{"version":"1.0.0","segments":[{"startTime":0.085,"endTime":0.165,"body":"\u003csilence\u003e"},{"startTime":5.455,"endTime":6.385,"body":"Well, thank you very much."},{"startTime":6.685,"endTime":8.185,"body":"Um, it's, it's lovely to be here."},{"startTime":8.185,"endTime":12.01,"body":"It's an absolute honor to be following some great speakers that have given this lecture"},{"startTime":12.01,"endTime":12.265,"body":"before."},{"startTime":12.765,"endTime":13.905,"body":"And thank you for coming."},{"startTime":14.065,"endTime":16.184,"body":"I know there's distractions tonight."},{"startTime":16.184,"endTime":19.225,"body":"There's an election, there's all kinds of things happening, and you, you've, you've come out"},{"startTime":19.225,"endTime":22.065,"body":"in the rain to a maths talk, so you know, you are my people."},{"startTime":22.115,"endTime":23.225,"body":"Thank you, \u003claugh\u003e."},{"startTime":23.405,"endTime":29.911,"body":"Um, so because it's a math lecture, I thought we should start with the maths"},{"startTime":29.911,"endTime":30.345,"body":"question."},{"startTime":30.445,"endTime":33.265,"body":"So here's a nice easy one just to kind of get you warmed up."},{"startTime":33.325,"endTime":36.825,"body":"So, and anybody in the room gonna tell me the answer to this?"},{"startTime":39.965,"endTime":41.265,"body":"Oh, okay."},{"startTime":41.455,"endTime":42.145,"body":"This isn't good."},{"startTime":42.145,"endTime":44.265,"body":"I've, I've obviously misjudged the audience a bit."},{"startTime":44.585,"endTime":49.585,"body":"\u003claugh\u003e."},{"startTime":44.885,"endTime":47.945,"body":"Um, okay, so let's spoil it."},{"startTime":47.945,"endTime":48.625,"body":"There's the answer."},{"startTime":48.865,"endTime":51.385,"body":"I mean, I, I'm, I'm, I'm a bit disappointed in you guys."},{"startTime":51.565,"endTime":53.145,"body":"I'm a bit disappointed you didn't know this one."},{"startTime":54.255,"endTime":54.545,"body":"Okay."},{"startTime":55.005,"endTime":57.745,"body":"Um, let's try something easier then."},{"startTime":57.745,"endTime":58.505,"body":"Let's try something easier."},{"startTime":59.045,"endTime":61.625,"body":"Um, who can tell me the answer to this one?"},{"startTime":64.084,"endTime":64.305,"body":"Yes."},{"startTime":64.455,"endTime":65.265,"body":"72."},{"startTime":65.295,"endTime":66.145,"body":"72."},{"startTime":66.215,"endTime":66.825,"body":"Fantastic."},{"startTime":67.255,"endTime":67.545,"body":"Okay."},{"startTime":67.955,"endTime":68.305,"body":"Right."},{"startTime":69.005,"endTime":69.625,"body":"That's good."},{"startTime":69.925,"endTime":74.785,"body":"Now, my claim is that these two questions are as easy as each other."},{"startTime":75.485,"endTime":77.385,"body":"My claim is, these are the same question."},{"startTime":78.245,"endTime":82.427,"body":"Um, and I want to try and explain why that's true, the sense in which"},{"startTime":82.427,"endTime":82.985,"body":"that's true."},{"startTime":83.405,"endTime":86.105,"body":"And you'll see there's a little bit of kind of subtle color coding here."},{"startTime":86.105,"endTime":87.785,"body":"There's some, some greens coming in here."},{"startTime":87.845,"endTime":90.465,"body":"The greens are there for a reason that we'll see as we go on."},{"startTime":90.965,"endTime":94.615,"body":"Um, but what I want to do is to try and explain why these are"},{"startTime":94.615,"endTime":95.345,"body":"the same questions."},{"startTime":95.565,"endTime":100.642,"body":"And to be honest, this was, I mean, I, I got the invitation to speak"},{"startTime":100.642,"endTime":104.705,"body":"at Gresham and it took me about, I wanna say 30 seconds."},{"startTime":104.805,"endTime":108.105,"body":"I'm not sure it was even that long to decide to accept, because one of"},{"startTime":108.105,"endTime":110.745,"body":"the reasons is that what I'm talking about kind of started here."},{"startTime":110.865,"endTime":115.915,"body":"I mean, not in this building because I, under, as I understand it, Gresham wasn't"},{"startTime":115.915,"endTime":120.965,"body":"in this building at the time, but some of what I'm gonna talk about was"},{"startTime":120.965,"endTime":122.985,"body":"done by Sarah's long distant predecessor."},{"startTime":123.125,"endTime":126.225,"body":"And another predecessor is professor of, of astronomy."},{"startTime":126.325,"endTime":128.745,"body":"So it's kind of almost kind of coming home with some of this stuff."},{"startTime":130.125,"endTime":134.498,"body":"So what I'm gonna talk about is, I'm gonna talk about logarithms and I'm gonna"},{"startTime":134.498,"endTime":135.665,"body":"talk about the applications."},{"startTime":136.045,"endTime":140.487,"body":"And one of the applications, and this is sort of slightly self, self-referential, is I'm"},{"startTime":140.487,"endTime":142.265,"body":"gonna talk about things going viral."},{"startTime":143.045,"endTime":146.83,"body":"So the last time I gave this lecture, I, I did the same question and"},{"startTime":146.83,"endTime":148.345,"body":"I did it with different numbers."},{"startTime":148.945,"endTime":151.787,"body":"'cause I, I, I thought somebody might have cheated and watched the lecture and, you"},{"startTime":151.787,"endTime":152.545,"body":"know, crib the answers."},{"startTime":152.705,"endTime":153.385,"body":"I changed the number."},{"startTime":153.885,"endTime":155.905,"body":"Um, but this is the number that was interesting."},{"startTime":155.905,"endTime":156.945,"body":"12.3 million."},{"startTime":157.255,"endTime":159.785,"body":"This was Oxford, put this on Instagram."},{"startTime":160.135,"endTime":164.065,"body":"This is how many views there were of Instagram, of me doing a maths lecture."},{"startTime":164.805,"endTime":172.019,"body":"So what I want to talk about also is not just logarithms, but things going"},{"startTime":172.019,"endTime":174.905,"body":"viral, things spreading things going fast."},{"startTime":175.605,"endTime":179.622,"body":"Now, we've obviously all had a little bit of experience of this lately, rather more"},{"startTime":179.622,"endTime":181.765,"body":"experienced of this than, than we might like."},{"startTime":182.265,"endTime":185.725,"body":"So I don't know who remembers this slide."},{"startTime":186.315,"endTime":190.645,"body":"This was maybe one of the most fateful slides of the whole pandemic."},{"startTime":190.705,"endTime":194.045,"body":"So this was the 21st of September, 2020."},{"startTime":194.105,"endTime":199.045,"body":"So this was just beginning to move into the second wave of the pandemic."},{"startTime":199.305,"endTime":205.245,"body":"And at this stage, um, Chris Witty and Patrick Valance gave a press conference."},{"startTime":205.275,"endTime":207.205,"body":"They were worried they could see what was coming."},{"startTime":207.275,"endTime":209.685,"body":"They gave a press conference and they presented this slide."},{"startTime":210.025,"endTime":214.136,"body":"So what they said was, okay, if doubling occurs every seven days, what would it"},{"startTime":214.136,"endTime":214.685,"body":"look like?"},{"startTime":216.425,"endTime":217.925,"body":"Now they're very clear here."},{"startTime":217.925,"endTime":220.005,"body":"They're not saying this is, this is a prediction."},{"startTime":220.005,"endTime":222.045,"body":"They're not saying this is definitely what's gonna happen."},{"startTime":222.045,"endTime":225.362,"body":"They're saying, if, you know, if it did double every seven days, this is what"},{"startTime":225.362,"endTime":225.805,"body":"would happen."},{"startTime":226.345,"endTime":230.955,"body":"Now, as far as we are concerned, there's nothing to argue with here, right?"},{"startTime":230.995,"endTime":231.835,"body":"I mean, this is, this is true."},{"startTime":231.915,"endTime":235.515,"body":"I mean, what they're saying is, okay, at this point, there are 3000 cases."},{"startTime":236.305,"endTime":240.784,"body":"This is the 15th of September when they've got the data up to you double"},{"startTime":240.784,"endTime":242.875,"body":"that every seven days, you double that."},{"startTime":242.875,"endTime":247.635,"body":"Four times you multiply by 16, you end up with this kind of scary growth."},{"startTime":248.935,"endTime":253.063,"body":"But I think from a kind of public relations communications point of view, I don't"},{"startTime":253.063,"endTime":254.715,"body":"think this was a good graph."},{"startTime":255.075,"endTime":258.478,"body":"I didn't think this was a good graph to use because you look at the"},{"startTime":258.478,"endTime":261.881,"body":"blue data up to this point, and you look at the red data that, okay,"},{"startTime":261.881,"endTime":265.285,"body":"it's not a, it's not a prediction, it's a projection, but still, it, it doesn't"},{"startTime":265.285,"endTime":268.235,"body":"look like there's a, there's much of a link between these two things."