1 00:00:12,290 --> 00:00:16,460 Welcome back to the Oxford Mathematics. Public Lectures Home Edition. 2 00:00:16,460 --> 00:00:22,430 My name is I don't go really and I'm in charge of external relations for the Mathematical Institute as usual. 3 00:00:22,430 --> 00:00:24,020 Special thanks to our sponsors. 4 00:00:24,020 --> 00:00:33,110 Equity markets across markets are leading quantitative driven electronic market maker with offices in London, Singapore and New York. 5 00:00:33,110 --> 00:00:39,530 The ongoing support in this time of crisis is crucial in providing you quality content. 6 00:00:39,530 --> 00:00:44,990 It is a great pleasure to me to welcome today and a second one of our youngest and brightest 7 00:00:44,990 --> 00:00:50,360 researchers in the Mathematical Institute in Oxford and read mathematics in Cambridge 8 00:00:50,360 --> 00:00:54,860 and receive a Ph.D. from Berkeley just over a year ago for which he was awarded the 9 00:00:54,860 --> 00:01:00,890 prestigious Richard de Primer prise from the Society for Industrial and Applied Mathematics. 10 00:01:00,890 --> 00:01:05,250 She's currently a research fellow with us. 11 00:01:05,250 --> 00:01:13,350 And that works in an exciting area of mathematics at the interface between pure mathematics, applied mathematics and data science. 12 00:01:13,350 --> 00:01:20,250 She uses algebraic and geometric tools to understand data and apply so our ideas to many interesting problem. 13 00:01:20,250 --> 00:01:29,400 And I have been long fascinated by your work. Every day we are confronted with a deluge of no and summer we have to navigate through them. 14 00:01:29,400 --> 00:01:33,000 While it's easy to understand that one number is bigger than another one, 15 00:01:33,000 --> 00:01:38,610 in many instances we are given a string of numbers to characterise a complex situation. 16 00:01:38,610 --> 00:01:42,060 This is especially true in the current crisis. 17 00:01:42,060 --> 00:01:50,770 For instance, you may be given the number of cases, number of deaths, number of hospitalisation prevalence, reproduction number and so on. 18 00:01:50,770 --> 00:01:55,860 Others one understand all these numbers at once. How do we combine them? 19 00:01:55,860 --> 00:02:06,390 Well, we need the appropriate tools, and we are very fortunate that Anna has brought the toolbox with us tonight in a talk ideas for Complex World, 20 00:02:06,390 --> 00:02:12,570 and I will share with you some of the mathematics that you can use to make sense of the world around you. 21 00:02:12,570 --> 00:02:17,840 So thank you very much, Annette, for doing this. Please start now. 22 00:02:17,840 --> 00:02:29,940 Hi, everyone, I'm Anna Siegel, and I research fellow at the Mathematical Institute in Oxford and a junior research fellow at the Queens College. 23 00:02:29,940 --> 00:02:40,590 It's an honour to be here this evening to give this public lecture, and I'd like to thank the organisers, Alan and Tyrell, for inviting me. 24 00:02:40,590 --> 00:02:44,180 And I'd like to thank you for coming today. 25 00:02:44,180 --> 00:02:55,390 I'm going to be talking about ideas for a complex world and how these ideas fit in with current topics in mathematical research. 26 00:02:55,390 --> 00:03:04,270 So let's get started. Firstly, what do I mean by ideas for complex world? 27 00:03:04,270 --> 00:03:16,090 Well, let's start with the complex world part. So here's our complex world with many things to understand this disease. 28 00:03:16,090 --> 00:03:25,020 Vaccinations, elections, racial justice, global warming and many more. 29 00:03:25,020 --> 00:03:35,340 And then as a society, we have quantitative tools, so here's a cartoon picture of our quantitative tools. 30 00:03:35,340 --> 00:03:48,180 And these tools enable us to approach these different topics and to understand them better, to build our understanding of the world. 31 00:03:48,180 --> 00:03:56,780 And these are quantitative tools because they're based on algorithms and models and data. 32 00:03:56,780 --> 00:04:07,340 All right. And then there's us, individual human beings, and we also encounter implicit complexity. 33 00:04:07,340 --> 00:04:20,390 We have many complex situations that appear in our day to day lives and we come up with ideas for how to approach them. 34 00:04:20,390 --> 00:04:23,300 So how do these three different pieces fit together? 35 00:04:23,300 --> 00:04:34,810 We've got our complex world, our quantitative tools and then us human beings in our day to day lives. 36 00:04:34,810 --> 00:04:41,260 Well, quantitative tools have an ever increasing impact on our lives, even a year ago, 37 00:04:41,260 --> 00:04:52,670 it would be difficult to imagine a mathematical model deciding if we're allowed to meet up with friends and family. 38 00:04:52,670 --> 00:04:57,540 But on the other hand, there's a disconnect. And these words seem very separate, 39 00:04:57,540 --> 00:05:09,530 the way we do things in our day to day lives seem very far removed from the inner workings of our quantitative toolbox. 40 00:05:09,530 --> 00:05:19,730 And with this disconnect, our impression of quantitative tools ends up being shaped by polarised opinions. 41 00:05:19,730 --> 00:05:28,520 So let's see some examples. What I mean by this? Here's a show from the recent movie Tenet. 42 00:05:28,520 --> 00:05:35,200 So here we have the protagonist here and a character called Priya. 43 00:05:35,200 --> 00:05:39,340 And I don't want to ruin the movie for anyone who hasn't seen it, 44 00:05:39,340 --> 00:05:45,400 but in the film we have an algorithm being portrayed in a very negative light so that 45 00:05:45,400 --> 00:05:53,300 the characters have to come up with a way to defend the world against this algorithm. 46 00:05:53,300 --> 00:05:58,760 I hope I haven't said too much. And here's another example here we have. 47 00:05:58,760 --> 00:06:06,530 Boris Johnson back in the summer saying, I'm afraid your grades were almost derailed by a mutant algorithm. 