I'm standing with Williamson in his office at the University of Sydney, on the seventh floor of the Carslaw Building, looking down at the campus's historic sandstone quadrangle. The Carslaw is renowned by staff and students as being the university's ugliest building, an antipodean approximation of a KGB regional headquarters. "The standard joke is that the best thing about working in Carslaw is that you don't have to look at it," Williamson says. His office is similarly unappealing – essentially a concrete bunker with bad carpet – and yet it has everything a mathematician might need; a whiteboard, marker pens, a computer, and most important of all, a couch. "When I moved in here, I told them 'I need a big couch!' " he says. "As opposed to things like medicine and science, which require specialised equipment – microscopes, X-rays – mathematicians can do most of their work with a pencil and paper. Really, you spend most of your time sitting around talking."

Fortunately, Williamson is a good talker – jaunty and light, his sentences tripping along before ending with an upward inflection, like a little trampoline kick-out off the final syllable. He's a little goofy. He smiles a lot; his eyes go wide. You get the sense that inside his head is a banging dinner party where all these brilliant ideas are elbowing one another to get out and roam around. Turning on his computer, he talks me through a slide display about representation theory – his area of expertise – and how it can, via spectral analysis of fundamental frequencies, explain why a whistle sounds different to a violin, and why, consequently, you'd rather listen to a concerto played by violins than a concerto played by whistles. An intriguing-looking textbook lies open on his desk, the pages crammed with cryptic glyphs and a photo of a Mayan pyramid. There's also a stack of shiny new books. "Our latest publication," he says, handing me one. I turn it over and read the back cover. "In this book," it says, "we conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block."

I want to ask: what is a "functor"? Who is Hecke? And why is the word diagrammatic in brackets? But instead, I ask: "Where can we get a sandwich around here?"

Williamson, 36, grew up in Canyonleigh, near Colo Vale in the Southern Highlands but has since 2008 been based overseas, at Germany's Max Planck Institute of Maths in Bonn, and at the Mathematical Institute at Britain's University of Oxford. His ability to use alternative methods to solve long-standing problems has won him a swag of awards, including the European Mathematical Society Prize, and the $100,000 New Horizons in Mathematics Prize. In May, he was elected the youngest living Fellow of the world's most prestigious scientific academy, the Royal Society (excluding 35-year-old Prince William, who is a Royal Fellow). Later this month, he will travel to London to accept his election, alongside South African engineer Elon Musk, the Tesla co-founder and billionaire. And although he downplays his chances, Williamson is also in the running to win a Fields Medal in August, considered the Nobel Prize of mathematics, which is awarded every four years by the International Congress of Mathematicians (ICM).

"Geordie is a world leader in his field, and his presence is much sought after at all the major mathematical centres," says Peter Sarnak, a professor of mathematics at America's Institute for Advanced Study, in Princeton. It was something of a surprise, then, to learn that Williamson had returned to Australia, late last year, to head up the University of Sydney's new mathematics research centre. Funded in part with a $5 million donation from the Simon Marais Foundation, it's hoped it will put Australia on the map as a place to do first-rate maths. Professor Anthony Henderson, the centre's executive director, says that "Australia is regarded as a great producer of mathematical talent – very few of the top-ranking institutions around the world wouldn't have an Australian mathematician. But we haven't had the institutions here to lure them back."

The centre, which launches in late 2018, is modelled on the Max Planck Institute and will host up to seven of the world's top mathematicians at a time, for anywhere between two months and a year. The idea is to provide a distraction-free atmosphere in which academics can concentrate on pure maths research, free of the burden of teaching or administration. "You benefit from the atmosphere when you get them all together," Henderson explains. "It means you can run seminars and less formal types of discussion where people exchange ideas, face to face." Because it's a "national" centre, it won't be able to hog the talent: regional institutions can apply to host one of the centre's guest academics.

