WEBVTT Kind: captions Language: en 00:00:01.080 --> 00:00:09.269 [Music] 00:00:11.200 --> 00:00:15.950 If you imagine that we're sitting here and you're listening to me talking, but 00:00:15.950 --> 00:00:21.740 also a car goes past, you can distinguish these two sounds in your in your mind: 00:00:21.740 --> 00:00:25.579 They're two separate entities. And similarly, if you're listening to an 00:00:25.579 --> 00:00:29.840 orchestra, you can hear the violin, you can hear the oboe, you can hear the voice. 00:00:29.840 --> 00:00:33.920 And if you think about what's actually happening in your brain, that's rather 00:00:33.920 --> 00:00:39.230 remarkable, because what's reaching your ear is just a single sound wave, and yet 00:00:39.230 --> 00:00:44.140 our brain is able to break the sound wave up into its composite parts, and 00:00:44.140 --> 00:00:51.859 that is an example of something that we call a Fourier transform. So, it's in some 00:00:51.859 --> 00:00:55.519 sense you're given some signal, and you want to distill it into its 00:00:55.519 --> 00:01:02.629 fundamental components. So, the theory of Fourier series is a very old 00:01:02.629 --> 00:01:06.230 and very beautiful subject. And in some sense, representation theory, which 00:01:06.230 --> 00:01:11.299 is my field, is some very vast generalization of the theory of 00:01:11.299 --> 00:01:18.340 Fourier series. I never found mathematics particularly deep or inspiring in school. 00:01:18.340 --> 00:01:24.860 For me it was something that I could do easily, but I didn't find it profound. I 00:01:24.860 --> 00:01:31.130 found poetry profound, I found philosophy profound. And then I remember in third 00:01:31.130 --> 00:01:37.939 year, in undergrad in Sydney, I went to a course on Galois theory, and I saw a 00:01:37.939 --> 00:01:41.479 subject that I just never ever would have come up with, had I thought 00:01:41.479 --> 00:01:47.479 about it for a long long long long time. And I was absolutely amazed at the depth 00:01:47.479 --> 00:01:50.719 of this subject, and in some sense this is a subject that is still very alive 00:01:50.719 --> 00:01:55.670 today, Galois theory. So this was definitely a turning point, to see the 00:01:55.670 --> 00:02:00.829 the depth that mathematics is capable of. I've actually spent quite a lot of time 00:02:00.829 --> 00:02:04.920 thinking about what motivates me in mathematics. 00:02:04.920 --> 00:02:10.709 I think that one aspect of it is just the incredible beauty of the theory. Both 00:02:10.709 --> 00:02:15.450 in its kind of broader sense, and also in the specificity of the 00:02:15.450 --> 00:02:18.989 examples and the fact that you can really work something out and see that a 00:02:18.989 --> 00:02:25.890 number is one or two. It's very exciting on the other hand: Barry Mazur 00:02:25.890 --> 00:02:31.709 calls mathematics humanity's longest conversation. And I think that 00:02:31.709 --> 00:02:36.239 this is a really beautiful description of how we function as a mathematical 00:02:36.239 --> 00:02:41.069 community. Thats where I kind of imagine us all sitting around this enormous 00:02:41.069 --> 00:02:47.040 table, and you work very hard to be able to say one little thing in this long 00:02:47.040 --> 00:02:52.319 conversation that is mathematics. And I find the social aspect of 00:02:52.319 --> 00:02:57.540 mathematics incredibly inspiring, and the exposure to many different ideas, many 00:02:57.540 --> 00:03:01.769 different people, the ability to travel, the ability to collaborate. These are 00:03:01.769 --> 00:03:05.519 extremely strong motivations for me as a mathematician. And I don't think 00:03:05.519 --> 00:03:09.359 that was so much the case when I began. When I began I just wanted to 00:03:09.359 --> 00:03:13.190 understand, and I wanted to sit in my room, and I wanted to work things out. 00:03:13.190 --> 00:03:18.770 Whereas now I feel much more like I'm part of an amazing community. 00:03:18.770 --> 00:03:24.370 [Music] I had two very difficult times in my 00:03:24.370 --> 00:03:31.489 career. One was as a PhD student. So, I arrived and I had the fantastic adviser, 00:03:31.489 --> 00:03:37.099 who saw the world in a very abstract way. He's a theory 00:03:37.099 --> 00:03:42.769 builder, not a problem solver as such, and he was explaining to 00:03:42.769 --> 00:03:47.299 me how I should go about solving various problems. I think by nature I'm much 00:03:47.299 --> 00:03:54.319 more of a problem solver. And at some point I decided to go to Denmark for a 00:03:54.319 --> 00:03:59.120 period of time, and in this period of time I decided I'm going to show him 00:03:59.120 --> 00:04:05.030 that he's wrong, that I can just solve this by hand, and I completely and 00:04:05.030 --> 00:04:11.690 utterly failed. Also it was incredibly dark in Denmark every day. I 00:04:11.690 --> 00:04:15.349 felt like there was no daylight, and I was riding my bike to and from the 00:04:15.349 --> 00:04:20.510 office in in the darkness, and at that point I vowed that I'm just going to 00:04:20.510 --> 00:04:23.750 finish this thesis however I can, and then leave mathematics and never ever 00:04:23.750 --> 00:04:29.660 come back. This was a really important time for me, because it made me 00:04:29.660 --> 00:04:34.940 realize that there are many problems in mathematics you can't just attack 00:04:34.940 --> 00:04:41.690 straight away. The other very difficult time was as a postdoc: I just found it a 00:04:41.690 --> 00:04:48.470 very lonely experience. I was working very hard, I didn't have many 00:04:48.470 --> 00:04:52.460 friends. You also move a lot as a postdoc, you're traveling a lot, so it's 00:04:52.460 --> 00:04:57.139 very difficult to establish meaningful relationships. I think that we 00:04:57.139 --> 00:05:03.200 underestimate the psychological burden, that we put on ourselves and our 00:05:03.200 --> 00:05:08.720 PhD students. And the more that I talk to mathematicians, I realize that often 00:05:08.720 --> 00:05:13.550 people have had periods of illness or of mental 00:05:13.550 --> 00:05:16.340 health issues or things like that, but we don't talk about that so much, and I 00:05:16.340 --> 00:05:22.940 think that's a pity. I definitely feel a genuine sadness that I can't tell people 00:05:22.940 --> 00:05:27.500 about some of the most beautiful mathematical theories that exist. 00:05:27.500 --> 00:05:32.000 So, one example of this is something called the Weil conjectures. For me, this is one 00:05:32.000 --> 00:05:35.060 of the most extraordinary achievements of human kind. 00:05:35.060 --> 00:05:39.740 I find it absolutely remarkable. It's just as remarkable as a moon 00:05:39.740 --> 00:05:45.950 landing, or something like that. And yet, almost nobody in the world knows 00:05:45.950 --> 00:05:52.650 about this, so it makes me sad and I also think that this can lead to loneliness. 00:05:53.140 --> 00:05:56.740 But, I also think that we can make an effort to communicate more. 00:05:56.960 --> 00:06:01.980 [Music] 00:06:02.220 --> 00:06:07.820 So, this discovery of this family of counter-examples to the expected bounds 00:06:07.820 --> 00:06:13.030 in Lusztig's conjecture was a very, very interesting process for me. I was in 00:06:13.030 --> 00:06:20.960 Hongkong visiting Xuhua He, and he was asking me to explain an enormous amount 00:06:20.960 --> 00:06:26.060 of the theory, and I basically spent hours and hours every day at the 00:06:26.060 --> 00:06:29.540 whiteboard in his office explaining. And he was asking extremely good questions 00:06:29.540 --> 00:06:36.139 and we were trying to work on it. I think, in some sense, he was allowing me 00:06:36.139 --> 00:06:40.340 to put together a whole lot of pieces of a puzzle, and after that I went to 00:06:40.340 --> 00:06:44.539 Beijing, and I arrived in Beijing, and I remember everything was kind of swirling 00:06:44.539 --> 00:06:48.650 around in my head. But I had no idea, that I was about to about to have this idea, 00:06:48.650 --> 00:06:53.990 and then I was in the shower, and I just suddenly realized. It's the only 00:06:53.990 --> 00:06:58.430 time in my life where I suddenly realized that something was possible. 00:06:58.430 --> 00:07:02.030 You know, at other points when I've solved something, it's a gradual process: 00:07:02.030 --> 00:07:05.360 You start feeling more and more optimistic, and things start to feel more 00:07:05.360 --> 00:07:10.370 and more solid. Whereas here, it was just boom! It came immediately, and of course I 00:07:10.370 --> 00:07:15.500 needed to check it, but though it was a real kind of moment, where suddenly it 00:07:15.500 --> 00:07:22.430 became clear in the shower. The Lusztig conjecture was a central open 00:07:22.430 --> 00:07:29.110 problem in our field, and it was so, it was put forward in 1980 by George Lusztig. And 00:07:29.110 --> 00:07:36.350 he proposed a way to solve this conjecture, which was carried out in the 00:07:36.350 --> 00:07:43.599 90s, but somehow it only said that it is true in the vast majority of cases. 