48 00:06:06,530 --> 00:06:12,020 So Boris Johnson here was referring to the A-level results algorithm, 49 00:06:12,020 --> 00:06:22,530 which was supposed to be a good replacement for the exams that students weren't able to take because of the pandemic. 50 00:06:22,530 --> 00:06:29,000 And here, Boris Johnson's phrasing puts the blame on the algorithm for being at fault. 51 00:06:29,000 --> 00:06:37,430 All right, and here we have Kamala Harris, the future vice president of the U.S., saying I trust the word of scientists. 52 00:06:37,430 --> 00:06:47,210 And she's saying this here in the context of whether she would trust Trump if he said COVID 19 vaccine was safe. 53 00:06:47,210 --> 00:06:51,710 And she's saying to position herself in opposition to people who don't trust 54 00:06:51,710 --> 00:06:58,910 scientists or people who trust other people like Trump for their medical information. 55 00:06:58,910 --> 00:07:04,640 So in all of these examples, we can see a polarised opinion emerging. 56 00:07:04,640 --> 00:07:15,240 Either we have quantitative algorithms being portrayed in quite a negative light as the body as something to blame when things go wrong. 57 00:07:15,240 --> 00:07:21,830 Or they're portrayed in a very positive light as something we trust, something we believe in and put our faith in. 58 00:07:21,830 --> 00:07:37,050 But both of these opinions are quite limiting. They disconnect us from the quantitative toolbox because they don't enable us to see how it works. 59 00:07:37,050 --> 00:07:38,100 But in fact, there are many, 60 00:07:38,100 --> 00:07:48,960 many connexions between us and the quantitative tools far beyond having these polarised opinions of viewing them as either good or bad. 61 00:07:48,960 --> 00:07:55,620 There are many, many ways that we're closely connected, and in this talk, 62 00:07:55,620 --> 00:08:09,470 we'll see how ideas from our day to day lives translates to give us some ideas in our quantitative toolbox. 63 00:08:09,470 --> 00:08:17,730 All right. So. Next, we'll think about a process that's familiar to us from our day to day lives and then 64 00:08:17,730 --> 00:08:25,330 later we'll see how this process translates over to give some quantitative to. 65 00:08:25,330 --> 00:08:25,750 All right. 66 00:08:25,750 --> 00:08:37,120 So here's a picture of a person think of someone, you know, maybe someone you know well, and lots of information may come to mind about that person. 67 00:08:37,120 --> 00:08:40,570 So maybe their name, their appearance, 68 00:08:40,570 --> 00:08:49,960 their likes and dislikes what they said to us recently or key memories that we've shared with this person in the past? 69 00:08:49,960 --> 00:08:57,100 And this is all complex data that we have in our heads about this person. 70 00:08:57,100 --> 00:09:06,070 So here's a picture of our complex data over here in cartoon format with these different coloured squiggles, 71 00:09:06,070 --> 00:09:13,810 and we can use this complex data to come up with some summary of the person. 72 00:09:13,810 --> 00:09:17,530 So for example, if I wanted to summarise this person's personality, 73 00:09:17,530 --> 00:09:26,110 I could think of all of the complex data that I know about them and come up with some traits to use to describe them. 74 00:09:26,110 --> 00:09:32,920 So maybe this person is creative and rebellious. All right. 75 00:09:32,920 --> 00:09:40,420 And we can do this not just for one person, but for more than one person as well. 76 00:09:40,420 --> 00:09:51,670 So maybe think of these six people or six people that you know, then for each of them, some complex data will come to mind. 77 00:09:51,670 --> 00:10:04,270 What we know about the different people and we can process these complex data to come up with summaries of all these different people's personalities. 78 00:10:04,270 --> 00:10:15,960 So maybe this person in orange over here is sociable and worldly, and this dark blue person here is empathetic and organised. 79 00:10:15,960 --> 00:10:24,940 Right, so as humans, we're quite good at coming up with summaries of complex data for other humans. 80 00:10:24,940 --> 00:10:33,610 But we can also do this for. Other things as well, like countries, so we should think about some countries that we know. 81 00:10:33,610 --> 00:10:39,720 Then again, some complex data will come to mind about these different countries, 82 00:10:39,720 --> 00:10:47,950 so we might be thinking about people we know there or time that we've spent that if we visited or. 83 00:10:47,950 --> 00:10:52,430 Maybe something we've heard about that country in the news or. 84 00:10:52,430 --> 00:11:05,570 The country's response to the pandemic, for example, and again, we can come up with some summary, so to fix on this example of COVID. 85 00:11:05,570 --> 00:11:12,620 Here's a summary of each of these countries that we're able to obtain from the complex data that we might know about. 86 00:11:12,620 --> 00:11:20,500 So. So, for example, New Zealand, over here in green, we know lots of things about New Zealand, maybe. 87 00:11:20,500 --> 00:11:27,240 And one thing is that there have been 25 deaths in total. 88 00:11:27,240 --> 00:11:31,110 Or we could think about Mexico, and maybe we know some different things about Mexico, 89 00:11:31,110 --> 00:11:42,640 and we can extract the key thing to bear in mind, which is that the COVID cases so far have peaked in August. 90 00:11:42,640 --> 00:11:50,830 All right, and we can use these summaries to think about differences and similarities between different countries. 91 00:11:50,830 --> 00:11:57,820 So we've seen this for people and for countries, but we can also do other examples like breeds of dog. 92 00:11:57,820 --> 00:12:05,230 If we're thinking about getting a dog. Or neighbourhoods of a city, if we're thinking about moving house. 