Nabbing Williamson was no small feat. The University of Sydney had to beat out academic rivals and a long list of hedge funds, which prize maths gurus for their ability to analyse markets and spot opportunities. Money was only part of the equation: it's thought that the university couldn't match the offers from the big international institutions, which typically pay well over $400,000. (Big hedge funds can pay at least double that.) More important to Williamson was the chance to come home and be close to his family; his fondness for the University of Sydney, where he studied in the early 2000s; and the fact that he will have carte blanche with visiting mathematicians, inviting whomever he wants, whenever he wants. Then there is the unprecedented opportunity he now has to change the image of Australian mathematics. "People see my decision to stay here as a kind of mild intellectual suicide," he tells me. "The centre is all about turning that perception around."

Describing what Williamson does is almost as difficult as doing what Williamson does.Representation theory is all about the interaction of symmetries – reflections, rotations – and the insights they provide into seemingly unrelated phenomena. Since first being mooted in the late 1890s, it has become a central part of modern mathematics, informing our understanding of everything from sound waves to computer science and high energy physics experiments, such as the Large Hadron Collider particle accelerator, which in turn allows us to better understand the creation of new particles and what happens when chemicals combine and the way planetary bodies move and, which, therefore, in a roundabout way, helps us find the answers to life, the universe and everything. "Representation theory is everywhere!" Williamson says. "It's pretty fundamental."

Williamson is playful, creative and casually nonconformist, a slacker-esque figure with a 100-megawatt brain. "Geordie has given surprising answers to very hard questions in unexpected ways," says Professor Jacqui Ramagge, head of the University of Sydney's School of Mathematics and Statistics. "What is different about his work is that lots of people had tried and failed where he succeeded." A good example of this is the Lusztig conjecture, which was, since first being posited in 1979, the most famous "open problem" in representation theory. The entire field was convinced the conjecture was true, but they didn't know why. Colossal amounts of computing power had been dedicated to the problem; lots of exceptionally clever people had devoted their careers to it. Then, in 2013, Williamson demonstrated that the conjecture was false. "I remember being at a conference in representation theory in Shanghai at the time," says Anthony Henderson, "When the news came through about Geordie's discovery, all discussion immediately ceased."

The intellectual stamina involved in such tasks is typical of top-end maths. To be any good at it requires what the writer Harold Robbins once called "ass glue", the ability to sit and think for long periods of time, years even, in the face of constant uncertainty and setbacks. "It's very psychologically exacting," says Williamson, who spent seven years on and off on the Lusztig conjecture. "Essentially you are dealing with the unknown. You wake up, and don't get anywhere, and repeat that for a year. And I'm not exaggerating." Mathematicians often talk of making "discoveries", not merely finding "solutions".

Identifying the material benefit of such an endeavour can be problematic. Williamson sometimes works on problems simply because he can. Other times he works on them in the vague anticipation that somewhere along the track they may actually prove useful. G.H. Hardy, an English mathematician, was immensely pleased that his work in number theory, during the early 20th century, was "completely useless". But it's used every day by people logging onto Wi-Fi or buying a coffee with their credit card.

Besides, to Williamson, the beauty of such work is self-evident. Sitting in the cafe, having lunch, it strikes me that none of the people around us would likely understand the first thing about his work. "I know," he says ruefully. "I'm not kidding when I say it's real, genuine sadness when I walk down the street and there is someone I can't explain my work to." He pauses, looks out the window. "It's the same if people were not aware of Beethoven, or '70s music. It's a similar thing. That is the gap I feel."

Williamson grew up on a vegetable farm, in a mudbrick house that his parents, Mark and Leigh, built in the early '80s using timber cut from the surrounding bush and bits and pieces salvaged from the local tip. Williamson takes me there, one freezing morning in May. The property is small – 20 hectares – and sits in a bowl between two hills and a steep escarpment. It is very still and spookily quiet. A stream, now largely dried up, wends through stands of she-oak and manna gums. There are no other buildings in sight: the nearest house is four kilometres away. It is also very rundown, with random piles of lumber here and there, and the odd abandoned car. No one has lived here since Mark left three years ago, to move in with his new partner. (Leigh died in a cycling accident in 2003.)

