One of the biggest stories in science is quietly playing out in the world of abstract mathematics. Over the course of last year, researchers fulfilled a decades-old dream when they unveiled a proof of the geometric Langlands conjecture ' a key piece of a group of interconnected problems called the Langlands programme. The proof ' a gargantuan effort ' validates the intricate and far-reaching Langlands programme, which is often hailed as the grand unified theory of mathematics but remains largely unproven. Yet the work's true impact might lie not in what it settles, but in the new avenues of inquiry it reveals. Proving the geometric Langlands conjecture has long been considered one of the deepest and most enigmatic pursuits in modern mathematics. Ultimately, it took a team of nine mathematicians to crack the problem, in a series of five papers spanning almost 1,000 pages1'5. The group was led by Dennis Gaitsgory at the Max Planck Institute for Mathematics in Bonn, Germany, and Sam Raskin at Yale University in New Haven, Connecticut, who completed his PhD with Gaitsgory in 2014....

A shard of smooth bone etched with irregular marks dating back 20,000 years puzzled archaeologists until they noticed something unique ' the etchings, lines like tally marks, may have represented prime numbers. Similarly, a clay tablet from 1800 B.C.E. inscribed with Babylonian numbers describes a number system built on prime numbers. As the Ishango bone, the Plimpton 322 tablet and other artifacts throughout history display, prime numbers have fascinated and captivated people throughout history. Today, prime numbers and their properties are studied in number theory, a branch of mathematics and active area of research today. Informally, a positive counting number larger than one is prime if that number of dots can be arranged only into a rectangular array with one column or one row. For example, 11 is a prime number since 11 dots form only rectangular arrays of sizes 1 by 11 and 11 by 1. Conversely, 12 is not prime since you can use 12 dots to make an array of 3 by 4 dots, with multiple rows and multiple columns. Math textbooks define a prime number as a whole number greater than one whose only positive divisors are only 1 and itself....

The growth of rose petals exploits a geometric trick previously unobserved in nature, physicists have found. Using theoretical analysis, computer simulations and experiments with rubbery plastic sheets, they established that as the petals curl outwards, mechanical feedback regulates their growth, leading to the formation of rolled edges and pointed corners at their tips. The findings, described in Science on 1 May1, could one day have applications in engineering and architecture. 'You might learn new principles and then implement them in manmade structures,' says Eran Sharon, an experimental physicist at the Hebrew University of Jerusalem. Geometric patterns are known to affect developing organisms. But in all previously observed cases, the principles had to do with the 'intrinsic' geometry of surfaces. Intrinsic features are those related to the distances between points on the surface, as would be measured by an ant walking on it. But a surface of a given intrinsic geometry can have multiple ways of existing in 3D ' or 'extrinsic' geometries. For example, a sheet of paper can be laid flat or curled up into a cylinder; and an ant walking on it would not notice any changes in the distances it covers, even though points on the sheet could be closer to or farther from each other in 3D space....

Are quantum computers worth the billions that are being invested in them' The answer is probably many years away. However, the machines could prove to be particularly suited to solving problems in mathematics ' especially in topology, the branch of maths that studies shapes. In a preprint posted on arXiv in March1, researchers at Quantinuum, a company headquartered in Cambridge, UK, report using their quantum machine H2-2 to distinguish between different types of knot on the basis of topological properties, and show that the method could be faster than those that run on ordinary, or 'classical', computers. Quantinuum chief product officer Ilyas Khan says that Helios, a quantum computer that the company expects to release later this year, could get much closer to beating classical supercomputers at analysing fiendishly complicated knots. Although other groups have already made similar claims of 'quantum advantage', typically for ad hoc calculations that have no practical use, classical algorithms tend to catch up eventually. But theoretical results2,3 suggest that for some topology problems, quantum algorithms could be faster than any possible classical counterpart. This is owing to mysterious connections between topology and quantum physics. 'That these things are related is mind-blowing, I think,' says Konstantinos Meichanetzidis, a Quantinuum researcher who led the work behind the preprint....