93 00:12:05,230 --> 00:12:17,860 We can think about some key similarities and differences between different people or countries or whatever it is. 94 00:12:17,860 --> 00:12:24,850 So what I'm trying to say here is that all of us process complex data in our day to day lives, 95 00:12:24,850 --> 00:12:33,550 but it becomes difficult when we want to process a thousand people or 100 countries as the number of things that we're trying to compare grows, 96 00:12:33,550 --> 00:12:38,840 it becomes more difficult for us to keep all of this information in mind. 97 00:12:38,840 --> 00:12:49,190 And that's why scientists use quantitative tools to scale things up to have some approach that will work when we're trying to understand many, 98 00:12:49,190 --> 00:13:00,750 many different things rather than just a few. All right, so where does mathematics come into all of this? 99 00:13:00,750 --> 00:13:10,130 Well, you might think mathematics lives over here, that mathematics is the quantitative tools that we use to understand the world. 100 00:13:10,130 --> 00:13:18,800 Or you might think mathematics slips over here in the world of people that mathematics is. 101 00:13:18,800 --> 00:13:24,260 Something that people have. Maybe it's a gift that only certain people have and certain people don't. 102 00:13:24,260 --> 00:13:35,270 But neither of these are true. Mathematics is about noticing a pattern and boiling it down to get at its key principles. 103 00:13:35,270 --> 00:13:44,690 So in this way, mathematics enables us to abstract human ideas into quantitative tools. 104 00:13:44,690 --> 00:13:58,540 But it does more than this. It also enables us to go the other way to use quantitative tools in order to gain insights that are helpful in our lives. 105 00:13:58,540 --> 00:14:04,630 So next, we'll see how to take this gun. 106 00:14:04,630 --> 00:14:10,780 Data processing that we're familiar with from day to day lives and how with the help of mathematics, 107 00:14:10,780 --> 00:14:18,820 we can turn that into a quantitative tool that will be very useful in many different applications. 108 00:14:18,820 --> 00:14:27,190 All right, so here was our processing data, a situation we have these six different people and complex data about each of them. 109 00:14:27,190 --> 00:14:35,350 And before we use this data to extract some key personality traits of each of the people. 110 00:14:35,350 --> 00:14:48,440 But we could also think about some particular personality traits and wonder for each person to what extent they match that personality trait. 111 00:14:48,440 --> 00:14:55,850 So, for example, we could think about nice and friendly then for each of our six people, 112 00:14:55,850 --> 00:15:00,860 we can think about how nice they are and how friendly they are. 113 00:15:00,860 --> 00:15:10,670 And maybe for simplicity, we'll say that we can summarise the niceness or their friendliness just by a single number. 114 00:15:10,670 --> 00:15:16,040 Then we could go a step further and think about plotting each of our people on this plot here. 115 00:15:16,040 --> 00:15:23,690 What the x axis is that niceness and the y axis is the friendliness. 116 00:15:23,690 --> 00:15:29,480 All right, so then for each person, I can think about their niceness, value and their friendliness value, 117 00:15:29,480 --> 00:15:35,630 and then I can plot the people over here and maybe I'll get something that looks a bit like this. 118 00:15:35,630 --> 00:15:46,200 So. Here are the six people on the plot, so for example, we can see that the red person is very nice and very friendly, 119 00:15:46,200 --> 00:15:54,120 and the orange person over here is pretty nice too, and maybe even a little bit more friendly. 120 00:15:54,120 --> 00:15:59,220 OK. And we can do this for other personality traits as well. 121 00:15:59,220 --> 00:16:01,470 If we wanted, for example, 122 00:16:01,470 --> 00:16:11,820 we could compare how shy someone is compared to how outgoing they are and plot that against how serious they are compared to how funny they are. 123 00:16:11,820 --> 00:16:19,350 So then we can go through the same process as before for each person we can think about where they lie on the x axis, 124 00:16:19,350 --> 00:16:28,500 how shy they are compared to how outgoing they are and then how serious they are compared to how funny their. 125 00:16:28,500 --> 00:16:35,580 So then each person will be plotted somewhere over here. 126 00:16:35,580 --> 00:16:42,420 So, for example, let's see, this green person here is a little bit shy and a little bit serious. 127 00:16:42,420 --> 00:16:48,860 And this orange bus in here is a little bit outgoing and quite funny. 128 00:16:48,860 --> 00:16:57,170 All right, and let's do this for one more personality trait, so we've got shy versus outgoing is still on the x axis, 129 00:16:57,170 --> 00:17:03,120 but maybe on the y axis we can plot how emotional someone is compared to how rational they are. 130 00:17:03,120 --> 00:17:07,880 OK, so we go through the same process of thinking for each person, 131 00:17:07,880 --> 00:17:16,460 what's their value on the sky versus outgoing axis and what's their value on the emotional versus rational axis? 132 00:17:16,460 --> 00:17:22,820 And then we can plot our people and maybe we get something like this. 133 00:17:22,820 --> 00:17:28,860 So we see that this yellow person, for example, is quite shy and also quite rational. 134 00:17:28,860 --> 00:17:33,990 All right, so why are we doing this? 135 00:17:33,990 --> 00:17:43,530 Well, one thing that we can see is that there are better and worse choices for which personality traits to choose. 136 00:17:43,530 --> 00:17:54,480 Let's say we want to identify differences between our friends in order to buy them personalised presents that match their personalities. 