},{"startTime":268.675,"endTime":271.817,"body":"I mean, you look at this, it doesn't feel like this is a trajectory we're,"},{"startTime":271.817,"endTime":273.075,"body":"we're about to go on immediately."},{"startTime":273.075,"endTime":273.355,"body":"Right?"},{"startTime":275.175,"endTime":279.681,"body":"Now, I think also from a kind of PR point of view, there's a sort"},{"startTime":279.681,"endTime":284.187,"body":"of slight bad choice of color scheme because some of you may, may have seen"},{"startTime":284.187,"endTime":288.693,"body":"this, this is disco stu from the Simpsons, um, who did a sort of similar"},{"startTime":288.693,"endTime":290.195,"body":"sort of extra extrapolation disco."},{"startTime":290.455,"endTime":295.04,"body":"Stu is sitting there saying, if disco record sales double throughout the 1970s, ah, you"},{"startTime":295.04,"endTime":296.875,"body":"know, all this kind of thing."},{"startTime":296.935,"endTime":300.675,"body":"So, um, is disco stu, right?"},{"startTime":300.735,"endTime":302.595,"body":"We, we we're witty and valance, right?"},{"startTime":302.595,"endTime":304.875,"body":"This is the kind of question that, that we'd like to answer."},{"startTime":305.655,"endTime":310.075,"body":"Now, obviously, we have the luxury of knowing what happened."},{"startTime":310.155,"endTime":311.715,"body":"I mean, we know that there was a second wave."},{"startTime":311.735,"endTime":316.595,"body":"We know that we, you know, we did go back into lockdown and all these"},{"startTime":316.595,"endTime":321.455,"body":"things did happen, but equally, um, witty and valances prediction, that wasn't a prediction, that"},{"startTime":321.455,"endTime":323.075,"body":"was a projection didn't happen."},{"startTime":323.275,"endTime":326.547,"body":"I mean, this was, I mean, this slide is slightly cheeky, and I'll say a"},{"startTime":326.547,"endTime":328.075,"body":"little bit about that in a second."},{"startTime":328.255,"endTime":331.795,"body":"But broadly speaking, the curve was meant to go up like this."},{"startTime":332.425,"endTime":335.36,"body":"This is, you know, various ways of, of doing the data, you know, reporting date,"},{"startTime":335.36,"endTime":338.296,"body":"specimen date, seven day average, you know, all this stuff that we used to kind"},{"startTime":338.296,"endTime":339.275,"body":"of worry about a lot."},{"startTime":339.575,"endTime":343.625,"body":"But what you can see is that, broadly speaking, the projections that it would take"},{"startTime":343.625,"endTime":346.595,"body":"off the actual data didn't, or not to the same extent."},{"startTime":347.895,"endTime":352.057,"body":"Now this slide comes from Carl Hannigan's group in Oxford, who, you know, he was"},{"startTime":352.057,"endTime":354.555,"body":"a kind of skeptic of all of this stuff."},{"startTime":354.555,"endTime":358.539,"body":"And so he was presenting this to say, well, what a ridiculous predict, you know,"},{"startTime":358.539,"endTime":358.805,"body":"prediction."},{"startTime":358.805,"endTime":363.838,"body":"This was obviously all overblown, but still, as I say, the, the, the, the second"},{"startTime":363.838,"endTime":364.845,"body":"wave did happen."},{"startTime":365.925,"endTime":368.679,"body":"I mean, one sense in which this is cheeky is he, he's run this on"},{"startTime":368.679,"endTime":369.965,"body":"a week further than he should have."},{"startTime":370.125,"endTime":373.762,"body":"I mean, he, we said that the projection would run out four weeks to the"},{"startTime":373.762,"endTime":374.005,"body":"14th."},{"startTime":374.075,"endTime":375.685,"body":"This would be 49,000."},{"startTime":376.075,"endTime":378.205,"body":"He's run it on another kind of eight or nine days."},{"startTime":378.225,"endTime":380.565,"body":"So it's, it's gone further than it should have done."},{"startTime":380.585,"endTime":383.725,"body":"So, you know, he's not even comparing with the projection that it's made."},{"startTime":383.725,"endTime":388.114,"body":"And you can see that that has the effect visually of compressing down the growth"},{"startTime":388.114,"endTime":389.285,"body":"that we did see."},{"startTime":389.745,"endTime":392.845,"body":"Um, but still, I mean, we can't argue with this."},{"startTime":393.005,"endTime":397.061,"body":"I mean, it's not the case that on this date when Witty and Valen said"},{"startTime":397.061,"endTime":399.765,"body":"there might be 49,000 cases that there were 49,000 cases."},{"startTime":401.345,"endTime":403.965,"body":"So kind of PR problem."},{"startTime":404.545,"endTime":406.085,"body":"So what should we have done?"},{"startTime":406.785,"endTime":411.638,"body":"So if, and I'm very glad I wasn't, but if I had been trying to"},{"startTime":411.638,"endTime":415.845,"body":"present this data, trying to present these numbers, here's what I would've done."},{"startTime":416.225,"endTime":417.565,"body":"Here's the graph I would've shown."},{"startTime":419.385,"endTime":423.045,"body":"So what we have here is we have the same thing."},{"startTime":423.045,"endTime":424.805,"body":"We have the data up to a certain point."},{"startTime":425.065,"endTime":429.171,"body":"So what we have is we have the particular days of data plotted as red"},{"startTime":429.171,"endTime":429.445,"body":"dots."},{"startTime":429.905,"endTime":433.108,"body":"And what you can see is that instead of having a kind of line that"},{"startTime":433.108,"endTime":435.885,"body":"goes flatten and swooshes up, I've got a straight line going through this."},{"startTime":437.025,"endTime":441.544,"body":"Now, this is kind of what the talk is about is you'll notice that what"},{"startTime":441.544,"endTime":446.063,"body":"I've done is I've done something weird with the y axis that if you go"},{"startTime":446.063,"endTime":450.582,"body":"back to this graph, you can see that the, the spacings are in some sense"},{"startTime":450.582,"endTime":455.101,"body":"additive that we go from zero to 25,000 is the same size as going from"},{"startTime":455.101,"endTime":456.005,"body":"25,000 to 50,000."},{"startTime":456.265,"endTime":458.485,"body":"The steps are additive amounts."},{"startTime":459.425,"endTime":462.005,"body":"My steps are different, my steps are multiplicative."},{"startTime":462.005,"endTime":466.285,"body":"So for example, my step from a thousand to 2000 is the same as my"},{"startTime":466.285,"endTime":470.565,"body":"step from 5,000 to a to 10,000, which is the same from 10,000 to 20,000."},{"startTime":470.875,"endTime":471.885,"body":"This is a log scale."},{"startTime":472.345,"endTime":473.805,"body":"And so this is what I want to talk about."},{"startTime":473.805,"endTime":477.776,"body":"This is what I want to evangelize for and to say log scales are the"},{"startTime":477.776,"endTime":479.365,"body":"right way to plot certain things."},{"startTime":479.745,"endTime":481.165,"body":"And this is an example of one of them."},{"startTime":482.185,"endTime":487.218,"body":"Now, one thing that you'll notice is that sort of for free, I got a"},{"startTime":487.218,"endTime":491.245,"body":"doubling time here that that Witt and balance made this particular number."},{"startTime":491.245,"endTime":493.485,"body":"They said, okay, seven days if it doubled every seven days."},{"startTime":493.705,"endTime":498.095,"body":"I'm not saying if I'm saying it was that, what I'm saying is that from"},{"startTime":498.095,"endTime":502.486,"body":"this data at this point, if I put a straight line through this, then the"},{"startTime":502.486,"endTime":506.877,"body":"amount of time it takes to go from this straight line being a 2000, being"},{"startTime":506.877,"endTime":509.805,"body":"at 2000 is turns out at the time 12.4 days."},{"startTime":509.905,"endTime":511.725,"body":"So it's slower, but it's still doubling."},{"startTime":511.865,"endTime":516.044,"body":"But what you can see is that because it's a straight line, it goes up"},{"startTime":516.044,"endTime":520.223,"body":"these doubling steps, it takes the same amount of time each time it takes 12.4"},{"startTime":520.223,"endTime":523.845,"body":"days down here, but it also takes 12.4 days up at the top."},{"startTime":525.835,"endTime":526.125,"body":"Okay?"},{"startTime":526.345,"endTime":528.885,"body":"So the obvious question is, well what happened?"},{"startTime":528.885,"endTime":531.925,"body":"You know, I said the witty valance thing didn't happen."},{"startTime":532.395,"endTime":534.165,"body":"What if I'd done the graph, what would've happened?"},{"startTime":534.275,"endTime":537.045,"body":"Well run it on, there's the data."},{"startTime":538.025,"endTime":542.97,"body":"So what you can see is that this kind of very crude log scale drew"},{"startTime":542.97,"endTime":547.916,"body":"a straight line through it projection that maybe was a projection or was a prediction,"},{"startTime":547.916,"endTime":549.565,"body":"I dunno what it was."