Geordie grew up here with his brother, James, who was 19 months younger. The family never had much money. Leigh taught at primary school and Mark was a market gardener. But they were relentlessly curious, not to mention persistent. "It took me four years to build that house," Mark says when I meet him at the property. He then points to a large, corrugated iron shack, about 10 metres away from the main building. "Until then, we lived in there, which was interesting – especially in winter, when it gets to minus 12."

Mark and Geordie walk me through the family home, across the kitchen with its giant slate flagstones and woodfired stove, where all the meals were cooked. The property had no mains water – it was all pumped from the creek – and no mains power (it still doesn't). There was a temperamental generator, "but when that broke we used candles", says Geordie. The boys lived in the same room as their parents, separated only by a curtain, until they were in their teens, when they took it upon themselves to build their own extension.

The isolation was not just physical, but cultural. The house had a small black-and-white television, but no reception, "so I could never talk to other kids about The Simpsons and stuff like that," says Geordie. There was, however, a VHS player. Once a week, their aunt, Meg, would deliver video recordings of the Italian soccer premier league, which the brothers loved. But their main entertainment was in the bush, especially the nearby Belanglo State Forest, where, together with Geordie's best friend, Flint Duxfield, they would camp for days by themselves, shooting rabbits and fishing for eels. "I wanted to get them to have a sense of independence and self-reliance," says Mark. "The bush teaches you that, because as soon as you are outside yelling distance you have to be responsible for yourself." (The Belanglo State Forest later became infamous for the Backpacker Murders, when seven young travellers were killed by Ivan Milat between 1989 and 1993. "We were hanging out there at pretty much the same time," says Duxfield.)

Geordie went to primary school in Moss Vale – Leigh would drive him – but he got bullied for being a "smart-arse". He then went to the Steiner school in Bowral, half an hour away, until year 7. He found maths a breeze, and got easily bored: "I wanted to know all this stuff that wasn't being taught." One day, on a four-hour car trip to Bathurst, he read a 656-page physics textbook. (He was 12.) "Another time, around the same age, I was on a bushwalk and I discovered one of the other people on the walk was a physics lecturer. I kept asking him questions until he told me pretty blankly to shut up." His high school teachers could barely keep up. "A true mathematician who keeps me on my toes," wrote his year 9 maths teacher at Bowral's Chevalier College. Williamson's senior year maths teacher, John Franzmann, scored him 15 out of 16 for one of his four-unit assignments. Franzmann then added four bonus points, taking Williamson's mark off the charts. "I guess I was pretty impressed," Franzmann tells me.

Williamson left school in 1999, with an ATAR of 99.45 (the highest score is 99.95). He enrolled at the University of Sydney in a Bachelor of Arts degree, with a double major in science and arts. Williamson chose pure maths as his science option, but only as a backup. "What really interested me was English," he says. "I actually loved English more than maths. I remember in first year uni, I was like, 'This maths stuff is just bookkeeping to make a solid theory.' English was deeply interesting in a way that maths wasn't."

That changed in second year, when he came across the work of Évariste Galois. Born in Paris in 1811, Galois was a flamboyant prodigy and radical republican who, at the age of 18, solved a mathematical problem that had baffled experts for 350 years. (He died at the age of 20 in a pistol duel over a lover.) "It took the next 50 years for people to figure out how he had done it," says Williamson. "The thing I loved about it was that he had used the most phenomenally beautiful working. It was the first time that I saw that mathematics can be really, really deep."

In 2003, Williamson completed his honours in pure maths; he also earned the University Medal. The following year, he went to university in Freiburg in south-west Germany to work on his PhD. "In Freiburg, it's impossible to survive without German," he says. "I went to see my supervisor on the first day, and we got a coffee.