137 00:17:54,480 --> 00:18:02,100 Then some choices of personality trait will be more useful than others to be able to tell them apart. 138 00:18:02,100 --> 00:18:10,620 So in this first example over here, where we plotted nice and friendly, everyone was pretty nice and friendly. 139 00:18:10,620 --> 00:18:23,510 So here we see that the data is all bunched up. In this second example here, where we plotted shy buses, outgoing and serious versus funny view, 140 00:18:23,510 --> 00:18:31,490 the data were a little bit more spread out, but still mostly concentrated along this line. 141 00:18:31,490 --> 00:18:40,100 And in this final example here, where we plotted shy versus outgoing against emotional versus rational or the six different 142 00:18:40,100 --> 00:18:49,250 people were quite spread out in all directions on this plot so that the well spread out. 143 00:18:49,250 --> 00:18:58,790 And this is our idea, but some personality traits are more useful than others because they allow us to see differences between people. 144 00:18:58,790 --> 00:19:06,470 If we were thinking about these personalised presents, it would be more helpful to keep these personality traits in mind when deciding 145 00:19:06,470 --> 00:19:11,810 which present to allocate to each different person than nice and friendly, 146 00:19:11,810 --> 00:19:15,470 where it would be difficult to tell them apart. 147 00:19:15,470 --> 00:19:19,800 Or then these traits over here, which are very closely related. 148 00:19:19,800 --> 00:19:34,070 So. In this third example, all our different people are well spread out, and we can identify the key personality traits. 149 00:19:34,070 --> 00:19:41,720 So, yeah, these first two are not so helpful when it comes to identifying differences between people. 150 00:19:41,720 --> 00:19:50,460 But this third example here is helpful because all are different people are spread out on the plot and, well, 151 00:19:50,460 --> 00:20:00,980 call these key measurements to highlight the fact that they're allowing us to see differences between the people. 152 00:20:00,980 --> 00:20:10,850 So our idea is that of a key measurement, a key measurement to something that spread to the data points out in all different directions, 153 00:20:10,850 --> 00:20:17,900 the data points are not bunched up, they're not along a line, the all spread out. 154 00:20:17,900 --> 00:20:26,330 And these offer a good way to summarise differences between people or countries or breeds of dog and so on. 155 00:20:26,330 --> 00:20:30,290 So all of us find key measurements all the time. 156 00:20:30,290 --> 00:20:34,010 We probably don't think about them in the context of a plot like this. 157 00:20:34,010 --> 00:20:44,100 But whenever we're thinking about the key way to describe a difference between two different people, we're thinking about key measurements. 158 00:20:44,100 --> 00:20:52,920 And key measurements is also the idea behind many different quantitative tools. 159 00:20:52,920 --> 00:20:58,890 So we can think about labelling our axis here by key measurement number one for the x axis and key 160 00:20:58,890 --> 00:21:08,000 measurement number two for the y axis to highlight the fact that we've identified some key measurements. 161 00:21:08,000 --> 00:21:14,820 All right. So I said before that quantitative tools are useful because they enable us to take 162 00:21:14,820 --> 00:21:20,300 a human process and scale it up so that it can be used at much bigger scales. 163 00:21:20,300 --> 00:21:29,930 So let's see how that would work for this example of looking at people's personality traits. 164 00:21:29,930 --> 00:21:36,860 All right, so here's our six people and their complex data again. 165 00:21:36,860 --> 00:21:44,370 And we can think about a personality trait like we did before, for example, 166 00:21:44,370 --> 00:21:53,060 nice and then for each person, we can think about a score to give them that says how nice they are. 167 00:21:53,060 --> 00:22:00,090 All right. And this column of numbers here is called Vector. 168 00:22:00,090 --> 00:22:03,600 And we can repeat this for other personality traits as well. 169 00:22:03,600 --> 00:22:04,320 For example, 170 00:22:04,320 --> 00:22:17,020 friendly so we could give each of our people a score that says how friendly they are and record it in this column of numbers here or this factor here. 171 00:22:17,020 --> 00:22:20,740 So, for example, this person in red is doing very well. 172 00:22:20,740 --> 00:22:26,540 They've received a nice score of five and a friendly score of five as well. 173 00:22:26,540 --> 00:22:37,180 All right, and then we can repeat this for our other personality traits, so outgoing versus shy, funny versus serious emotional vs. rational. 174 00:22:37,180 --> 00:22:44,530 And these columns of numbers all together form what's called a matrix. 175 00:22:44,530 --> 00:22:51,130 All right, so each row of the matrix corresponds to a particular person. 176 00:22:51,130 --> 00:22:57,190 So, for example, the first row here corresponds to the red person. 177 00:22:57,190 --> 00:23:06,590 And each column is a personality trait or something that we've measured about the person. 178 00:23:06,590 --> 00:23:12,530 And then we saw before that we can plot certain columns of this matrix against each other. 179 00:23:12,530 --> 00:23:20,210 So in this plot over here, we've plotted shy versus outgoing against emotional versus rational, so we've plotted. 180 00:23:20,210 --> 00:23:25,580 Column number three against column number five. 181 00:23:25,580 --> 00:23:33,440 And these we identified as our key measurements because they exhibited a nice amount of spreads 182 00:23:33,440 --> 00:23:41,020 between all the different people that enabled us to see the similarities and differences. 