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A FEW corners of the internet still rock something of a 1998 vibe, and Princeton University’s server has one of them. The text is in Times New Roman and comes in clashing shades of red, blue, mauve and orange. Down the left side is a list of seemingly obscure phrases like “Impossibility theorems”, “Spin glass” and “Separatrix separation”.

This is the website for , which, as the name suggests, is a list of the most mind-bending unsolved maths conundrums in physics. Crack any one of these beasts and you would probably earn yourself a Fields medal, the maths equivalent of a Nobel prize. That, and the editor of the website will post a cartoon explosion next to the problem bearing the word SOLVED!

One of these problems has had mathematicians stumped for years, with legions of them having made only piecemeal progress. Few people outside physics have heard of the quantum Hall conductance problem, but it is intimately connected with experiments that get us closer than ever to harnessing the mysterious power of quantum technology. You can imagine the anticipation, then, when a newcomer named Spyridon Michalakis claimed he had the answer. Fittingly, however, his solution to this impossible problem was itself impossible to understand – or at least nearly impossible.

Michalakis grew up in Greece, spending his summers on the island of Lesbos, playing beach volleyball in the sun with his two brothers. In the evenings, his siblings were glued to screens playing video games while he tinkered with maths puzzles. In 1994, when one older brother returned dejected from a prestigious national maths competition, 14-year-old Spyridon swiped the exam paper and solved the lot. It took him three days, not the 3 hours allocated in the competition, but he had always been stubborn when it came to mathematics.

He went on to get a doctorate in quantum information science, but the story of his monumental proof begins a little later, in a sushi restaurant in 2008. A couple of days after Michalakis arrived in town for his first academic job, at the Los Alamos National Laboratory in New Mexico, he met up with his new supervisor, , for lunch. As Michalakis poured the tea, Hastings asked: “So, would you like to work on something interesting or just something to get you warmed up?” Michalakis said he was ready for a challenge.

Later that day, Hastings sent him a link to that and explained that the problems had been collected between 1998 and 1999 by the mathematical physicist Michael Aizenman (see “5 of the world’s toughest unsolved maths problems”). Michalakis could see that only one had the cartoon explosion next to it, although in truth it was and is only partially solved. He would later discover that even this partial solution had earned one mathematician a Fields medal in 2006, with another awarded in 2010 to a different contributor.

Hastings wanted him to try his hand at the quantum Hall conductance problem. Michalakis had never heard of it. And since Hastings was keen to let his protégé stand on his own two feet, what followed was months of hair-pulling frustration. “I was freaking out,” says Michalakis. “It’s like saying to somebody: ‘Alright, you have the right characteristics to be a Hollywood star – just do it’.”

“Spyridon spent his days playing beach volleyball and his evenings doing maths”

To get started, he had to understand what the quantum Hall effect is. Discovered in 1879 by the physicist Edwin Hall, it occurs when an electrical current is passed through a sheet of metal while a magnetic field is applied at right angles. Hall noticed that a small amount of current was deflected off to the side.

Hall’s discovery was an experimental tour de force, coming 18 years before the electron, the particle that gives us electricity, was even discovered. But the effect came into its own in 1980, when physicist Klaus von Klitzing repeated the experiment, this time running the current through a thin interface between a metal and a semiconductor that were cooled to almost absolute zero. Under these conditions, the electron flow is essentially two-dimensional, and as we now know from materials like the ultra thin graphene, this totally changes its properties. Von Klitzing showed that if you slowly increase the magnet’s power, the deflected current doesn’t grow steadily as you would expect, but makes sharp jumps separated by plateaus, like steps on a staircase.

In other words, the current was acting much like particles do in the quantum realm, where they can only take on certain energy values. This, however, was quantum behaviour visible at an everyday scale.

Even more impressive than that, imperfect experiments behaved just the same way. Using a different type of semiconductor, or one that contained lots of impurities, made not a jot of difference to the steps. This made a splash because quantum effects are usually notoriously picky, only showing themselves in carefully choreographed experiments.

“This was quantum behaviour visible on an everyday scale”

Interest in the effect deepened in 1982, when researchers at what was then AT&T Bell Laboratories refined the experiments. In experiments with stronger magnetic fields and even lower temperatures, they found new, equally consistent steps between the old ones. They gave this new phenomenon a name of its own: the fractional quantum Hall effect.

Michalakis quickly found out this much, but it took him longer to understand how theorists had begun to scratch at why the quantum Hall effect itself happened. A crucial aspect of the story, he discovered, came in the 1980s when physicist David Thouless and others began to paint a picture of the quantum Hall effect that involves something called topology.

Topology is the mathematical study of the global properties of objects. Think of it as zooming out further than geometry. It becomes useful when you are interested in the properties of shapes that are unaffected by small changes. Imagine squeezing a bagel, for example: it is no longer ring shaped, but there is no question that it is still a bagel.

If you want to describe this sort of thing mathematically, two properties come in handy. First, the genus, which says how many holes an object has. And second, the winding number, which describes how many times one object can be looped around another (see “The maths of holes and loops”). Whole numbers are important in this game – good luck making an extra half-hole in a bagel.

Thouless and his collaborators treated the electrons in the quantum Hall effect like a continuous sea and used topological language to describe it. Doing so explained everything von Klitzing had observed. Like a squashed bagel, their topological description of the electron sea didn’t change noticeably if you subtly altered the electrons, as impurities in the semiconductor would do, for example. This provided mathematical clarification for why the effect was precise even when the experiment wasn’t. It also turned out that winding numbers, which can only be a whole number like 1 or 2, were inextricably linked in the maths to the voltage at which the steps appeared.

