 Let's talk again back to your time at the University of Auckland when you were an undergraduate majoring in physics and mathematics, double major. Can you describe those years, how you remember them as your development? I wasn't the model student, shall we say. I did have at least one year where I sort of went off the rails. But my career didn't get age and everything. Ultimately that turned out to be one of the most amusing things about my entire career which is that I was in a year cohort of students who were really quite exceptional. So my couple of non-A grades made my record sort of not look all that spectacular compared with some of my fellow students. So when it came to scholarship time to go on overseas the other guys had taken all of the things and there was essentially nothing left for me. So it looked like my career was over at the end of my undergraduate studies or at least my masters. But fortunately there was a guy in the department, you know, Paul Hafner, and he knew about a Swiss government scholarship for students specifically from New Zealand and another one for Australia. And I had for a couple of other reasons I was kind of interested in going to Geneva. There was a guy there that was doing mathematical foundations of quantum mechanics and I had read his book and I was kind of interested in staying with him. And Paul told me about the scholarship so I applied for that and they were delighted because they never got anyone applying from sciences before. And so I got that and I went to Switzerland and my career sort of blossomed from there I'd say. It's really incredible isn't it? It's just, you know, who do you talk to? Yeah right, so I can't say how grateful I am to Paul for that, you know, without his intervention that stage as I said my career might have been over. I was also playing rugby reasonably well. While you were doing it? Yeah. Oh right, okay. It was still true that when I went to Geneva from Auckland I began in the physics department doing my PhD so it's always been very close in between the two, physics and maths. The actual path well, I was good at it and I enjoyed it and eventually people started to pay me to do it so that's sort of how it happened. And of course that led to you being awarded a very prestigious award called Fields Medal. And the Fields Medal is awarded by the International Congress of Mathematicians every four years and it is considered by mathematicians as an equivalent of a Nobel Prize for the sciences. Can you please describe in a few sentences your work that was so highly recognized? It was kind of an unusual sequence of events. I was working in a definitely physics related area, namely phonomenalgebras which is part of the mathematical background for quantum mechanics and quantum field theory. Fairly abstract stuff, phonomenalgebras are supposed to represent the observables of a physical system and that's the mathematical model for it. And these phonomenalgebras, these algebra observables have a lot of structure in their own right and I was working on that structure. And I defined this so-called invariant of a phonomenalgebra which was actually a subalgebra. You have one of these phonomenalgebras sitting inside a bigger one and I invented this number which describes just how much bigger the big one is than the smaller one. It's called the index. And what I observed was that this index itself is somehow quantized just like if you have the hydrogen atom, the wavelengths of the light emitted from a hydrogen atom are quantized. You don't just get a continuously varying family of wavelengths, you get these discrete bursts and it turns out that this index was the same. Instead of getting arbitrary number, which is sort of what phonomen would have predicted, you get these discrete bursts of indices. Because of the proof of this result, there was actually a connection which was totally unexpected with the theory of knots. So this is real knots, just ordinary knots like the ones you tie in your shoelaces. And it turned out that because of various structure that exists in knots, when you try to braid your hair, because I don't have too much left to braid, but if you braid your hair or the braiding that is in ropes, for instance, is a structure called the braid group. And that particular structure had actually quite coincidentally been part of my work on the phonomen algebras. And from that I managed to discover something new in knot theory. So it was kind of an unusual event where some event in one part of mathematics, physics, actually has a non-trivial impact in a completely different branch of mathematics and physics. And that's the kind of thing that makes waves and people stand up and notice and that's sort of how it happened. How do you do mathematical research? What do you actually do? Do you sit down, scratch your head? Do you go through sort of aha moments? Can you tell us a little bit about it? Well, yeah, I mean, it's a big mixture of hard work. You have something that you think is true and you'll want to prove it, figure out whether it is or find a counter example. And you can get really, really stuck. I mean, I know that in my, for instance, in this work, this index for subfactors work, I think I spent at least 80% of my time in that, at least 80% trying to prove something that I thought was true and turned out to be wrong. The final breakthrough in that was that what I was trying to do was actually completely and utterly wrong. The truth was the opposite of it. And that was the hard thing because you're trying to prove something and you try all kinds of methods but you're sort of going in the wrong direction. The actual truth is over here and you only get sort of forced over into that corner after a long time and a lot of hard work. Perseverance, no doubt. And then there's the other thing where you do have these aha moments. With the not polynomial, it was exactly like this. I was, I've been thinking about it for about three years and I met up with Joan Berman who was a not theory braided person in New York and a long discussion with her and it was pretty depressing because it didn't seem like anything that I had done was actually going to be any use for anything. But she told me a lot of things and one of those things just kept nagging away at my brain and then about a week afterwards all of a sudden I actually, well I just woke up in the middle of the night thinking, oh my God, that's all I have to do. I was trying to do something much harder. So I went downstairs, did a few calculations. In the middle of the night. In the middle of the night, yeah, and then I went back to sleep. Of course you have to be aware, as I'm sure you are, that most of the time when you get a great idea right before you go to sleep, you wake up in the morning and it's raw, but this is one very occasion it was right. Well, incredible feeling I'm sure. Yeah it was, it was a big high.