183 00:23:41,020 --> 00:23:46,450 OK. And all of us translate complex data to key measurements all the time, 184 00:23:46,450 --> 00:23:56,020 we don't make a plot like this one over here and even more certainly we don't build a matrix like this one in the middle, 185 00:23:56,020 --> 00:24:04,630 so we miss out these intermediate steps, but we're often taking complex data to think about some key measurements. 186 00:24:04,630 --> 00:24:14,500 But now we'll see how these intermediate steps enable us to scale this process up to turn it into a quantitative tool. 187 00:24:14,500 --> 00:24:23,980 All right. So we started with six people. Then we built a matrix, which had six rows, one row for each person. 188 00:24:23,980 --> 00:24:28,510 And then we obtained a plot which had six points on it. 189 00:24:28,510 --> 00:24:36,580 So each person was a single point on our plot, but there's nothing special about the number six. 190 00:24:36,580 --> 00:24:41,200 We can equally do this for a thousand people, at least in principle. 191 00:24:41,200 --> 00:24:52,450 Then we have a matrix with a thousand rows and then we'd have plots with 1000 points on it. 192 00:24:52,450 --> 00:25:01,710 So let's see what this would look like is a thousand people or an artist's impression of a thousand people. 193 00:25:01,710 --> 00:25:12,330 And from these a thousand people, we can build a matrix where each row of the matrix corresponds to a person and each column is some measurement, 194 00:25:12,330 --> 00:25:21,660 something we've measured about each of our people. So before these measurements were personality traits, but they could be something different. 195 00:25:21,660 --> 00:25:27,520 And in most applications are very likely to be something different than a personality trait. 196 00:25:27,520 --> 00:25:32,350 All right, and this is our victor matrix, 197 00:25:32,350 --> 00:25:40,300 and then we can build a plot of each of our points so we can plot key measurement number one on the x axis and key 198 00:25:40,300 --> 00:25:48,610 measurement number two on the y axis until we start to see the importance of having this plot because for six people, 199 00:25:48,610 --> 00:25:57,640 we can think about the similarities and differences between each of the people. But for a thousand people, it becomes a lot more difficult. 200 00:25:57,640 --> 00:26:01,810 And with this plot of our key measurements, this allows us to see well to start, 201 00:26:01,810 --> 00:26:08,920 to see maybe groups of people that exist or some people that are more similar than others. 202 00:26:08,920 --> 00:26:17,470 But an important question at this point is how do we choose the key measurements in our example before we chose them 203 00:26:17,470 --> 00:26:24,670 by trying out a few different personality traits and seeing which ones looked the most spread out on the plot? 204 00:26:24,670 --> 00:26:30,670 But how would we do this at these much bigger scales? 205 00:26:30,670 --> 00:26:41,290 Well, one of the most widely used tools for finding key measurements is called principal component analysis, 206 00:26:41,290 --> 00:26:52,240 and principal component analysis finds them using linear algebra the theory of matrices. 207 00:26:52,240 --> 00:27:03,400 And the key measurements will be some combinations of columns of our matrix that enable us to spread out the data the most. 208 00:27:03,400 --> 00:27:09,730 So, for example, key measurement number one would be the combination of columns, 209 00:27:09,730 --> 00:27:15,880 the combination of measurements that spread out the data, the best and key measurement. 210 00:27:15,880 --> 00:27:23,680 Number two will be the combination of columns that spread all the data the second best, and we can use linear algebra, 211 00:27:23,680 --> 00:27:36,190 the algebraic theory of matrices of grids, of numbers like this to enable us to find these key measurements. 212 00:27:36,190 --> 00:27:42,190 Right, so let's see some applications of this. 213 00:27:42,190 --> 00:27:48,640 First, we'll think about treating disease or coming up with different ways to treat disease. 214 00:27:48,640 --> 00:27:55,210 So now maybe our 1000 people are hospital patients. 215 00:27:55,210 --> 00:28:01,180 Then our matrix of data, he could record their genetic information. 216 00:28:01,180 --> 00:28:09,040 So this matrix will have a thousand rows if we have a thousand hospital patients and if we have a recording for each of their genes, 217 00:28:09,040 --> 00:28:20,740 it will have 20000 columns. So this is now a really huge matrix that it would be impossible to understand and find structuring by hand. 218 00:28:20,740 --> 00:28:31,260 We have to use some tool to be able to. Extract information from this matrix and then we can find our key measurements. 219 00:28:31,260 --> 00:28:35,400 So for this example, they'll be key genes, so we'll have key genes. 220 00:28:35,400 --> 00:28:40,980 Number one on the x axis and key genes number two on the y axis. 221 00:28:40,980 --> 00:28:49,350 And I say jeans rather than Gene, because our key measurements will be some combinations of the columns of our matrix, 222 00:28:49,350 --> 00:28:56,610 they may not be one exact column plotted against another exact column. 223 00:28:56,610 --> 00:29:03,060 All right, and then on this plot, we can start to see similarities and differences between the different hospital patients. 224 00:29:03,060 --> 00:29:19,320 So maybe we can identify a key group over here circled in purple, and this group of points on the plot corresponds to some rows of our matrix. 225 00:29:19,320 --> 00:29:24,810 And that then corresponds to some people in our cohort of hospital patients. 226 00:29:24,810 --> 00:29:32,040 And maybe these people are patients who may be suitable for a new treatment. 227 00:29:32,040 --> 00:29:38,520 So our key measurements shouldn't be relied on entirely to suggest patients for a new treatment. 