Beauty in Numbers
Geordie Williamson talks maths, poetry and launching SMRI
By Laura Watson
Professor Geordie Williamson has had a busy year. Not only has he been setting up the new Sydney Mathematical Research Institute (SMRI) at the University of Sydney, but he’s been gathering awards and accolades too, most recently the Australian Mathematical Society Medal.
Acknowledged as the world leader in modular representation theory, Professor Geordie Williamson’s work addresses the most fundamental questions in representation theory, transforming the field and opening new questions. For the past half century, fundamental yet unproved conjectures, including Lusztig’s and James’ conjectures, underpinned much of the work in representation theory.
His paradigm-breaking series of counterexamples to the expected bounds in Lusztig’s 1980 conjecture about reductive groups also invalidated James’ 1990 conjecture. In collaboration with others, he followed up this breakthrough with the remarkable achievement of a new character formula for irreducible representations of reductive groups in characteristic p, replacing Lusztig’s disproved conjecture, and providing an unconditional proof of a correction of Lusztig’s character formula whenever p is at least 2h–2.
This extraordinary result provided a new framework for thinking about these conjectures, thus opening the field of modular representations to new ideas. On another front, Williamson has collaborated with Lusztig to develop a new approach for such questions. Their conjecture, supported by the available computer evidence, would imply the striking consequence that the modular decomposition numbers for the symmetric groups grow at least exponentially.
Geordie’s breakthroughs have not gone unnoticed – in 2018 alone, he was elected both Fellow of the Royal Society (where he is also the youngest living Fellow) and Fellow of the Australian Academy of Science, and was the first mathematician working in Australia to give a plenary address at the International Congress of Mathematicians.
In 2017 he shared the New Horizons in Mathematics Prize, worth $US100,000, with Ben Elias from the University of Oregon, and in 2016 he received the Clay Research Award, the European Mathematical Society Prize and the Chevalley Prize of the American Mathematical Society.
After a number of years with Max Planck Institute in Bonn, Geordie returned to Australia in 2017 to head the new SMRI after accepting a mathematics professorship with the University of Sydney. Following a similar researcher-in-residence model as the Max Planck Institute, the institute will increase the number of international mathematicians researching in Australia.
IT WOULD BE FAIR TO SAY 2018 HAS BEEN A BIG YEAR FOR YOU, WHAT DO YOU CONSIDER YOUR BIGGEST HIGHLIGHT?
Giving a talk in front of so many great mathematicians in Rio was a major highlight. Barry Mazur calls mathematics “the world’s longest conversation”, and I love this phrase. Sometimes we work for decades just to be able to make one small remark at the big round table of mathematics.
Probably the biggest highlight is launching the new Sydney Mathematical Research Institute. It has been a dream of mine for some time to be involved in such an institute in Australia. That it happened so fast was a big, and very pleasant, surprise. I think it has great potential … now we need to get to work!
CAN YOU TELL US ABOUT ONE OF THE OUTSTANDING PROBLEMS IN YOUR FIELD?
We have this tremendously useful and versatile theory of representations of finite groups over the real or complex numbers. However, over fields of positive characteristic (like finite fields) the classical theory breaks down. For about a century, a few visionaries have suspected that there is also a beautiful theory there, that lies much deeper than the theory over the complex numbers. Potentially we are on the verge of knowing what this theory is. Fascinating connections to number theory are also emerging.
HOW IMPORTANT IS COLLABORATION TO YOUR WORK IN MATHEMATICS?
Collaboration is an extremely important part of my research. I’ve learnt more from my collaborators than I’ve learnt from books or papers. It is an honour to be able to work with such amazing people. Certainly, the most useful parts of conferences for me are the coffee breaks and walks.
WHAT ARE THE BENEFITS OF BRINGING INTERNATIONAL MATHEMATICIANS TO AUSTRALIA FOR EXTENDED RESEARCH?
The benefits will be threefold: mathematical research in Australia will grow and develop; the reputation of Australian mathematics world-wide will increase; and the institute will help foster integration within Australian mathematics. The Institute supports visits to any Australian university to carry out research, and visits to multiple destinations are encouraged.
WHAT DO YOU THINK NEEDS TO BE DONE TO TRANSFORM THE WAY MATHS IS VIEWED BY THE AVERAGE AUSTRALIAN?
My feeling is that a cultural change needs to take place. A friend commented recently to me that many are taught in school to hate maths. Perhaps there is some truth to that, sad as it may be. How do we x this? Many people have thought more about this than I have, but my two cent’s worth: one can work on showing the general public the joy of mathematics (with small sudoku-like problems, with fun and humour) at the same time as showing the bigger picture (how does google work? How secure is the internet?).
WOMEN ARE UNDER-REPRESENTED IN THE MATHEMATICAL SCIENCES. HOW CAN THIS BE IMPROVED?
This is a cultural phenomenon. For example, the representation of women in the mathematical sciences does vary dramatically between different countries. How does one x this? One obvious potential x is to hire more women in senior roles in academia. There are numerous examples showing that this works. Fostering inclusiveness is also a wonderful thing (and this goes beyond gender issues, and beyond mathematics).
YOU HAVE LIKENED MATHS TO POETRY. DOES THAT MAKE MATHEMATICIANS THE BARDS OF SCIENCE? DO YOU HAVE A FAVOURITE POET?
One of my heroes, Yuri Manin, has advocated the theory that human language did not advance linearly, but rather came in jumps. There is an enormous creative leap involved in naming the unnamed. Poetry is the language of the things slightly beyond our capacity to express them. Mathematicians occupy a similar space in the sciences. Before Galois, Ruffini and Lagrange we spent millennia not knowing what a group is. However, once the concept was expressed it is difficult to imagine the mathematical world without it. My favourite poet? I love the Australian poet Les Murray, I also love Auden and Borges. The way that Beilinson writes mathematics borders on poetry sometimes.
Professor Geordie Williamson is Director of the newly-opened Sydney Mathematical Research Institute at the University of Sydney
Published in 2019, this article first appeared in Issued 8 of AMSI Update. Read the full issue online at amsi.org.au/wp-content/uploads/2019/02/the-update-8th-edition-web.pdf