was some very big mystery, and until we solve that we won’t be able to do the full proof. I thought it would take decades to prove it, and suddenly they cracked it.”
– Dennis Gaitsgory, director of the Max Planck Institute for mathematics in Bonn, Germany, who worked closely with Raskin on his team
“Mathematical research isn’t necessarily geared towards big problems, but it’s geared towards incremental progress and understanding things a little bit better. And sometimes you have a new idea which is interesting, and you play with it; if you get really lucky, then it connects to some big stuff.”
– Professor Sam Raskin, who led the team proving the geometric conjectures
Ultimately, the aim of Langlands’ conjectures is to unite the various continents of mathematics into one, united ‘Pangea’, and thanks to decades of research from dedicated mathematicians, perhaps a Grand Unified Theory is a little closer.
Find out more about how the Geometric Langlands Conjecture was proved here
The rise of AI: The International Mathematical Olympiad’s most significant silver medallist?
Earlier this year in the British city of Bath, over 600 students from more than 100 countries came to compete at the 65th International Mathematical Olympiad. But while the US broke China’s five-year winning streak, despite China’s star mathlete Haojia Shi achieving a perfect score for the second year in a row, there were other, perhaps more seminal, headlines leaving the county of Somerset this summer.
Two machines developed by Google DeepMind also entered the competition, charmingly named AlphaProof and AlphaGeometry 2. Despite taking much longer on some problems than the allotted time for competing students, together the programs solved four of the six problems and achieved a score of 28 out of 42, the level of a silver medalist. Only 60 young mathematicians from around the world scored higher.
AlphaGeometry 2 solved the geometry problem correctly in just 19 seconds, while AlphaProof was able to solve one number theory and two algebra problems, including one that only five human contestants managed to crack. AlphaProof took as long as 60 hours to solve some problems, significantly more than the 9 hours students have available to them, and while the students would have undoubtedly scored more highly given this amount of time, this is still a massive breakthrough in the performance of theorem solvers, and as efficiency gains are made this time is bound to come down.
The model failed to solve the two combinatorial problems (finding the number of ways to arrange, combine, or select objects from a set according to certain constraints), possibly because they are very difficult to translate into programming languages.
AlphaProof works in the same way as the algorithms that mastered board games like chess and Go, using reinforcement learning to compete against itself and improve step by step. This is fairly simple to implement for board games, as the AI makes a number of moves and if it does not win it tries new strategies the next time. However, in mathematics, the program not only has to check it has found the correct answer, but that each of its reasoning steps are also correct. To do this, AlphaProof used a proof assistant, an algorithm which checked the steps in the AI’s logical reasoning. These have been around for decades, however as there are very few maths problems written in a language, such as AlphaProof’s native Lean, their use has been limited.
However, the internet is an endless resources of maths problems humans have solved step-by-step in human languages. The DeepMind team trained the large language model Gemini to translate a million problems into the Lean programming language to be used by the proof assistant to train AlphaProof. “When presented with a problem, AlphaProof generates solution candidates and then proves or disproves them by searching over possible proof steps in Lean,” say DeepMind. AlphaProof gradually learns the proof steps that are useful and those which are not, improving its ability to solve more complex problems.
Geometry problems, however, require a very different approach. AlphaGeometry 2 is an evolution of a model released by DeepMind in January that could successfully solve IMO geometry problems. This model uses a large set of geometric premises, for example a triangle with given lengths and points, and uses a ‘deduction engine’ to infer properties of the figure. Experts trained the AI on a training dataset of theorems and proofs, coupled with a large language model that sometimes uses ‘auxiliary constructions’, for example extending a line to form a perpendicular.
“We might be close to having a program that would enable mathematicians to get answers to a wide range of questions. Are we close to the point where mathematicians are redundant? It’s hard to say.”
– Timothy Gowers, Fields medallist and previous IMO gold medallist, one of the mathematicians who scored the models’ performance
This is surely only the beginning of AI advancements in mathematics, and goes to show how AI continues to take astonishing leaps in capability, and perhaps the once-thought-unconquerable field of Olympiad maths problems is slightly closer to their reach.