},{"startTime":549.905,"endTime":553.69,"body":"But that by squashing the axis up in this particular way by representing the data"},{"startTime":553.69,"endTime":557.475,"body":"so that the way that it grows is on a straight line, you can actually"},{"startTime":557.475,"endTime":561.26,"body":"see that it did a pretty good job of picking out what would happen into"},{"startTime":561.26,"endTime":561.765,"body":"the future."},{"startTime":562.425,"endTime":566.19,"body":"So it's not saying that we would end up at 50,000, it said we would"},{"startTime":566.19,"endTime":567.445,"body":"end up at at 20,000."},{"startTime":567.865,"endTime":569.125,"body":"But the point is that was enough."},{"startTime":569.405,"endTime":570.525,"body":"I mean, doubling is doubling."},{"startTime":570.605,"endTime":574.52,"body":"I mean, it's maybe taking slightly longer to get to the stage where you run"},{"startTime":574.52,"endTime":578.435,"body":"out of hospital beds, but still it, you're gonna get there eventually and this sort"},{"startTime":578.435,"endTime":581.045,"body":"of projection is telling you how fast that's gonna happen."},{"startTime":581.625,"endTime":583.165,"body":"So that's the kind of theme of the lecture."},{"startTime":585.145,"endTime":588.685,"body":"Now, I wanted to kind of go back a little bit in history on this."},{"startTime":588.785,"endTime":591.965,"body":"So first of all, I wanted to tell you what a algorithm is."},{"startTime":592.005,"endTime":594.825,"body":"I mean, I know you are here in the rain for a math lecture, but"},{"startTime":594.825,"endTime":597.645,"body":"I just want to kind of just check that we're all on board with this."},{"startTime":598.305,"endTime":601.565,"body":"So here is a simpler problem than the one that I did at the start."},{"startTime":601.705,"endTime":603.725,"body":"So we had one that people could do."},{"startTime":603.745,"endTime":607.685,"body":"Here's another one that hopefully everybody can do four times 80, 32."},{"startTime":607.755,"endTime":609.445,"body":"Hopefully everybody's on board with that."},{"startTime":609.625,"endTime":609.845,"body":"Yes."},{"startTime":609.915,"endTime":610.205,"body":"Good."},{"startTime":610.275,"endTime":610.565,"body":"Okay."},{"startTime":610.625,"endTime":611.485,"body":"I'm seeing some dots."},{"startTime":611.745,"endTime":616.892,"body":"Now the, obviously we know that because we learned our times tables, but there is"},{"startTime":616.892,"endTime":622.039,"body":"another way to think about this, which is that, again, I've been slightly cunning here"},{"startTime":622.039,"endTime":627.186,"body":"that when I say four, the way I'm thinking about four is I'm thinking about"},{"startTime":627.186,"endTime":629.245,"body":"four as two times two, okay?"},{"startTime":630.135,"endTime":635.225,"body":"Which obviously, and again, you'll notice this is in green, I'm thinking of as two"},{"startTime":635.225,"endTime":635.565,"body":"squared."},{"startTime":635.745,"endTime":638.445,"body":"So there are two factors of two in my four here."},{"startTime":639.305,"endTime":643.885,"body":"The eight, again, I'm lucky or I cheated depending on how you think about it."},{"startTime":644.345,"endTime":646.965,"body":"The eight is, again, it's multiples of two."},{"startTime":647.015,"endTime":648.645,"body":"Eight is two times two times two."},{"startTime":648.645,"endTime":650.805,"body":"There are three factors of two in the eight."},{"startTime":651.265,"endTime":654.365,"body":"So what I have is that I have two cubed."},{"startTime":654.505,"endTime":655.685,"body":"So again, in green."},{"startTime":656.705,"endTime":660.495,"body":"So now when I come to do my sum, when I come to do this"},{"startTime":660.495,"endTime":664.286,"body":"multiplication, then what you can see is that that to to, to know that the"},{"startTime":664.286,"endTime":665.045,"body":"answer is 32."},{"startTime":665.355,"endTime":669.322,"body":"When I take four times eight, what I need to do is to say, well,"},{"startTime":669.322,"endTime":673.289,"body":"I've got two factors of two from the four, I've got three factors of two"},{"startTime":673.289,"endTime":675.405,"body":"from the eight, two plus three is funna."},{"startTime":675.405,"endTime":678.445,"body":"In total I've got funna factors of two in, in the product."},{"startTime":679.185,"endTime":684.796,"body":"And so what I've managed to do is I've managed to turn a problem about"},{"startTime":684.796,"endTime":689.285,"body":"multiplication, which is hard into a problem about addition, which is easy."},{"startTime":689.945,"endTime":693.549,"body":"So this is the idea here that, that what we've done is just that, just"},{"startTime":693.549,"endTime":697.154,"body":"by counting the factors of two, seeing how many of these things there are, what's"},{"startTime":697.154,"endTime":700.759,"body":"driving this, and you can see this in green, is the two to the two"},{"startTime":700.759,"endTime":704.364,"body":"times two to the three is two to the two plus three, which is two"},{"startTime":704.364,"endTime":705.085,"body":"to the five."},{"startTime":705.305,"endTime":706.965,"body":"So that's what's underlying this equation."},{"startTime":707.025,"endTime":709.005,"body":"The four times eight is 32 that we know."},{"startTime":709.345,"endTime":712.613,"body":"We can also think about it as being an equation that's to do with with"},{"startTime":712.613,"endTime":713.485,"body":"without adding things up."},{"startTime":714.025,"endTime":716.902,"body":"So what we can do is we can focus just on these numbers in green,"},{"startTime":716.902,"endTime":719.205,"body":"we can focus on the two and the three and the five."},{"startTime":719.705,"endTime":720.965,"body":"And these are the logarithms."},{"startTime":720.965,"endTime":725.451,"body":"This is, this is, you know, if, if I was to define to you what"},{"startTime":725.451,"endTime":727.845,"body":"a logarithm is, then you know it's there."},{"startTime":727.895,"endTime":732.041,"body":"These, these are the things formally speaking, I would say it was the, the inverse"},{"startTime":732.041,"endTime":734.805,"body":"of the exponential function and all this kind of thing."},{"startTime":734.875,"endTime":736.325,"body":"It's the thing that you put in the power."},{"startTime":736.515,"endTime":739.676,"body":"It's the thing that you put on the top of the, of the power to,"},{"startTime":739.676,"endTime":742.205,"body":"to to, to make the, to give the answer that you want."},{"startTime":742.785,"endTime":747.096,"body":"And so the rule is this, that if the numbers multiply the logarithms add, so"},{"startTime":747.096,"endTime":751.407,"body":"the, the fact that it's five, it's two plus three where you add together the"},{"startTime":751.407,"endTime":752.845,"body":"log, the, the, the logarithms."},{"startTime":754.385,"endTime":760.971,"body":"And so my silly maths question at the beginning, I cheated, I found online there"},{"startTime":760.971,"endTime":764.045,"body":"is a table of powers of two."},{"startTime":764.585,"endTime":767.365,"body":"Um, I looked up what is two to the 31."},{"startTime":767.365,"endTime":770.605,"body":"Well, two to the 31 is the first of these big numbers I wrote down."},{"startTime":771.045,"endTime":772.365,"body":"I looked up two to the 41."},{"startTime":772.365,"endTime":774.125,"body":"It's the second of these big numbers that I wrote down."},{"startTime":774.425,"endTime":778.49,"body":"So this scary looking product that I had at the beginning, I can represent it's"},{"startTime":778.49,"endTime":782.556,"body":"two to the 31 times two to the 41, which is two to the, and"},{"startTime":782.556,"endTime":784.725,"body":"we had the answer at the back 72."},{"startTime":785.425,"endTime":787.445,"body":"So it's a cheat."},{"startTime":788.085,"endTime":788.645,"body":"I cheated."},{"startTime":788.885,"endTime":795.453,"body":"I mean I, I I, you know, clearly there's a sense in which I have"},{"startTime":795.453,"endTime":797.205,"body":"slightly deceived you here."},{"startTime":798.235,"endTime":800.205,"body":"Most numbers aren't powers of two."},{"startTime":800.235,"endTime":806.605,"body":"Most of the time I can't do this trick except I sort of can."},{"startTime":807.465,"endTime":812.208,"body":"Um, because you know, even if it isn't a perfect power, then similar things will"},{"startTime":812.208,"endTime":812.525,"body":"work."},{"startTime":812.825,"endTime":818.667,"body":"If you'd given me two big scary numbers in theory, I could have gone back"},{"startTime":818.667,"endTime":821.005,"body":"to my powers of two table."},{"startTime":821.485,"endTime":822.125,"body":"I could have looked up."},{"startTime":822.125,"endTime":825.487,"body":"It wouldn't have been exactly a whole number, but, but the numbers would've been two"},{"startTime":825.487,"endTime":828.85,"body":"to the something, the, the numbers I could represent as two to the, something I"},{"startTime":828.85,"endTime":831.765,"body":"could look up, I could solve, I could find what the somethings are."