We chatted for about 10 minutes in English. He asked, 'Have you learnt some German?' And I said, 'Oh, yeah, guten tag, blah blah.' Then he said, 'Okay, so let's go back to my office and discuss maths.' And he then started to speak in German, and I understood essentially nothing of what he was saying. So it became very clear from him that I needed to learn German as fast as I could." He is now fluent in German and French, and is currently learning Spanish, as well as having a smattering of Mandarin.

He completed his PhD in 2008 and moved to the University of Oxford, where he became a postdoctoral research fellow. Then, in 2010, out of the blue, he learnt that his brother James had died. His death made no sense: James had always been extremely fit. As a boy, he had been a keen mountain-bike rider: in his early 20s he had begun competing in mountain-bike endurance events, quickly becoming Australian champion, then world champion in 2006.

He was competing in South Africa when he died, in his sleep, of an undiagnosed heart condition at the age of 26. "That was hard on Geords," says his friend Duxfield. "Having your mum and then your brother die so close to one another was a huge thing. But Geordie took a lot out of Jimbo's death."

Williamson says he was "destroyed by what happened. People at Oxford would say, 'Oh, it'll get better,' which sounded completely unbelievable, like a joke. But after I came back from it, I had a real feuer im arsch, which is a German phrase which means 'fire in the arse'. It all became very real to me, that idea that you may not be here forever."

Williamson regularly goes rock-climbing, which he took up as a kid, and practises yoga three days a week. "It really helps me, because I'm not very good at quieting my mind," he says. "And I'm not supercool under pressure." In 2017, he also began meditating. He would do it every day, for 10 to 20 minutes, first thing in the morning. "I got into this beautiful rhythm," he says. "I got up and did it straight away, before anything else, before I looked at my phone or had breakfast." Immediately after he stopped meditating, he would write down a couple of sentences. After a year, he had compiled a log of random thoughts, fears and preoccupations. Some of it was "very personal stuff", concerns he had with a family member or a friend. "I look back at it every now and again," he tells me. "It's incredibly valuable. I began to see that I was really worried about this particular thing, but that it passed, and it might come back, but then it would pass again. And it made me less afraid of stuff, because I realised that it would all pass."

The meditation log encouraged a certain intellectual adventurousness, and a willingness to experiment. "It's a huge asset in maths," he says. "People can get in the road of themselves. They become afraid of trying something, for fear of failure. If you're a tightrope walker, fear of failure is legitimate. If you're a mathematician, it's not. I notice this a lot with students. You put a certain problem in front of the class, and half of them will go, 'It's impossible, I don't get it.' But some of them will play with it and learn and maybe find a solution."

He is likewise concerned about high school maths, where standards and participation rates have been declining for more than a decade. The latest report from the Trends in International Mathematics and Science study, released in 2016, ranks Australia below Kazakhstan, Poland and Bulgaria. The mere mention of the word "maths" induces in many students an almost allergic reaction, if not outright antipathy. "You couldn't get away with saying, 'Oh, I hated English in school,' and just walking away from it," the University of Sydney's Jacqui Ramagge tells me. "But people do it all the time with maths."

Williamson feels a similar frustration. "With sport, in Australia, there's an understanding that it takes years of training and toil to get to the very top level," he says. "But there's this general perception that a so-called 'genius' or a great artist simply waves their hands and the magic happens. It's dangerous for kids, because they have that feeling that you're either good at maths or bad at maths, rather than keeping on going and working at it."

Perhaps the new maths centre will be a circuit breaker. Apart from anything else, it can be viewed as an affirmation of ambition, and a multimillion-dollar bet on Australian mathematics. There is also the no small matter of Williamson getting a new office, away from his concrete cubicle in the Carslaw Building. There's only one item he'll be taking with him. The couch.

The story Maths prodigy comes home to establish $5 million world-class maths centre was originally published on The Sydney Morning Herald.