00:07:43.599 --> 00:07:48.129 But there was always this question about these special cases. So, for us, 00:07:48.129 --> 00:07:55.059 this is for small small prime numbers, and I discovered that the conjecture 00:07:55.059 --> 00:07:59.529 was not correct, and it was very interesting, because I was in Beijing at 00:07:59.529 --> 00:08:05.919 the time and just after that there was a conference in Zhangjiajie, 00:08:05.919 --> 00:08:10.809 where Lusztig was actually present. And I was going to announce the fact that this 00:08:10.809 --> 00:08:16.269 conjecture was incorrect in certain cases. I told him this, and he was 00:08:16.269 --> 00:08:23.649 very very shocked initially, but very very soon he understood what I 00:08:23.649 --> 00:08:28.719 was saying and was also able to explain it to others. And I was amazed at the speed 00:08:28.719 --> 00:08:33.370 at which he took on these new ideas. I have to say that, psychologically, for me 00:08:33.370 --> 00:08:38.680 it was quite shocking, because we, as a field, had been operating under the 00:08:38.680 --> 00:08:41.800 assumption that it was true. And I had also been operating under the assumption 00:08:41.800 --> 00:08:46.689 that it was true, and that we would prove it. So, to suddenly realize that the 00:08:46.689 --> 00:08:51.040 situation was much much much more complicated than we first thought, it 00:08:51.040 --> 00:08:58.060 felt like a step backwards for me, and I think other people felt this too. 00:08:58.060 --> 00:09:02.110 And so I remember, that for a few weeks afterwards, I would actually wake 00:09:02.110 --> 00:09:05.199 up in the middle of night, almost having a nightmare about the fact 00:09:05.199 --> 00:09:09.910 that it was not true. There's one thing that I would like to say with the Lusztig 00:09:09.910 --> 00:09:13.720 conjecture stuff that I find fascinating. So, you know, I was working very hard on 00:09:13.720 --> 00:09:16.509 this problem for a long period of time, from various different points of view, and 00:09:16.509 --> 00:09:24.250 at some point I did these calculations with my collaborator Ben Elias, and I 00:09:24.250 --> 00:09:28.779 was very very happy with four of them. I put them up on my wall in Bonn, 00:09:28.779 --> 00:09:32.740 and I was looking at them every day, and I'd had them up for about two years 00:09:32.740 --> 00:09:37.839 before I went to Hong Kong, and then I had this idea, and then I came back. 00:09:37.839 --> 00:09:43.569 The remarkable thing that was, that it was incredibly close to one of the 00:09:43.569 --> 00:09:48.040 pictures on the wall, so, in some sense, for a very long period of time 00:09:48.040 --> 00:09:52.170 I'd been staring at the answer and not realised it. 00:09:55.510 --> 00:10:01.230 Yes. So, a few times I've suddenly had the feeling that something is solved. 00:10:01.230 --> 00:10:07.899 It's an incredibly beautiful feeling, and there's a standard joke amongst 00:10:07.899 --> 00:10:10.959 mathematicians: That, when you get this feeling, you should just go and have a 00:10:10.959 --> 00:10:14.760 holiday, because it's so nice and, almost certainly, you actually haven't solved it. 00:10:14.760 --> 00:10:19.990 Almost certainly you're making a mistake, and I have to say, that of the ten kind 00:10:19.990 --> 00:10:24.339 of Eureka moments that I felt in my in my life, eight of them have been wrong. I 00:10:24.339 --> 00:10:28.360 never admit to myself that I cannot solve it, but I admit to myself that I do 00:10:28.360 --> 00:10:35.529 need a break. So, I think it's extremely healthy for me. Particularly, I've 00:10:35.529 --> 00:10:41.350 discovered that it's very healthy for me to move between problems, and so I think 00:10:41.350 --> 00:10:44.709 I'm quite good at being able to say: Okay, stop thinking about that. I'm not always 00:10:44.709 --> 00:10:48.790 good at that, but I try: Stop thinking about that and start thinking about this. And you're 00:10:48.790 --> 00:10:52.600 allowed to come back to this later, but at the moment you're stuck. We have 00:10:52.600 --> 00:10:56.529 something within ourselves that's very special, which is some way of 00:10:56.529 --> 00:11:01.510 thinking about things. If I look at mathematicians that I admire, you can see 00:11:01.510 --> 00:11:07.089 that often they use the same technique throughout their lives, and they're very 00:11:07.089 --> 00:11:11.290 good at that technique, and that is somehow inherent to them. And I think 00:11:11.290 --> 00:11:19.089 that it's very important to try not to stifle what is in you, but I'm 00:11:19.089 --> 00:11:25.460 not sure how one manages to do this. [Music] 00:11:25.460 --> 00:11:30.120 Have a small peek at this incredibly rich world of modular representation 00:11:30.120 --> 00:11:38.640 theory, that I think is waiting for us. I would love to understand some... 00:11:38.640 --> 00:11:42.780 There's very interesting connections with number theory and I would love to 00:11:42.780 --> 00:11:51.330 understand these connections more deeply. I would love to understand what's going 00:11:51.330 --> 00:11:55.980 on in this conjecture with Lusztig. I thought about it very hard and banned 00:11:55.980 --> 00:11:59.550 myself from thinking about it for some time, because I made almost no progress 00:11:59.550 --> 00:12:04.800 on it. And I think I'm just really 00:12:04.800 --> 00:12:08.710 looking forward to continuing exploring the mathematical world. 00:12:08.710 --> 00:12:14.049 [Music]