},{"startTime":832.025,"endTime":833.765,"body":"And then the multiplication, I could do the same thing."},{"startTime":833.845,"endTime":835.005,"body":"I could multiply 'em together."},{"startTime":836.825,"endTime":840.668,"body":"Now, I mentioned this already, but this is where I'm, again, I'm using green as"},{"startTime":840.668,"endTime":840.925,"body":"code."},{"startTime":841.545,"endTime":847.405,"body":"Um, so green now was for logarithms, green is also going to be for Gresham."},{"startTime":847.465,"endTime":849.165,"body":"So here are a couple of people."},{"startTime":849.305,"endTime":853.222,"body":"So generally I think if you talk to mathematicians and say, who invented logarithms, I"},{"startTime":853.222,"endTime":857.139,"body":"think if anybody was to mention a name, the name people would come up with"},{"startTime":857.139,"endTime":858.445,"body":"would, would probably be Napier."},{"startTime":858.445,"endTime":861.285,"body":"So Napier is, is on the left here."},{"startTime":861.945,"endTime":866.476,"body":"But actually somebody that sort of maybe doesn't get as much credit as they should"},{"startTime":866.476,"endTime":867.685,"body":"is, is is Briggs."},{"startTime":867.685,"endTime":872.895,"body":"So Briggs was, as I say, the first Gresham professor of, of geometry, um, who"},{"startTime":872.895,"endTime":878.105,"body":"actually, if, if you, and the, the Wikipedia page is very good on this, uh,"},{"startTime":878.105,"endTime":883.315,"body":"nap p's logarithms aren't logarithms as we know them, that, that they don't satisfy exactly"},{"startTime":883.315,"endTime":888.525,"body":"the relationship they have instead of being the, the, um, the logarithm of the product"},{"startTime":888.525,"endTime":893.735,"body":"is the sum of the law algorithms, it's, there's an extra term, there's an extra"},{"startTime":893.735,"endTime":895.125,"body":"weird factor in there."},{"startTime":895.145,"endTime":897.605,"body":"So actually it was, it was Briggs that straightened this out."},{"startTime":897.605,"endTime":900.653,"body":"It was Briggs that realized there was a better way of representing it, that what"},{"startTime":900.653,"endTime":902.685,"body":"we were aiming for was this kind of product rule."},{"startTime":902.865,"endTime":904.765,"body":"And so Briggs was the person that figured this out."},{"startTime":905.225,"endTime":906.725,"body":"So that's one Gresham connection."},{"startTime":907.665,"endTime":912.446,"body":"The other Gresham connection, um, some people in the audience, let's do a show of"},{"startTime":912.446,"endTime":912.765,"body":"hands."},{"startTime":913.345,"endTime":915.285,"body":"Who has used a slide rule?"},{"startTime":916.025,"endTime":919.737,"body":"Yes, I'm, I'm, I'm, I'm just about missed this, but you know, a lot of"},{"startTime":919.737,"endTime":921.965,"body":"people in this room have used a slide rule."},{"startTime":922.115,"endTime":925.581,"body":"What you'll notice is that if you look at a slide rule, it's exactly like"},{"startTime":925.581,"endTime":929.048,"body":"my log scale on the graph that what we have here, the step from one"},{"startTime":929.048,"endTime":932.515,"body":"to two is the same as the step from two to four, which is the"},{"startTime":932.515,"endTime":934.365,"body":"same as the step from four to eight."},{"startTime":934.745,"endTime":938.99,"body":"So what a slide rule does, is it, it's, it's a physical device that has"},{"startTime":938.99,"endTime":940.405,"body":"a log scale in it."},{"startTime":940.825,"endTime":944.085,"body":"And so you multiply numbers using a slide rule."},{"startTime":944.585,"endTime":946.805,"body":"You are basically using these laws of logarithms."},{"startTime":946.805,"endTime":949.776,"body":"You're using exactly the same property that I did with the number that I, that"},{"startTime":949.776,"endTime":950.965,"body":"I, that I came up with."},{"startTime":951.425,"endTime":953.685,"body":"And so again, there's aggression connection."},{"startTime":953.685,"endTime":957.741,"body":"So the, the idea of a log scale was invented by Gunter, who was, I"},{"startTime":957.741,"endTime":961.797,"body":"think the third Gresham professor of astronomy came up with this idea that actually, you"},{"startTime":961.797,"endTime":965.853,"body":"know, instead of plotting the numbers themselves, you could plot the logs that this was"},{"startTime":965.853,"endTime":967.205,"body":"an interesting thing to do."},{"startTime":967.705,"endTime":970.445,"body":"And outed turned this into a physical device."},{"startTime":970.575,"endTime":973.613,"body":"Outed said, well, okay, what we could do is if we had a piece of"},{"startTime":973.613,"endTime":976.651,"body":"wood that was log X long and we had a piece of wood that was"},{"startTime":976.651,"endTime":979.689,"body":"log y long, then we could put those two pieces of wood next to each"},{"startTime":979.689,"endTime":982.525,"body":"other, the, the, the length of wood be the log of, of the product."},{"startTime":982.985,"endTime":986.36,"body":"And so it, it, it's a sort of nice chain of ideas that sort of"},{"startTime":986.36,"endTime":989.285,"body":"says, okay, we start off with this thing that Napier came up with."},{"startTime":989.665,"endTime":993.605,"body":"Briggs refined it, Gunter invented the log scale out, turned it into a device."},{"startTime":993.605,"endTime":997.038,"body":"It's a nice story of how sort of mathematics works often that these things build"},{"startTime":997.038,"endTime":997.725,"body":"on one another."},{"startTime":997.725,"endTime":1001.867,"body":"That, that it's often not just the case that one person has an idea, but"},{"startTime":1001.867,"endTime":1003.525,"body":"there's, there's often sort of teamwork."},{"startTime":1008.515,"endTime":1008.805,"body":"Okay?"},{"startTime":1008.865,"endTime":1013.72,"body":"So what I'd like to do now is to sort of try to explain to"},{"startTime":1013.72,"endTime":1017.605,"body":"you why this trick with the log scale and the pandemic worked."},{"startTime":1018.745,"endTime":1024.854,"body":"So my claim is, and I had innumerable arguments with people on Twitter about this"},{"startTime":1024.854,"endTime":1026.483,"body":"for about two years."},{"startTime":1027.145,"endTime":1031.605,"body":"My claim is that a log scale is the right way to represent a pandemic."},{"startTime":1032.204,"endTime":1035.504,"body":"I mean, we sort of saw this already from the fact that we plotted the"},{"startTime":1035.504,"endTime":1038.365,"body":"numbers, they lay along a line, but, you know, was that a coincidence?"},{"startTime":1038.365,"endTime":1039.885,"body":"You know, what, what's the reason for that?"},{"startTime":1040.635,"endTime":1046.342,"body":"Well, on some level, the point is that the way that epidemics work is they"},{"startTime":1046.342,"endTime":1047.483,"body":"tend to multiply."},{"startTime":1047.635,"endTime":1051.645,"body":"There's a sort of multiplicative nature built into them."},{"startTime":1051.645,"endTime":1056.57,"body":"And this is why they're scary that the idea is that roughly speaking, each person"},{"startTime":1056.57,"endTime":1059.525,"body":"who is infected infects a similar number of people."},{"startTime":1059.645,"endTime":1063.572,"body":"I mean, this is this famous R number that we used to worry about a"},{"startTime":1063.572,"endTime":1067.499,"body":"lot, that if the R number is three, if I'm infected on average, I will"},{"startTime":1067.499,"endTime":1068.285,"body":"infect three people."},{"startTime":1068.335,"endTime":1070.66,"body":"Those three people will go on and infect three people each."},{"startTime":1070.66,"endTime":1072.725,"body":"So it's three times three, they will infect three."},{"startTime":1073.025,"endTime":1075.445,"body":"And and so you get this exponential growth coming out of it."},{"startTime":1076.745,"endTime":1082.109,"body":"Now, the, the next stage of the argument that I have on Twitter then goes,"},{"startTime":1082.109,"endTime":1085.685,"body":"people say, oh, but exponential growth doesn't go on forever."},{"startTime":1085.865,"endTime":1087.085,"body":"And it's like, well, no, it doesn't."},{"startTime":1087.605,"endTime":1089.365,"body":"I mean, clearly it, it, it, it can't do."},{"startTime":1089.405,"endTime":1093.168,"body":"I mean, you know, you you, you take these numbers, if I infect three people,"},{"startTime":1093.168,"endTime":1094.925,"body":"they infect three people and so on."},{"startTime":1094.925,"endTime":1099.575,"body":"You can quickly see that in a certain number of generations you will run outta"},{"startTime":1099.575,"endTime":1099.885,"body":"people."},{"startTime":1099.885,"endTime":1105.785,"body":"There aren't enough people in the world before this thing starts to, um, starts to"},{"startTime":1105.785,"endTime":1106.965,"body":"run outta steam."