Calming the sea

But beneath the surface, Thouless’s description was never realistic. According to quantum theory, the electrons in that quantum sea can each harbour any amount of energy, and some of them should have had enough to push the conductance upwards before each observed step. But when anybody tried to portray the electrons in this way, the maths became intractable. So Thouless artificially smoothed out the energy. This made calculations far easier by averaging out the energy fluctuations. Unfortunately, however, there were no good grounds for making this assumption and so the problem remained unsolved.

This was the state of play when Michalakis was tearing his hair out in Los Alamos. But as he began to digest more about the problem, something began to click. There must have been a reason why Hastings set him this task. He began scrutinising his boss’s back catalogue of research and soon found tools with promise for solving the problem.

From then on, his lack of experience turned into an advantage, allowing him to bypass many of the dead ends that had stopped more acclaimed experts in their tracks. Gently guided by Hastings, he gradually moulded a new understanding of the problem.

The essential ingredient Michalakis and Hastings brought was a more nuanced understanding of the topology of the problem. The best existing description of the quantum Hall effect at the time used a tool called adiabatic evolution, developed over the previous decade. This aimed to explain how the electrons in the quantum sea don’t suddenly have enough energy to jump to the next step of the staircase as we turn the dial on the magnetic field, but must instead wait until they get to the foot of the subsequent step. But this still relied on a Thouless-style averaging out of the energy fluctuations.

“The proof was in limbo: too important to reject, too impenetrable to accept”

The duo replaced this with a refined version of one of Hastings’s tools called quasi-adiabatic evolution, which, unlike its predecessor, was itself topological. This allowed them to see the sea’s overall energy landscape and level it out by massaging all the peaks and troughs – after all, topology isn’t fussed about such details.

It may sound like cheating, but Michalakis believed it provided a mathematical riposte to the questions that had lingered ever since Thouless’s work. Fourteen sleep-deprived months after their sushi lunch, Michalakis and Hastings completed the proof and . The problem was solved.

Or was it? When they tried to publish the work officially, the academic referees couldn’t make head or tail of it. “Hastings and Michalakis’s paper was extremely technical,” says , a physicist at the Technion technology institute in Haifa, Israel. “From all the leaves, I couldn’t see the forest.” Avron is one of those who originally proposed adding the problem to the Princeton website in 1999 and who therefore had the power to declare it solved.

“I may have been the only one for quite a while who believed that their arguments were essentially sound and complete,” says Bruno Nachtergaele, Michalakis’s former mentor at the University of California, Davis. He says neither of the duo have the patience to write out well-presented mathematical proofs. So the paper sat in limbo for more than three years, too important to reject, too impenetrable to accept.

That didn’t stop others from trying to wrap their minds around it. Six months after the paper was finished, mathematical physicist Yoshiko Ogata at the University of Tokyo invited Michalakis to explain the solution. Ogata got six epic 90-minute lectures. But even after this, the audience remained largely flummoxed.

Hastings and Michalakis went through the proof one last time in 2013, making it as tight and elegant as possible. But the still faced resistance from the referees, one of whom was Avron. He talked with the pair on the phone, but struggled to grasp the maths. “When I would ask them something, I think they misunderstood my question and I misunderstood their answer,” he says. As well as speaking to them both multiple times over two years, he read and reread the paper, and attended their lectures. Eventually, from his own comprehension of certain parts, and from the word of experts who understood other parts, he was satisfied.

No more limbo

That allowed the paper to be published in 2015, but Avron wouldn’t mark the problem as solved on the Princeton web page until he could fully grasp it himself. That came shortly afterwards, when other researchers posted . One of the first was Roma Tre University in Italy and his collaborators, who used more conventional mathematical tools to analyse the problem. Their work provided an alternative proof that the Hall effect is quantised, while also confirming critical assumptions Michalakis and Hastings had made in their work.

Soon afterwards, Avron finally marked the problem “SOLVED!”, nearly 19 years after it was posted and almost a decade after Hastings and Michalakis had first completed their proof.

Not everyone agrees this is the final word. “They proved the topological aspect of the quantum Hall effect. They didn’t prove that you have these plateaus in conductivity,” says , a physicist at the Georgia Institute of Technology known for his work on the quantum Hall effect. But Michalakis disputes this and Bellissard does say he thinks the work is a breakthrough that will be part of the eventual solution.

Regardless of that, the proof is starting to make waves because it helps point the way to practical uses of the quantum Hall effect. Just look at where Hastings now works: Station Q, Microsoft Research’s lab in California, where the company is trying to build a new kind of quantum computer.

Most quantum computers aim to take advantage of the fact that subatomic particles can be in different states at the same time, a property that could make certain types of intractable computations dead easy. The trouble is that quantum bits, or qubits, based on particles are delicate, with the merest vibration breaking the spell. Qubits governed by topology, however, are much more robust, like the quantum Hall effect itself. “We have a new kind of qubit that Microsoft is trying to build,” says Hastings. “A topological qubit.”

“The proof is helping us build a new type of quantum computer”

The qubits Microsoft is attempting to make are based on tiny imperfections in the sea of electrons, called anyons. Hastings and Michalakis’s proof offers insights that could help create them.

So how about that Fields medal? Maths’ top prize is given out every four years to up to four mathematicians under the age of 40. Michalakis was 29 and Hastings 36 when they originally posted their proof, but the near-decade it has taken for the scientific community to accept it means neither will be eligible by the time the prize next rolls around in 2022. That doesn’t bother Michalakis though. He now blends research with science communication, notably as a consultant on the Ant-Man movies. “I don’t care about prizes any more,” he says. “For me, the real pleasure of figuring things out is in sharing the journey with the next generation.”