By DAVID MYTON
If Australian academia has a star right now then it has to be Geordie Williamson. The University of Sydney mathematician has been in the news, with reports and colour spreads in mainstream media, as well as coverage in the more specialised publications.
At the age of 36, he was recently elected the youngest living Fellow of The Royal Society – one of seven Australians and 50 distinguished scientists globally – and was also named a Fellow of the Australian Academy of Science.
Is he a celebrity academic?
“No,” he replies firmly. “I don’t feel like a star, not at all. All the attention will be over in a couple of weeks.
“It is strange because the recognition comes a long time after you’ve done the work. People are talking about things I kind of did five years ago.
“I’m excited about the things I’m doing now.”
Williamson’s rare ability to use “alternative methods to solve long-standing problems” has seen him win numerous awards, including the European Mathematical Society Prize, the New Horizons in Mathematics Prize, the Chevalley Prize of the American Mathematical Society, and the European Mathematics Society Prize.
In 2020 he will lead a special year on Representation Theory at The Institute for Advanced Study in Princeton, New Jersey.
When I spoke to Williamson he had just returned from the Royal Society function in London.
“It was an amazing three days,” he says. “Some of the work that was discussed was incredibly inspiring.”
This month, in yet another first, Williamson became the only Australian-based mathematician to deliver an address to the quadrennial International Congress of Mathematicians (ICM), in Rio de Janeiro.
Williamson, along with colleague Professor Anthony Henderson, has been appointed director of Sydney University’s new maths research institute – the first of its kind in Australia and which has been likened to Germany’s prestigious Max Planck Institute for Mathematics.
“It’s one of my dreams to be involved in this,” he says. “It has great potential. Our goal is to increase Australia’s visibility as a place to do first-rate mathematics.”
The poetry of the sciences
Williamson describes maths as “the poetry of the sciences”. His major area is representation theory – the study of linear symmetry – a branch of pure mathematics, and he also researches in algebra, geometry and number theory.
Williamson’s achievements include his proof (with Ben Elias) of the Kazhdan-Lusztig positivity conjecture, algebraic proof of the Jantzen conjectures, and his discovery of counter-examples to the expected bounds in the Lusztig conjecture in modular representation theory.
His work centres on what he calls “complicated structures that only exist in a higher dimension”.
“A very nice analogy is to think about mathematical research as exploring new terrain – and a breakthrough is like finding a new path into a new valley,” he says.
“It gives you pieces of knowledge about the landscape and so you try to put those together. You also have conjectures to guide you, and you might use heuristics and make guesses – for example, ‘imagine that the following is true’ – and then you’ll explore the consequences of that guess.”
He references the Hungarian mathematician George Polya:
“Polya said that if there is a problem you can’t solve, there is an easier one that you can solve – so find it. And I love that because when you solve the easier problem it gives you a little hint about where to go next.
“The work that I do may perhaps have concrete application for the real world in the years to come, or perhaps not – but it certainly has important consequences within modern pure mathematics.”
An incredibly beautiful idea
Williamson grew up in the Colo Vale area south of Sydney. After attending local schools in Moss Vale and Bowral (achieving an ATAR of 99.45) he went on to the University of Sydney, graduating with a first-class BA in 2003, winning the University Medal in the process.
It was in his second year of university that his deep interest in maths began to take root.
“I was attending a class on Galois Theory and in that lecture I saw an idea that was incredibly beautiful and something I could never have come up with myself. Until then, maths had seemed to me to be just the formal manipulation of symbols,” he recalls.
In 2004 he moved to the University of Freiburg in Germany from which he graduated in 2008 with a PhD in pure maths. He then spent time at Germany’s Max Planck Institute of Maths in Bonn, and at the Mathematical Institute at the University of Oxford.
He was appointed professor of mathematics at Sydney in 2017. He is also a visiting research fellow at Hausdorff Centre for Mathematics in Bonn.
A day in the life
A typical day begins with him sitting in a cafe “thinking a little bit about what I want to do that day”, followed by four or five hours in the morning trying to make progress in research – “and this involves sitting at my desk until my brain is fried” – and then writing, proof reading, submitting papers, and checking on his students’ work.
He estimates he spends about 30 per cent of his time “basically sitting down with pen and paper, tying to think”; about 40 per cent “talking to others or trying to do calculations with others”; and the rest writing computer code “and running tests on super computers” in Sydney and Bonn.
He acknowledges he is “incredibly lucky” in that he has “enormous amounts of time to dedicate to research”.
However, he “absolutely loves” teaching and engaging with students … “but I feel people are often overloaded with teaching and that takes away the pleasure of it”.
I ask Williamson about how he came to be called Geordie – where I come from it usually refers to someone who hails from the northern England city of Newcastle.
But in this case it’s all down to the great American singer and musician Joan Baez – his late mother Leigh was entranced by Baez’s version of the traditional English folk song Geordie that begins:
As I walked out over London bridge
One misty morning early,
I overheard a fair pretty maid
Was lamenting for her Geordie.