},{"startTime":1107.745,"endTime":1112.651,"body":"But the point is that it can go on exponentially for, for, for much longer"},{"startTime":1112.651,"endTime":1117.557,"body":"than we'd like that it does run outta steam eventually, but particularly in the sort"},{"startTime":1117.557,"endTime":1122.463,"body":"of early waves of the pandemic before we had any vaccines, the exponential growth could"},{"startTime":1122.463,"endTime":1123.445,"body":"be uncomfortably long."},{"startTime":1124.865,"endTime":1127.325,"body":"And so we'd like to understand how this happens."},{"startTime":1127.505,"endTime":1131.125,"body":"And actually, it turns out nearly a hundred years ago we understood this."},{"startTime":1131.345,"endTime":1138.64,"body":"So there is this classic SIR paper of 1927 that sets out some mathematical equations"},{"startTime":1138.64,"endTime":1142.045,"body":"that essentially says how, how pandemics evolve."},{"startTime":1143.145,"endTime":1145.485,"body":"And so it's a toy model."},{"startTime":1145.795,"endTime":1147.645,"body":"It's, it's, it's not a model that's perfect."},{"startTime":1147.795,"endTime":1153.287,"body":"It's not a model that captures everything, but it's a model that works pretty well,"},{"startTime":1153.287,"endTime":1155.485,"body":"you know, for a long time."},{"startTime":1155.985,"endTime":1157.765,"body":"It gives you a good idea of what's going on."},{"startTime":1158.145,"endTime":1160.95,"body":"So the idea is this, that what we think about is we think about, there"},{"startTime":1160.95,"endTime":1161.885,"body":"are three types of people."},{"startTime":1161.885,"endTime":1166.165,"body":"There are people who are susceptible, so that means they haven't been infected yet."},{"startTime":1166.975,"endTime":1170.568,"body":"There are people who are infected, so they've got it, they, you know, they're in"},{"startTime":1170.568,"endTime":1172.485,"body":"danger of, of giving it to other people."},{"startTime":1172.735,"endTime":1174.125,"body":"There are people who are recovered."},{"startTime":1175.185,"endTime":1179.913,"body":"Now what you'll see is that as a story for the first few months of"},{"startTime":1179.913,"endTime":1181.805,"body":"the pandemic, this is pretty good."},{"startTime":1182.745,"endTime":1186.285,"body":"The issue with this is that, for example, this doesn't allow for reinfection."},{"startTime":1186.465,"endTime":1189.733,"body":"So the way that this model works is when you're recovered, you're immune, you've got"},{"startTime":1189.733,"endTime":1190.605,"body":"this, you've done forever."},{"startTime":1191.345,"endTime":1193.965,"body":"So for some diseases, this would work as a model for covid."},{"startTime":1193.965,"endTime":1197.325,"body":"Whereas we now know, unfortunately, you know, reinfection and things are possible."},{"startTime":1198.555,"endTime":1202.282,"body":"It's not perfect, but, you know, early on in terms of the dynamics, it's, it's,"},{"startTime":1202.282,"endTime":1203.525,"body":"it's, it's, it's not bad."},{"startTime":1204.065,"endTime":1209.268,"body":"So what they did, Mack and McKendrick who wrote this paper, um, wrote down this"},{"startTime":1209.268,"endTime":1214.471,"body":"set of equations that say how the number of infected, susceptible, and recovered people change"},{"startTime":1214.471,"endTime":1215.165,"body":"over time."},{"startTime":1217.145,"endTime":1221.174,"body":"Now, the interesting one, I mean they're all interesting, but the, the most interesting one"},{"startTime":1221.174,"endTime":1225.204,"body":"is probably this one in the middle, this equation here, this is saying how the"},{"startTime":1225.204,"endTime":1227.085,"body":"number of infected people changes with time."},{"startTime":1227.305,"endTime":1228.325,"body":"That's the left hand side."},{"startTime":1228.445,"endTime":1231.525,"body":"DI by dt, how does the number of infected people change with time?"},{"startTime":1232.185,"endTime":1234.485,"body":"And what you'll see is that there are two effects here."},{"startTime":1234.485,"endTime":1237.285,"body":"There's a, there's a, a positive term and a negative term."},{"startTime":1237.985,"endTime":1241.445,"body":"So the negative term is here, there's a minus gamma."},{"startTime":1241.485,"endTime":1245.907,"body":"I, so what that's saying is that people get better, A proportion of people, a,"},{"startTime":1245.907,"endTime":1249.445,"body":"a, a certain fixed fraction of people who are infected get better."},{"startTime":1249.985,"endTime":1254.035,"body":"So there's a minus gam RI people drop outta the infected class and they turn"},{"startTime":1254.035,"endTime":1254.845,"body":"up over here."},{"startTime":1255.025,"endTime":1260.365,"body":"The DR by DT is positive that people move from infected to recovered."},{"startTime":1260.745,"endTime":1263.205,"body":"As you know, the symptoms wear off, you know, 10 days or whatever."},{"startTime":1263.205,"endTime":1267.245,"body":"It's the more interesting term is the first term."},{"startTime":1267.315,"endTime":1271.125,"body":"This is the term by which people get, um, infected."},{"startTime":1271.585,"endTime":1275.485,"body":"And this is kind of representing a sort of degree of interaction here."},{"startTime":1276.065,"endTime":1281.309,"body":"So what you can think about is that, that at a particular time, there are"},{"startTime":1281.309,"endTime":1286.205,"body":"eye people that have this disease eye, people have this disease left to themselves."},{"startTime":1286.475,"endTime":1290.507,"body":"They would infect beta people say there is some number beta of people that they"},{"startTime":1290.507,"endTime":1291.045,"body":"would infect."},{"startTime":1292.385,"endTime":1297.335,"body":"So you would multiply it by beta, but not all the people that they could"},{"startTime":1297.335,"endTime":1298.325,"body":"infect are infect."},{"startTime":1299.155,"endTime":1302.869,"body":"Some of the people that they meet have got it already or have had it"},{"startTime":1302.869,"endTime":1303.365,"body":"been recovered."},{"startTime":1303.705,"endTime":1307.43,"body":"So what you see is that there is this term here, s over N, which"},{"startTime":1307.43,"endTime":1311.156,"body":"is kind of the fraction of people that they meet who are capable of being"},{"startTime":1311.156,"endTime":1311.405,"body":"infected."},{"startTime":1311.825,"endTime":1315.098,"body":"So left of themselves, they would infect beta times I, but actually they only meet,"},{"startTime":1315.098,"endTime":1318.372,"body":"you know, the proportion of people that they meet who are, who could get it"},{"startTime":1318.372,"endTime":1319.245,"body":"are s over N."},{"startTime":1319.585,"endTime":1323.353,"body":"So in total, what you get is you get beta times I times s over"},{"startTime":1323.353,"endTime":1323.605,"body":"N."},{"startTime":1323.625,"endTime":1327.082,"body":"So that's the sort of term that says this is, this is the rate at"},{"startTime":1327.082,"endTime":1328.005,"body":"which people get infected."},{"startTime":1328.605,"endTime":1330.405,"body":"I dunno what beta is, I dunno what gamma is."},{"startTime":1330.595,"endTime":1331.445,"body":"They're, they're numbers."},{"startTime":1331.585,"endTime":1334.645,"body":"But the point is that, that these should stay the same over time."},{"startTime":1334.645,"endTime":1335.845,"body":"These should stay roughly the same."},{"startTime":1338.025,"endTime":1341.175,"body":"Now, if we go back to this equation, if you think about it a little"},{"startTime":1341.175,"endTime":1344.325,"body":"bit more, you know, I've kind of written it out here as, as an equation."},{"startTime":1344.865,"endTime":1348.165,"body":"I'm doing my best with Microsoft's word equation editor."},{"startTime":1348.865,"endTime":1351.855,"body":"But the point is that what we can do is we can divide this equation"},{"startTime":1351.855,"endTime":1354.845,"body":"by I, on the right hand side, there is an I and there is an"},{"startTime":1354.845,"endTime":1355.045,"body":"I."},{"startTime":1355.865,"endTime":1359.525,"body":"So we could divide through the whole equation by I."},{"startTime":1360.105,"endTime":1363.125,"body":"So if we do that, then on the right hand side, we've got something simpler."},{"startTime":1363.295,"endTime":1366.285,"body":"We've got beta times s over N minus gamma, okay?"},{"startTime":1366.345,"endTime":1368.445,"body":"So on the right hand side, that's become simpler."},{"startTime":1369.665,"endTime":1374.19,"body":"And the point is that by doing this, people that are doing a level and"},{"startTime":1374.19,"endTime":1378.715,"body":"doing calculus will know this, that the derivative of the log is the di by"},{"startTime":1378.715,"endTime":1380.525,"body":"DT divided by the I itself."},{"startTime":1381.145,"endTime":1384.568,"body":"So people that aren't doing a level or did a level a long time ago,"},{"startTime":1384.568,"endTime":1387.991,"body":"I'll let you off if you don't know this, just trust me on this, that"},{"startTime":1387.991,"endTime":1390.045,"body":"dividing through, we get the derivative of the log."