228 00:29:38,520 --> 00:29:46,830 But they allow us to see similarities and differences between the different patients and to identify key groups 229 00:29:46,830 --> 00:29:57,620 where we can then use our or someone else's medical expertise to interpret what these key groups might be. 230 00:29:57,620 --> 00:30:01,760 OK. And it's difficult to do this by hand. 231 00:30:01,760 --> 00:30:09,890 As I said, but it's important to be able to do the because we don't want to be trialling new treatments on everyone, 232 00:30:09,890 --> 00:30:19,240 we want to identify some particular group of patients who who may respond well to it. 233 00:30:19,240 --> 00:30:22,300 All right, and let's see another example. 234 00:30:22,300 --> 00:30:35,050 So exam results prediction now maybe a thousand people are students, and now our matrix of data is some information about the schoolwork. 235 00:30:35,050 --> 00:30:43,690 So we have schoolwork information no one plotted on the x axis and schoolwork information number two plotted on the y axis. 236 00:30:43,690 --> 00:30:47,680 And now we've identified some group of students over here. 237 00:30:47,680 --> 00:30:52,780 And maybe these are students whose predicted grades may be too low. 238 00:30:52,780 --> 00:30:57,130 So it's useful to be able to identify this group of students because we can have a second 239 00:30:57,130 --> 00:31:07,690 look at the predicted grades and make some judgement over whether that is indeed too low. 240 00:31:07,690 --> 00:31:12,550 And this is not practical to do for all the all the students on the on the plot. 241 00:31:12,550 --> 00:31:21,170 We want to identify some particular group of students who we think need some extra scrutiny. 242 00:31:21,170 --> 00:31:26,290 Shh. All right. And here's a third example in personalised advertising. 243 00:31:26,290 --> 00:31:36,130 So now maybe a thousand people are some social media users or in this example, it's likely to be many more people. 244 00:31:36,130 --> 00:31:42,940 And then the data matrix is information about what each of these users like to click on. 245 00:31:42,940 --> 00:31:47,440 So now our plot over here, the key measurements are key clicks. 246 00:31:47,440 --> 00:31:55,090 We have the key kicks number one on the x axis and key clicks number two on the y axis. 247 00:31:55,090 --> 00:32:06,970 And this plot enables us as before to identify perhaps some important groups of different users in terms of how they interact with this platform. 248 00:32:06,970 --> 00:32:17,200 And maybe we think that these users over here may respond particularly well to an advert, 249 00:32:17,200 --> 00:32:21,850 and that's useful because it costs much less to advertise to fewer people. 250 00:32:21,850 --> 00:32:28,750 So you want to make sure you select people who are likely to be persuaded by a particular 251 00:32:28,750 --> 00:32:34,120 advert rather than going to the cost of advertising to many different people. 252 00:32:34,120 --> 00:32:43,270 OK, so I said before that quantitative tools are often either labelled as being good or bad. 253 00:32:43,270 --> 00:32:50,230 So what about principal component analysis? Is it good or bad? 254 00:32:50,230 --> 00:32:54,550 Well, we've seen various context in which it could be applied. 255 00:32:54,550 --> 00:33:03,160 So we saw that it could be used to give personalised presents to people based on their personalities. 256 00:33:03,160 --> 00:33:13,660 That seems quite nice. Or it could be used to identify patients who may be suitable for a new disease treatment. 257 00:33:13,660 --> 00:33:22,920 This seems like a good use of the quantitative tool, although I would need to be taken to make sure that it was. 258 00:33:22,920 --> 00:33:34,680 Are able to be interpreted medically. And we also saw that it could be used to identify groups of students who predicted grades may be too low. 259 00:33:34,680 --> 00:33:43,200 Again, this is a situation where we would find it helpful to have the input of some quantitative tool, 260 00:33:43,200 --> 00:33:50,010 but we have to make sure it wasn't unfairly prejudiced against certain students. 261 00:33:50,010 --> 00:34:00,450 And then we saw the example of personalised advertising of identifying users who may respond well to a particular advert, 262 00:34:00,450 --> 00:34:06,150 and this might seem like a good way to use this tool. 263 00:34:06,150 --> 00:34:13,080 It's helpful to see adverts for things that we might be interested in buying. 264 00:34:13,080 --> 00:34:23,550 Or it could be seen as a problem, as a potential issue with data privacy and has the potential to be used in a way that we might not like so much. 265 00:34:23,550 --> 00:34:25,410 So, for example, 266 00:34:25,410 --> 00:34:36,180 principal component analysis was said to be a key tool that was used by Cambridge Analytica in Donald Trump's 2016 presidential campaign 267 00:34:36,180 --> 00:34:52,750 to send targeted adverts to people based on their social media platform habits to be able to send them very persuasive adverts. 268 00:34:52,750 --> 00:35:02,410 And whether you think of this as a tool for psychological manipulation or as just a good way to get across relevant 269 00:35:02,410 --> 00:35:11,710 political information maybe depends on which campaign you support or which campaign is using this approach. 270 00:35:11,710 --> 00:35:21,190 So already we've seen quite a spread of possible uses of principal component analysis along this spectrum of good to bad. 271 00:35:21,190 --> 00:35:31,960 But it can get even worse. So principal component analysis originated from a theoretical perspective in this paper called 272 00:35:31,960 --> 00:35:41,590 on lines and planes of closest fit to a systems of points in space by Karl Pierson in 1981. 273 00:35:41,590 --> 00:35:46,660 So here's a picture of Karl Pierson here, and here's a quote by him. 274 00:35:46,660 --> 00:35:53,080 He says In Germany, a vast experiment is in hand, and some of you may live to see its results. 