“Mum loved that song,” he says.
Deep understanding of the world
Williamson doesn’t see himself as being good at maths – rather he just “really, really loves the subject”.
“I have some natural desire to apply the laws of mathematics and physics to understand the world. It’s some very natural instinct,” he says.
At school he “absolutely loved” literature and reading.
“That was my source of a really deep understanding of the world, and it was my complete passion until about second year at university – but now maths to some extent has replaced that. But I still love reading and I read a lot.”
Does he have empathy for people who struggle with maths?
“I understand that for many people maths is a real difficulty… I was talking recently to an Uber driver who said the biggest pain in his life was mathematics.
“I find that really sad. For me that says you have never had a good teacher.”
Williamson contends that with a good education “almost anybody can appreciate the beauty of mathematics”.
However, he argues that learning maths is different to other subjects “in that it builds on itself the whole time”.
“If you somehow lose contact with your English class it’s easier in Years 9 and 10 to get back once you start working again or once you have a better teacher; whereas in maths, if you miss algebra, it’s very difficult to keep going. I think this is a big issue.”
Maths teaching in Australia
Williamson is concerned that Australia’s performance in mathematics and sciences education at school level lags behind many other nations.
“I think it’s a real problem,” he says. “Perhaps I’m biased, but I feel the whole world is moving in a more mathematical direction, and so maths is very useful and practical to help us to understand the modern world.
“At the same time there’s evidence that in Australia our skills in mathematics are declining, and I think this is a real issue for our society.”
He is at pains to point out that he is not criticising teachers. It’s more about the way maths and sciences are perceived.
“I feel very, very strongly that if we can just change the narrative a little bit around mathematics, science and computer science – if people think about it as being a fascinating place where you can be enormously creative, to picture it like an adventure – I think that could change the way people relate to mathematics and science.”
He also feels there is a need to improve the numbers and quality of university maths postgraduate programs.
“Traditionally any really good student is told to go to the US to do their PhD. And to some extent that’s still the case, but I think that now – given the improved quality of mathematics in Australia at university level – it’s no longer so clear that it’s the best idea to go to one of the top US institutions.
“There are extremely good mathematicians here in Australia that could supervise students, but somehow we don’t have the structures yet to rival the graduate schools in the US. I would say that nowadays the faculty is up to standard but the structures are not.
Williamson says he has concerns about Australia’s three-year PhD program.
“This makes it very difficult to compete with the US because they have a-five year PhD in mathematics.
“And so the PhD students just have a lot more time to digest and produce a good thesis.
“The other difficulty is that in the US a PhD student basically gets a fully-funded position involving teaching – you don’t pay fees, you teach, and you get a stipend based on your teaching and this is enough to live from. That structure is very clean and very clear.”
He is excited by the prospects for Sydney University’s new maths research centre.
“It has great potential. I think that to have dedicated research institutes in Australia is the next natural step and is very important”.
Value of the arts and humanities
Williamson believes the arts and humanities are “extremely important” and are also underrated in Australia.
“Our society seems to love things that we can attach numbers to,” he says.
“We love measurements – GDP, blood pressure, calories – and I think the arts suffer because we can’t attach a number to them. That’s a big pity.”
Good writing, he says, is also a vital ability for mathematicians – the job is not just about scribbling formulas all day long.
“Writing well is very important in communicating your work.”
He cites as examples of fine writers the mathematical physicist Edward Witten from Princeton University – “he is an exceptional writer” – and the University of Chicago’s Alexander Beilinson … “some of his papers I think are very, very close to great poetry”.
“I feel that the mental process you go through in understanding a poem is not dissimilar to what you go through when understanding a deep mathematical theory.”
‘The most beautiful word that I know’
Humanities scholars may have favourite books, but do mathematicians have favourite numbers?
They do. Williamson loves the Italian word for 15 – quindici.
“That’s purely because I love the way it sounds – I think it’s the most beautiful word that I know actually.”
Other favourites are Pi (3.141592 etc) E (Euler’s “irrational number” – 2.718281828 etc) i (the “imaginary number” and square root of -1) and 1,000,003 – “it’s a prime number and it’s useful”.
“Sometimes on a computer I need a large prime number for various things, and it’s good to know that this is prime and it’s easy to remember”.
‘We are carving out a small path into the unknown’
I ask Williamson if there may be a limit to what can be known in mathematics as suggested by Godel’s Incompleteness Theorems and philosophers such as Soren Kierkegaard.
This view can be summed up in the Latin phrase often quoted by scholars of previous eras: ignoramus et ignorabimus – “we do not know and will not know”.
The renowned German mathematician David Hilbert in a famous radio address in 1930 declared against this, asserting: “In place of the foolish ignorabimus let stand our slogan: We must know, We will know.”
Williamson says: “For me the statement that there’s always something unknowable is a bit like saying the sky is blue. Of course.
“I think the unknowable is everywhere, and we are carving out a small path into the unknown.”