},{"startTime":1390.265,"endTime":1394.017,"body":"So if we look at what this story is telling us, what this story is"},{"startTime":1394.017,"endTime":1397.769,"body":"telling us is the, the, the, the log of the number of infected people is"},{"startTime":1397.769,"endTime":1401.521,"body":"the right unit in which to be thinking that instead of thinking about the rate"},{"startTime":1401.521,"endTime":1405.273,"body":"of the, the rate at which the number of infected people changes, we think about"},{"startTime":1405.273,"endTime":1408.525,"body":"the rate at which the log of the number of infected people changes."},{"startTime":1408.625,"endTime":1409.885,"body":"That's a more natural thing."},{"startTime":1410.785,"endTime":1416.005,"body":"And the point is that roughly speaking, this right hand side is constant."},{"startTime":1418.125,"endTime":1423.35,"body":"I mean, it's not, but for a long time it is right that roughly speaking,"},{"startTime":1423.35,"endTime":1426.485,"body":"the proportion of susceptible people doesn't change very much."},{"startTime":1426.545,"endTime":1431.09,"body":"You know, how many people were susceptible today is probably similar to the number of"},{"startTime":1431.09,"endTime":1432.605,"body":"people that were susceptible yesterday."},{"startTime":1432.725,"endTime":1436.625,"body":"I mean, if you run it out, say three months into the future, then no,"},{"startTime":1436.625,"endTime":1440.525,"body":"but you know, for a few weeks at a time, roughly speaking particularly early on"},{"startTime":1440.525,"endTime":1443.125,"body":"in the pandemic, the s over n is roughly constant."},{"startTime":1443.625,"endTime":1446.868,"body":"So roughly speaking, what this is saying is that the right hand side is roughly"},{"startTime":1446.868,"endTime":1447.085,"body":"constant."},{"startTime":1447.345,"endTime":1449.565,"body":"So the derivative of the log is roughly constant."},{"startTime":1449.865,"endTime":1452.805,"body":"So this should be roughly a straight line, and that's what we saw."},{"startTime":1453.185,"endTime":1458.135,"body":"So that's the kind of argument from the kergan, um, McKendrick paper to go back"},{"startTime":1458.135,"endTime":1463.085,"body":"to that, to say, broadly speaking is what we expect is exponential growth or decay."},{"startTime":1463.585,"endTime":1468.351,"body":"So just to kind of illustrate this, to go back to real covid data again"},{"startTime":1468.351,"endTime":1472.165,"body":"here is this was Northwest hospitals up to similar sort of dates."},{"startTime":1472.165,"endTime":1475.565,"body":"The 19th of September doesn't look too bad, right?"},{"startTime":1475.685,"endTime":1479.094,"body":"I mean, you know, we, we, we have this level, you know, in the first"},{"startTime":1479.094,"endTime":1480.685,"body":"wave we hit 3000 people in hospital."},{"startTime":1481.855,"endTime":1483.085,"body":"We've gone right down, okay?"},{"startTime":1483.085,"endTime":1485.747,"body":"It started growing a little bit, but it doesn't seem like it's anything to worry"},{"startTime":1485.747,"endTime":1485.925,"body":"about."},{"startTime":1486.025,"endTime":1489.405,"body":"It doesn't seem like looked at like that."},{"startTime":1489.465,"endTime":1491.965,"body":"It doesn't look like this is, this is something that's gonna trouble us."},{"startTime":1493.525,"endTime":1498.198,"body":"I was going quietly spare about this because I was on Twitter plotting this stuff"},{"startTime":1498.198,"endTime":1499.445,"body":"on a log scale."},{"startTime":1499.505,"endTime":1502.447,"body":"So what I've done is I've taken the same numbers and instead of plotting it"},{"startTime":1502.447,"endTime":1504.605,"body":"on a linear scale, I'm plotting it on a log scale."},{"startTime":1504.655,"endTime":1508.108,"body":"Again, you can see that these, these spacings, uh, have this property that you have,"},{"startTime":1508.108,"endTime":1511.561,"body":"I've done this, it's maybe not super visible, but here's a red line, here's a"},{"startTime":1511.561,"endTime":1515.014,"body":"blue line that's sort of saying this is, this is what the real data is"},{"startTime":1515.014,"endTime":1515.245,"body":"doing."},{"startTime":1516.025,"endTime":1519.613,"body":"And what I said is that, well, this is the level we got to before,"},{"startTime":1519.613,"endTime":1522.005,"body":"the level we got to before was, was about 3000."},{"startTime":1522.705,"endTime":1525.725,"body":"And so what I said is that we're doubling every 12 days."},{"startTime":1526.515,"endTime":1530.165,"body":"This second graph shows that we hit the previous peak at Halloween on this rate."},{"startTime":1531.825,"endTime":1533.845,"body":"And so that was a kind of quantifiable prediction."},{"startTime":1533.845,"endTime":1537.445,"body":"That was a, you know, if we keep going at this rate, this is what"},{"startTime":1537.445,"endTime":1537.925,"body":"we'll do."},{"startTime":1538.015,"endTime":1540.725,"body":"We'll hit the level at Halloween."},{"startTime":1541.905,"endTime":1547.519,"body":"Now, I don't know who remembers this, but Halloween 2020 coincidentally was the date on"},{"startTime":1547.519,"endTime":1549.765,"body":"which the second lockdown was announced."},{"startTime":1549.885,"endTime":1555.096,"body":"I mean, I'm not claiming that I got it to within a day, but you"},{"startTime":1555.096,"endTime":1560.307,"body":"know, roughly speaking, these kind of, you know, qualitative predictions came true, were much more"},{"startTime":1560.307,"endTime":1562.045,"body":"accurate than you would've liked."},{"startTime":1562.465,"endTime":1566.02,"body":"And so this kind of picture off the dashboard where, okay, it'd gone down, it's"},{"startTime":1566.02,"endTime":1567.205,"body":"going up a little bit."},{"startTime":1567.475,"endTime":1570.965,"body":"What we actually saw was precisely this kind of exponential growth."},{"startTime":1570.995,"endTime":1572.885,"body":"It's, it's, it's slow and then it's fast."},{"startTime":1573.185,"endTime":1576.569,"body":"If you plot it on a linear scale, it kind of hides the fact that"},{"startTime":1576.569,"endTime":1579.953,"body":"this thing is gonna take off, it's going to go, going to get bad in"},{"startTime":1579.953,"endTime":1580.405,"body":"the future."},{"startTime":1580.955,"endTime":1586.485,"body":"Whereas on a log scale that it's always the same degree of bad."},{"startTime":1586.515,"endTime":1587.645,"body":"It's, it's the same slope."},{"startTime":1587.645,"endTime":1589.525,"body":"You're just going up, you're just following a straight line."},{"startTime":1589.905,"endTime":1594.356,"body":"So my contention is, is is that if more people had plotted things on log"},{"startTime":1594.356,"endTime":1598.807,"body":"scales in early stages of the wave, then we would've had a better appreciation of"},{"startTime":1598.807,"endTime":1600.885,"body":"of, of where this stuff was going."},{"startTime":1604.555,"endTime":1604.845,"body":"Okay?"},{"startTime":1604.845,"endTime":1606.125,"body":"That's not very cheerful, is it?"},{"startTime":1606.655,"endTime":1609.005,"body":"Let's talk about something slightly more cheerful."},{"startTime":1609.785,"endTime":1616.67,"body":"Um, what I do as my day job, what I research on is an area"},{"startTime":1616.67,"endTime":1618.965,"body":"of math called Information Theory."},{"startTime":1618.985,"endTime":1625.721,"body":"So a few years ago I was very lucky to be invited to a meeting"},{"startTime":1625.721,"endTime":1632.457,"body":"in San Jose and the institute that hosts this is run by somebody who collects"},{"startTime":1632.457,"endTime":1633.805,"body":"old maths things."},{"startTime":1633.905,"endTime":1640.488,"body":"He has a copy of Newton's print Kier signed by Stephen Hawking, that's quite called"},{"startTime":1640.488,"endTime":1641.805,"body":"Two Lucasian Professors."},{"startTime":1642.025,"endTime":1646.245,"body":"He also has this, this is the Bell System technical journal of 1948."},{"startTime":1647.065,"endTime":1652.881,"body":"Now, if you are me, this is almost as exciting and possibly more exciting than"},{"startTime":1652.881,"endTime":1654.045,"body":"Newton's pre kpia."},{"startTime":1654.125,"endTime":1655.285,"body":"I mean, what am I looking at?"},{"startTime":1655.285,"endTime":1659.085,"body":"What I'm looking at is I'm looking at Claude Shannon's paper from 1948."},{"startTime":1659.435,"endTime":1664.58,"body":"This is a mathematical theory of communication from 1948 that was published in here that"},{"startTime":1664.58,"endTime":1667.325,"body":"created the whole of information theory almost overnight."},{"startTime":1667.515,"endTime":1668.485,"body":"This is what I do."},{"startTime":1668.865,"endTime":1673.112,"body":"It was in there, those two, those two articles in, in in that journal created"},{"startTime":1673.112,"endTime":1674.245,"body":"the whole field overnight."},{"startTime":1675.065,"endTime":1675.885,"body":"So what did he do?"