275 00:35:53,080 --> 00:35:56,530 If it fails, it will not be for want of enthusiasm, 276 00:35:56,530 --> 00:36:04,820 but rather because the Germans are only just starting the study of mathematical statistics in the modern sense. 277 00:36:04,820 --> 00:36:18,060 So this was Pearson in 1934 praising the Nazis for the eugenics programme, which culminated in the Holocaust. 278 00:36:18,060 --> 00:36:25,140 And Pearson, as well as being a mathematician and a statistician, was also a eugenicists, a study of eugenics, 279 00:36:25,140 --> 00:36:36,420 the racist and discredited attempts to improve the human race by identifying racially superior groups. 280 00:36:36,420 --> 00:36:50,410 And Pearson made these comments at around the same time my grandfather cut, pictured here in the light blue, was leaving Germany as a refugee. 281 00:36:50,410 --> 00:36:59,080 So we've seen that principal component analysis, this same tool can be used for good and it can be used for bad. 282 00:36:59,080 --> 00:37:10,330 And in this middle ground between giving people presents and eugenics, how can we identify if it's a suitable tool to use? 283 00:37:10,330 --> 00:37:18,300 Well, we can see this from looking at its limitations, the limitations of a tool. 284 00:37:18,300 --> 00:37:28,830 Very informative, and they can be seen from the limitations of the underlying idea of the idea that gave rise to that tool. 285 00:37:28,830 --> 00:37:39,150 So for example, here we have our tool is principal component analysis, and the idea is that of a key measurement. 286 00:37:39,150 --> 00:37:49,000 So what's the limitations of a tool and how can we see this from the limitations of the idea? 287 00:37:49,000 --> 00:37:56,050 Well, to go back to our example with the people, we had these six people and complex data about each of them, 288 00:37:56,050 --> 00:38:02,050 and we can encode that information into a matrix. 289 00:38:02,050 --> 00:38:11,770 So this is our matrix. Each row of the matrix corresponds to a person, and each column is a measurement. 290 00:38:11,770 --> 00:38:19,480 So here we can imagine that this is some grid filled with numbers, and we saw before that. 291 00:38:19,480 --> 00:38:26,950 Our key measurement is some combination of columns of this matrix. 292 00:38:26,950 --> 00:38:31,240 And here we can start to see a possible problem. 293 00:38:31,240 --> 00:38:39,310 Let's imagine that instead of allocating presents to our friends, we're allocating jobs to people. 294 00:38:39,310 --> 00:38:47,320 Then the complex data that we have in mind for each of the individual people may fall into some different types. 295 00:38:47,320 --> 00:38:58,270 So let's say that the data has comes in blue, green and red, so the blue information is about someone's personality. 296 00:38:58,270 --> 00:39:04,360 Maybe the green information is about the. 297 00:39:04,360 --> 00:39:11,620 CV, the professional experience or educational background, and then the red part is the demographics. 298 00:39:11,620 --> 00:39:18,280 So for example, that gender, all that race, then in our matrix of data over here, 299 00:39:18,280 --> 00:39:23,380 some of our columns will correspond to these different types of data. 300 00:39:23,380 --> 00:39:29,200 So maybe the first few columns over here are the personality of each person. 301 00:39:29,200 --> 00:39:34,990 And then next, we have the professional details about someone. 302 00:39:34,990 --> 00:39:43,480 And then finally, the demographic information so we can think about splitting our matrix into these three different pieces. 303 00:39:43,480 --> 00:39:51,900 We've got the personality parts, the CV part and the demographics. 304 00:39:51,900 --> 00:39:59,580 All we could imagine that these A-level students were trying to come up with a prediction 305 00:39:59,580 --> 00:40:08,490 of their grades of what they would have got in the exam so we can use that as the grade, 306 00:40:08,490 --> 00:40:17,220 then maybe the first part of information is about the individual student and then the green 307 00:40:17,220 --> 00:40:26,070 information could be about the teacher and the Reds information could be about the school. 308 00:40:26,070 --> 00:40:32,640 So all of these different types of information are important to keep in mind. 309 00:40:32,640 --> 00:40:35,340 But do we want to keep them together or separate? 310 00:40:35,340 --> 00:40:45,810 We could also imagine separating the data, so we have student information completely separate from teaching information and school information. 311 00:40:45,810 --> 00:40:50,880 So this is our problem that the key measurement is a measurement of which data. 312 00:40:50,880 --> 00:40:55,440 Is it a key measurement we get by combining all our different measurements together? 313 00:40:55,440 --> 00:41:00,900 So we may combine school and teacher and student all all together? 314 00:41:00,900 --> 00:41:04,860 Or do we want to find individual key measurements of these individual pieces? 315 00:41:04,860 --> 00:41:11,640 And then we'll end up with many different key measurements and not this nice plot of just key measurement number one against key measurement. 316 00:41:11,640 --> 00:41:21,690 Number two will have key measurements one and two for blue, green and red, and it all gets much more complicated. 317 00:41:21,690 --> 00:41:28,680 And combining all of these different types of data together into our key measurement is a big problem, 318 00:41:28,680 --> 00:41:34,620 because then we may end up deciding what grade someone should have got in that A-levels, 319 00:41:34,620 --> 00:41:40,050 not based on information about the student, but based on information about the school. 320 00:41:40,050 --> 00:41:47,700 But on the other hand, we don't want to throw away that contextual information because we'll lose a lot of information that way. 321 00:41:47,700 --> 00:42:00,110 All right, so we need new tools that can overcome these limitations, and luckily, there's a wide rich well beyond principal component analysis. 