Find out more about these AIs’ achievements here
How random! Michel Talagrand receives the ‘Nobel Prize of Mathematics’
Earlier this year, French mathematician Michel Talagrand was awarded the 2024 Abel Prize for his “groundbreaking contributions” to describing and predicting the universe’s randomness.
“There was a total blank in my mind for at least four seconds. If I had been told an alien ship had landed in front of the White House, I would not have been more surprised.”
– Talagrand describing hearing the news of his award
“It’s like a piece of art. The magic here is to find a good estimate, not just a rough estimate.”
– Hedge Holden, a mathematician at the Norwegian University of Science and Technology and the Abel committee chair
The Abel Prize, named after Norwegian mathematician Niels Hendrik Abel, is the closest equivalent to a Nobel Prize in the world of maths and one of the most prestigious awards in the field, awarded annually by the King of Norway. The winner is chosen by a committee of five internationally recognised mathematicians and comes with a 7.5 million kroner (£858,941) cheque.
Talagrand’s work focused on systems which model random variables in a given time and space, known as stochastic systems, for example the height of a flowing river, stock prices, the number of patients at a hospital, the movement of gas molecules and even the path of a stumbling drunkard. The Frenchman makes sense of these systems using mathematical inequalities to improve the characterisation of their limits, turning complex systems into geometrical terms and yielding precise estimates.
“It turned out the solution was not that difficult. But of course, you couldn’t get up in the morning and figure it out. There has to be a lot of humble work.”
– Talagrand
Using Talagrand’s methods may inform where to safely build a house along a rushing waterway, or how to anticipate the growth of a bacterial population.
The Abel committee also commended another aspect of his work that shows even random systems have some element of predictability, for example rolling a die 600 times will predictably result in having rolled six around 100 times.
The 72-year-old’s work is even more impressive considering that, due to a genetic condition, he became blind in his right eye at the age of 5 and blind in his left at the age of 15. During a lengthy treatment at hospital, his father taught him the discipline of maths. A self-confessed average student before the second incident, Michel returned to school with a sharper mind for maths.
“It turned out the solution was not that difficult,” Talagrand tells the New York Times. “But of course, you couldn’t get up in the morning and figure it out. There has to be a lot of humble work.”
The mathematician, now 72 years old, has followed a similar approach to his own life. At the age of 5 he became blind in his right eye after its retina detached, and a decade later, the same thing happened with his left eye, as a result of a genetic condition. He underwent a long treatment in a hospital, where his father, a university math professor, taught young Talagrand the discipline. A self-described average student before that second incident, he returned to school with a sharper mind for math.
His advice for the students of today: “You can fail to solve a problem 10 times, but that doesn’t matter if you succeed on the 11th try.”
“I’m not able to learn mathematics easily. I have to work. It takes a very long time and I have a terrible memory. I forget things. So I try to work, despite handicaps, and the way I worked was trying to understand really well the simple things. Really, really well, in complete detail. And that turned out to be a successful approach.”
– Talagrand
Talagrand retired in 2017 after working at the French National Centre for Scientific Research (CNRS) for 43 years, and has become the 27th, and fifth French recipient of the Abel Prize since it was founded in 2003.
Find out more about Talagrand’s work here
Packing spheres randomly is best?
In the year that we held our Malteser-Teaser Experiment, where people had to guess the number of spherical chocolates in a jar given certain details, new advances were made on the mathematically-best ways to pack spheres, but in higher dimensions.
Sphere packing is certainly a lot easier to understand than the Langlands conjectures. The question is simple: How do you arrange identical spheres to fill as much volume as possible without overlapping? We have been studying the best way to pack spheres packings since Johannes Kepler posed in 1611 that the most efficient way to pack equally-sized spheres is to place them in a pyramid, like you see with oranges at the shops today. This may seem simple, but this was not settled until 1998, when Thomas Hales finally proved Kepler’s conjecture with 250 pages of mathematical arguments. Maryna Viazovska was revered for being the first to find the optimal latices to pack spheres in more than three dimensions, in eight and 24 dimensions specifically, achievements that earnt her the prestigious Fields Medal in 2022, only the second woman and second Ukrainian to do so.