},{"startTime":1676.075,"endTime":1683.312,"body":"What he showed was that information we can understand as a physical resource, that information"},{"startTime":1683.312,"endTime":1690.55,"body":"is something that we can measure, we can quantify it, we can compress it, we"},{"startTime":1690.55,"endTime":1693.445,"body":"can send it over noisy channels."},{"startTime":1693.705,"endTime":1695.285,"body":"All this stuff came back to Shannon."},{"startTime":1695.825,"endTime":1700.256,"body":"So what he did was he gave the fundamental limits for lots of things to"},{"startTime":1700.256,"endTime":1702.325,"body":"do with how our life is now."},{"startTime":1702.625,"endTime":1708.012,"body":"So when you buy a phone, when you buy a memory stick, you are talking"},{"startTime":1708.012,"endTime":1711.245,"body":"Shannon's language, things to do with bits and bites."},{"startTime":1711.905,"endTime":1713.405,"body":"That's to do with Shannon."},{"startTime":1713.475,"endTime":1718.05,"body":"That goes back to Shannon's contribution was to, to kind of develop this language, to"},{"startTime":1718.05,"endTime":1718.965,"body":"quantify these things."},{"startTime":1720.065,"endTime":1725.302,"body":"And the interesting thing is, this is related to what I've just been talking about,"},{"startTime":1725.302,"endTime":1730.539,"body":"and the way to think about it is to think about the key idea is"},{"startTime":1730.539,"endTime":1732.285,"body":"a, a source of randomness."},{"startTime":1732.905,"endTime":1737.507,"body":"So for example, if you think about an enigma machine, if you think about somebody"},{"startTime":1737.507,"endTime":1742.109,"body":"in the desert in North Africa sending messages on an enig, on an enigma machine,"},{"startTime":1742.109,"endTime":1746.405,"body":"they are typing things that we can kind of think of as being random."},{"startTime":1746.675,"endTime":1750.905,"body":"That the person typing things into the enigma machine is a kind of source of"},{"startTime":1750.905,"endTime":1753.725,"body":"randomness that they're generating something that we'd like to understand."},{"startTime":1755.885,"endTime":1757.965,"body":"I mean, a more natural one is a source of randomness."},{"startTime":1757.965,"endTime":1759.005,"body":"Think about flipping a coin."},{"startTime":1759.425,"endTime":1760.605,"body":"So think about tossing a coin."},{"startTime":1760.635,"endTime":1763.125,"body":"Each time you toss a coin, you get head or a tail."},{"startTime":1763.595,"endTime":1765.045,"body":"That is a source of randomness."},{"startTime":1765.385,"endTime":1768.085,"body":"And of course we know it's a source of randomness because that's how you start."},{"startTime":1768.745,"endTime":1771.765,"body":"You know how, how, how you decide who bats first in the test match."},{"startTime":1771.765,"endTime":1773.805,"body":"That's how you decide who goes to lunch first."},{"startTime":1773.805,"endTime":1776.982,"body":"You know, if you, if you want to decide something, we know that coins are"},{"startTime":1776.982,"endTime":1779.525,"body":"random and that flipping coins is a good way to do it."},{"startTime":1780.385,"endTime":1786.365,"body":"Now implicitly, we, we are building into that the idea that the coin is fair."},{"startTime":1788.625,"endTime":1792.843,"body":"Now, by fair what I mean is that it's equally likely to be heads and"},{"startTime":1792.843,"endTime":1793.125,"body":"tails."},{"startTime":1793.505,"endTime":1795.925,"body":"So what I'd like you to imagine is that that may not be true."},{"startTime":1796.625,"endTime":1799.725,"body":"So we might imagine that there might be some coins that are very biased."},{"startTime":1799.825,"endTime":1804.845,"body":"We might imagine coins that are, um, very, very likely to come up."},{"startTime":1804.845,"endTime":1807.245,"body":"Heads not at all likely to come up tails."},{"startTime":1807.985,"endTime":1811.565,"body":"Now my claim is that the fair coins are unpredictable."},{"startTime":1811.565,"endTime":1812.565,"body":"That's why we use them."},{"startTime":1812.565,"endTime":1814.765,"body":"We toss a coin, you can't tell which way it's gonna come up."},{"startTime":1814.765,"endTime":1816.605,"body":"You can't tell whether it's gonna be heads or tails."},{"startTime":1817.425,"endTime":1821.165,"body":"The biased coin, you can't predict perfectly."},{"startTime":1821.425,"endTime":1827.208,"body":"But if it's biased towards heads, if you, you guess heads you'll be right more"},{"startTime":1827.208,"endTime":1828.365,"body":"often than not."},{"startTime":1828.865,"endTime":1831.005,"body":"So coins that are biased are more predictable."},{"startTime":1831.025,"endTime":1833.165,"body":"And so that's a sort of way into this."},{"startTime":1833.865,"endTime":1839.285,"body":"Now what Shannon did was that he came up with a formula for this."},{"startTime":1841.065,"endTime":1844.125,"body":"So what he said was, okay, so let's, let's think about an event."},{"startTime":1844.465,"endTime":1845.765,"body":"It doesn't matter what the event is."},{"startTime":1845.765,"endTime":1850.181,"body":"Let's think about an event a and with suppose that this happens with probability P"},{"startTime":1850.181,"endTime":1853.125,"body":"of a, there is some probability that this event happens."},{"startTime":1853.145,"endTime":1857.117,"body":"So for example, tossing a coin, the chance that I get a tossing a fair"},{"startTime":1857.117,"endTime":1859.765,"body":"coin, the chance that I get ahead is a half."},{"startTime":1860.865,"endTime":1867.33,"body":"Now what Shannon asked, the question he asked is, if a happens, how much information"},{"startTime":1867.33,"endTime":1869.485,"body":"do we gain from that?"},{"startTime":1870.345,"endTime":1874.821,"body":"Now his first insight was that, well, it should be a function of the probability"},{"startTime":1874.821,"endTime":1879.297,"body":"that there should be some function f such that the amount of information that we"},{"startTime":1879.297,"endTime":1881.685,"body":"gain should be some function of the probability."},{"startTime":1881.685,"endTime":1885.805,"body":"There should be some function f such that we gain some function of the probability."},{"startTime":1887.225,"endTime":1892.893,"body":"Now he thought about this some more and what he realized is that unlikely events"},{"startTime":1892.893,"endTime":1894.405,"body":"bring us more information."},{"startTime":1894.905,"endTime":1898.781,"body":"So I mean, if you're a football fan, man city won the league again, you"},{"startTime":1898.781,"endTime":1901.365,"body":"know, yawn, we, we all knew that was gonna happen."},{"startTime":1901.865,"endTime":1905.947,"body":"Our view of the world, nothing has changed by learning that man city won the"},{"startTime":1905.947,"endTime":1908.125,"body":"league again, the year Leicester won the league."},{"startTime":1908.125,"endTime":1911.245,"body":"That was really interesting, you know, that that was all of a sudden, you know,"},{"startTime":1911.245,"endTime":1914.365,"body":"our whole view of football and who was good and who wasn't to change massively."},{"startTime":1914.785,"endTime":1917.565,"body":"So it's the rare events that are the, are the interesting ones."},{"startTime":1917.565,"endTime":1920.245,"body":"So Shannon said, okay, so we wanna function."},{"startTime":1921.105,"endTime":1925.125,"body":"So it should be the case that the unlikely events have got more information."},{"startTime":1925.745,"endTime":1930.882,"body":"So that suggests that f should be a decreasing function, that the smaller p is"},{"startTime":1930.882,"endTime":1933.965,"body":"the bigger F, FP should be, but which one?"},{"startTime":1935.795,"endTime":1939.688,"body":"Okay, if you weren't in a talk about logarithms, then it maybe be hard to"},{"startTime":1939.688,"endTime":1941.765,"body":"guess you are in a talk about logarithms."},{"startTime":1942.225,"endTime":1944.605,"body":"So the answer is gonna be the logarithm, but why?"},{"startTime":1945.195,"endTime":1949.485,"body":"Well, this is Shannon's other insight is to think about independent events."},{"startTime":1950.185,"endTime":1955.092,"body":"So to think about, for example, if somebody over that side of the room tosses"},{"startTime":1955.092,"endTime":1959.999,"body":"a coin, somebody over that side of the room throws a dice, there is no"},{"startTime":1959.999,"endTime":1963.925,"body":"relationship between what, what we get out of either of them, right?"},{"startTime":1964.865,"endTime":1970.31,"body":"So the amount that we, that well, the probability of seeing a head and a"},{"startTime":1970.31,"endTime":1975.755,"body":"five should be the product of the probability of seeing ahead and the probability of"},{"startTime":1975.755,"endTime":1976.845,"body":"seeing a five."},{"startTime":1977.665,"endTime":1983.565,"body":"So what we know is that independent events, the, um, the probabilities should multiply."