322 00:42:00,110 --> 00:42:07,460 So here's our data matrix here, we've got our six rows again correspond to our six different people. 323 00:42:07,460 --> 00:42:15,830 And then we have different columns of the Matrix, which are different information for each of the people. 324 00:42:15,830 --> 00:42:20,570 We've got the school information teacher information on school. 325 00:42:20,570 --> 00:42:25,970 Sir, student information, teacher information and school information. 326 00:42:25,970 --> 00:42:31,730 All right, and we want to keep these different pieces together, but also separate. 327 00:42:31,730 --> 00:42:40,790 And one way we can do that is to separate them into three matrices, three grids of numbers like this. 328 00:42:40,790 --> 00:42:51,860 And in many cases, it's possible to combine this together to get a three dimensional grid of numbers, which is called a tensor. 329 00:42:51,860 --> 00:42:59,450 So here's our tents over here, where each row of our three dimensional grid is still a person, 330 00:42:59,450 --> 00:43:09,400 but now we have three layers corresponding to the school information, teacher information and student information. 331 00:43:09,400 --> 00:43:20,580 All right. And we saw before that we can design tools for matrix data using linear algebra, using the theory of matrices. 332 00:43:20,580 --> 00:43:31,120 And similarly, we can design tools for tensor data using multimedia algebra, the theory of tenses. 333 00:43:31,120 --> 00:43:39,010 OK, so it might seem a little bit strange to store data in this way in the form of this three dimensional grid. 334 00:43:39,010 --> 00:43:43,960 But there are some examples of tenses which are very, very familiar to us. 335 00:43:43,960 --> 00:43:55,390 For example, colour photographs. So in a colour photograph, we have different pixels and each pixel is made up of three different numbers. 336 00:43:55,390 --> 00:44:00,460 We've got a number for how red it is, how green it is and how blue it is. 337 00:44:00,460 --> 00:44:07,540 So a data structure like this is really a colour photograph. Now, instead of the rows corresponding to people, 338 00:44:07,540 --> 00:44:18,610 they correspond to the vertical location of a pixel and the columns correspond to the horizontal location of a pixel and then each location. 339 00:44:18,610 --> 00:44:25,990 We have three numbers how blue, how green and how is. 340 00:44:25,990 --> 00:44:31,210 OK, so the tension enables us to keep these different types of information together, 341 00:44:31,210 --> 00:44:38,560 but also separated, and we can design tools using the theory of tenses. 342 00:44:38,560 --> 00:44:44,920 But maybe we want to understand more complex interactions between these different types of information. 343 00:44:44,920 --> 00:44:51,820 So maybe we think that school has some complex influence on the teacher and then subsequently on the student. 344 00:44:51,820 --> 00:45:00,580 And in this more general setting, we can design tools for data using applied algebra. 345 00:45:00,580 --> 00:45:06,610 OK, so these different areas of linear algebra and then multi linear algebra and applied algebra 346 00:45:06,610 --> 00:45:16,540 are ongoing topics of interest in the mathematical community to me and other people. 347 00:45:16,540 --> 00:45:24,670 But at this point, you might be wondering, Well, who are these other people who are also interested in types of algebra? 348 00:45:24,670 --> 00:45:30,250 And I'm happy to report that there's a world beyond call Pearson. 349 00:45:30,250 --> 00:45:36,400 So here's a photo of my colleagues and me from the Society for Industrial and 350 00:45:36,400 --> 00:45:44,560 Applied Mathematics Conference on Applied Algebraic Geometry from July 2019. 351 00:45:44,560 --> 00:45:51,910 So in the world of mathematics has come a long way since the days of Karl Pearson, 352 00:45:51,910 --> 00:46:08,090 and it's a fun and diverse and sociable community of people with an ongoing project to improve its diversity over time. 353 00:46:08,090 --> 00:46:14,090 So to anyone young who might be watching, I highly recommend a career as a mathematician, 354 00:46:14,090 --> 00:46:19,430 we get to travel around different parts of the world and meet up with each other. 355 00:46:19,430 --> 00:46:25,880 So, for example, this photo was taken in Switzerland. 356 00:46:25,880 --> 00:46:36,030 Well, now we have virtual conferences as well, but hopefully we'll be able to travel and see each other in person soon. 357 00:46:36,030 --> 00:46:47,610 All right, so in summary, we've seen how these three different parts of our world, how our Human Day to day lives and our quantitative toolbox. 358 00:46:47,610 --> 00:46:52,230 And then applications in the world all connects together. 359 00:46:52,230 --> 00:47:03,330 So well, specifically, we've seen how an idea via mathematics can be translated to some quantitative tool. 360 00:47:03,330 --> 00:47:15,240 And then that quantitative tool can be used in applications. And insights from these applications can help us to better inform our new ideas. 361 00:47:15,240 --> 00:47:21,570 And we saw this in a particular example where the idea was that of a key measurement. 362 00:47:21,570 --> 00:47:29,270 And then the mathematics that we used to scale it up to a quantitative tool was linear algebra. 363 00:47:29,270 --> 00:47:34,690 And then the quantitative tool was principal component analysis. 364 00:47:34,690 --> 00:47:37,270 But the story is by no means finished, 365 00:47:37,270 --> 00:47:50,910 so we're in need of many new ideas and new areas of mathematics in order to turn these ideas into a new quantitative tools. 366 00:47:50,910 --> 00:48:16,084 And that's all I wanted to say, thank you very much for listening.