However, mathematicians have also wanted a general solution giving a way to densely pack spheres in arbitrarily high dimensions, even if the packing is not completely optimal. After all, mathematicians love to generalise concepts into higher dimensions. You arrange squares like a chequerboard. You arrange cubes like moving boxes. This generalises to higher dimensions.
However, packing spheres is much harder. The known optimal sphere packings in two, three, eight and 24 dimensions are or are similar to lattices, full of patterns and symmetry but in every other dimension, the best packings might be totally chaotic.
This year a team of four mathematicians made the first major advance on this aspect of the sphere packing problem in over 75 years, improving 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.
There were other sphere packing developments this year, with two mathematicians, including Thomas Hales, proving the worst configurations to pack spheres.
Maybe packing spheres in as many as 24 dimensions won’t help future Malteser-Teaser contestants win a jar of chocolates, but it is inherently useful in the error-correcting codes used by mobile phones, space probes and the Internet to send signals through noisy channels.
Find out more about sphere packing in higher dimensions here
I like big primes: Largest ever prime number discovered!
Ever since Euclid established over 2,300 years ago that there are an infinite number of prime numbers, that can only be divided by themselves and one, the hunt has been on for mathematicians to find the largest primes they can find, mustering vast amounts of computing power and algorithmic ingenuity in the hope their name ends up in the history books.
The latest record holder for world’s largest prime is Luke Durant from California, who’s discovery contains an uncomprehendable 41,024,320 digits. Just to put that into context, that is around half a million times the number of atoms in the observable universe (around 80 digits), and would take around 230 days to read.
Prime numbers play a major role in pure mathematics, are the most important numbers in number theory, and in the real world, for example in encryption algorithms.
Durant’s success in part came from new software from the Great Internet Mersenne Prime Search, and his own heavy-duty hardware. He assembled a supercomputer in the cloud across 17 countries, attempting to break a prime record held for six years by ending a long tradition of personal computers discovering primes.
Until 1951, primes were found by hand, but computers quickly took over the search. As amateurs looked to find larger and larger prime numbers, they started looking at specific types of prime. Mersenne primes, named after the 17th century French theologian who studied them, Marin Mersenne, are prime numbers generated by multiplying two by itself a prime number of times, and then subtracting one (2n-1 where n is prime). This does not always yield a prime, but we have a quick method known as the Lucas-Lehmer primality test for checking whether numbers generated by this formula are prime.
This test fuelled the Great Internet Mersenne Prime Search, which since 1996 has allowed amateurs to download free code that searches for Mersenne primes on their computer. The seven largest-known primes are all Mersenne primes found by amateur enthusiasts.
Do you want to etch your name into the prime number history books? Are you ready to up the ante even further next year? Well, you’ll need plenty of computing power behind you.
Find out more about how to find these enormous prime numbers here
More stories from the world of maths this year
A Year in Maths Society: Badges, Maltesers and our podcast’s pilot
While it has been a big year for the world of maths in general, it has been an especially special year for all of us at Maths Society. Whether it was our AGM in March, our aforementioned Malteser-Teaser stall, recording the pilot episode of our podcast, launching our newsletter, or even just publishing an article or holding a meeting, this last year has been unforgettable. But, as ever when looking back at the old year, the most exciting part is knowing everything that is possible the next year and remembering to thank everyone who has backed and supported you along the way. So, to everyone who is reading this feature, a massive thank you from everyone at Maths Society.
Well, what a year 2024 was. If you are considering a career in academia, or are simply a fan of maths and mathematicians, hopefully as we head into a new year full of opportunity, you can be inspired by these stories, only a select few of the events of this year, and appreciate what really is possible in the world of maths.