},{"startTime":1983.565,"endTime":1987.231,"body":"So the probability of A and B should be the probability of A times the"},{"startTime":1987.231,"endTime":1987.965,"body":"probability of B."},{"startTime":1989.305,"endTime":1992.645,"body":"Now, what Shannon realized is that the information from those things should add up."},{"startTime":1992.645,"endTime":1997.078,"body":"If, if, if I learn the head and I learn the , it's like I've"},{"startTime":1997.078,"endTime":1997.965,"body":"learned two things."},{"startTime":1998.465,"endTime":2002.022,"body":"So it should be the case that the amount of information that I've learned is"},{"startTime":2002.022,"endTime":2003.445,"body":"the sum of the two things."},{"startTime":2003.785,"endTime":2007.61,"body":"So it should be that the information in A and B should be the sum"},{"startTime":2007.61,"endTime":2010.925,"body":"of the information in a and the sum of the information in B."},{"startTime":2011.425,"endTime":2016.21,"body":"And if you put those two things together, we want a decreasing function, we want"},{"startTime":2016.21,"endTime":2018.125,"body":"a function that satisfies this property."},{"startTime":2018.355,"endTime":2020.605,"body":"Essentially the logarithm is the only thing that works."},{"startTime":2020.605,"endTime":2024.355,"body":"There's a sort of paper that studied this that says that essentially the logarithm is"},{"startTime":2024.355,"endTime":2025.605,"body":"the only thing that works."},{"startTime":2026.265,"endTime":2029.285,"body":"Um, technically it's minus the log."},{"startTime":2029.285,"endTime":2031.125,"body":"We have to take the function as minus the log."},{"startTime":2032.065,"endTime":2036.823,"body":"Um, and if we do that, if we take it to base two, so if"},{"startTime":2036.823,"endTime":2041.581,"body":"our logarithms are based on powers of two, then if you do that, then the"},{"startTime":2041.581,"endTime":2043.485,"body":"unit of information is the bit."},{"startTime":2043.865,"endTime":2049.052,"body":"So the bit turns out to come from using the logarithm, using the, the logarithm"},{"startTime":2049.052,"endTime":2052.165,"body":"to base two is what gives us the bit."},{"startTime":2052.905,"endTime":2054.085,"body":"So it was pretty impressive."},{"startTime":2054.085,"endTime":2058.173,"body":"So Shannon, you know, came up with all of this just by thinking about sort"},{"startTime":2058.173,"endTime":2061.445,"body":"of simple things, thinking about coin tosses, thinking about combining things together."},{"startTime":2062.985,"endTime":2064.685,"body":"And he went on from there, he did a lot more."},{"startTime":2064.685,"endTime":2066.804,"body":"So for example, he defined something called entropy."},{"startTime":2067.065,"endTime":2074.706,"body":"So entropy is the, it is the amount of information that we, that we expect"},{"startTime":2074.706,"endTime":2075.725,"body":"to gain."},{"startTime":2076.105,"endTime":2080.704,"body":"Now this is a sort of slightly more complicated thing, but um, we can think"},{"startTime":2080.704,"endTime":2085.304,"body":"about if I got all of the air in this room and I kind of"},{"startTime":2085.304,"endTime":2089.904,"body":"squashed it down and I squashed it down, there would be a point beyond which"},{"startTime":2089.904,"endTime":2090.824,"body":"I can't go."},{"startTime":2090.824,"endTime":2094.036,"body":"There's a point beyond which, you know, the kind of molecules would be bashing into"},{"startTime":2094.036,"endTime":2094.465,"body":"each other."},{"startTime":2094.465,"endTime":2096.344,"body":"There's a sort of minimum volume."},{"startTime":2096.895,"endTime":2100.025,"body":"What Shannon showed is that there is a similar thing for information."},{"startTime":2100.405,"endTime":2103.731,"body":"He showed that, that if you take a file, you compress it, you know, you"},{"startTime":2103.731,"endTime":2107.057,"body":"take a song, you turn it into an MP three, there is a point beyond"},{"startTime":2107.057,"endTime":2107.945,"body":"which you can't go."},{"startTime":2107.995,"endTime":2111.025,"body":"There is a limit to how far you can, you can compress this stuff."},{"startTime":2111.605,"endTime":2115.947,"body":"And he showed that not only is there a limit, but there's a limit that's"},{"startTime":2115.947,"endTime":2117.105,"body":"given by his entropy."},{"startTime":2117.105,"endTime":2121.455,"body":"And his entropy is defined to be the amount of information that you expect to"},{"startTime":2121.455,"endTime":2121.745,"body":"learn."},{"startTime":2122.255,"endTime":2123.985,"body":"It's again, it comes from the log."},{"startTime":2124.405,"endTime":2128.235,"body":"So it's kind of remarkable really that, that through these experiments, through this kind of"},{"startTime":2128.235,"endTime":2132.065,"body":"thinking about this thing, he was able to to, to come up with this quantity."},{"startTime":2132.685,"endTime":2137.157,"body":"And the kind of key slogan is here that the more unpredictable the source, the"},{"startTime":2137.157,"endTime":2139.245,"body":"more space you need to summarize it."},{"startTime":2139.705,"endTime":2144.94,"body":"So fair coins, you can't compress the outcome of a sequence of fair coins, heads,"},{"startTime":2144.94,"endTime":2146.685,"body":"tails, heads, tails, heads, tail."},{"startTime":2146.745,"endTime":2147.925,"body":"You have to give the whole sequence."},{"startTime":2147.925,"endTime":2149.565,"body":"There isn't a better way to represent it."},{"startTime":2149.985,"endTime":2153.365,"body":"The sequence that comes up heads all the time."},{"startTime":2153.915,"endTime":2158.276,"body":"What we can do is we can do things like, we could say, well, okay,"},{"startTime":2158.276,"endTime":2161.765,"body":"the first seven tosses, you go seven tosses to the first tail."},{"startTime":2162.305,"endTime":2164.045,"body":"You go 11 tosses to the next tail."},{"startTime":2164.045,"endTime":2166.085,"body":"After that you go 13, you know, whatever it is."},{"startTime":2166.085,"endTime":2170.885,"body":"So, so you can count on the head, there's ways of representing these predictable sequences"},{"startTime":2170.885,"endTime":2174.405,"body":"more efficiently than there are of representing heads and tails sequences."},{"startTime":2174.595,"endTime":2174.885,"body":"Okay?"},{"startTime":2175.265,"endTime":2182.738,"body":"So that's the log again, the logarithm here is underlying this entropy, which is underlying"},{"startTime":2182.738,"endTime":2186.725,"body":"this sort of, um, idea of compression, right?"},{"startTime":2186.985,"endTime":2190.296,"body":"One more thing I want to talk about, and for some people in this room,"},{"startTime":2190.296,"endTime":2192.725,"body":"this is gonna be very important, I wanna talk about Instagram."},{"startTime":2194.545,"endTime":2201.15,"body":"Now, when you talk about log scales, one reason that people like log scales is"},{"startTime":2201.15,"endTime":2205.995,"body":"people like log scales when the numbers vary by huge amounts."},{"startTime":2206.495,"endTime":2211.199,"body":"So numbers that vary by massive factors taking the logs somehow tames them, it brings"},{"startTime":2211.199,"endTime":2213.395,"body":"them sort of back down to earth."},{"startTime":2214.295,"endTime":2218.168,"body":"Now, you may not even realize that some of these things are log scales, but"},{"startTime":2218.168,"endTime":2220.235,"body":"the, the Richter scale is a log scale."},{"startTime":2220.255,"endTime":2224.698,"body":"The Richter scale for earthquakes, pH scale is a log scale, decibels are a log"},{"startTime":2224.698,"endTime":2224.995,"body":"scale."},{"startTime":2225.015,"endTime":2229.082,"body":"All of these things are log scales because taking these things and and taking the"},{"startTime":2229.082,"endTime":2233.15,"body":"log turns them into numbers that we can handle into numbers that we can, we"},{"startTime":2233.15,"endTime":2234.235,"body":"can, we can understand."},{"startTime":2235.615,"endTime":2239.755,"body":"So for example, the Richter scale is set up with logs to base 10."},{"startTime":2239.825,"endTime":2244.406,"body":"Each time you go up on the Richter scale, each step up on the Richter"},{"startTime":2244.406,"endTime":2247.155,"body":"scale represents 10 times more energy in the earthquake."},{"startTime":2248.695,"endTime":2252.16,"body":"So, I mean, you know, Rick to nine doesn't sound, I mean it sounds bigger"},{"startTime":2252.16,"endTime":2255.395,"body":"than Rick to two, but it doesn't sound so much bigger than Rickter two."},{"startTime":2255.855,"endTime":2259.976,"body":"But the point is that going from Rick to two to Rick to nine, you're"},{"startTime":2259.976,"endTime":2261.075,"body":"going up seven steps."},{"startTime":2261.255,"endTime":2264.195,"body":"You're going up seven steps, each of which are a factor of 10."},{"startTime":2264.495,"endTime":2266.595,"body":"So you are going by 10 to the power of seven."},{"startTime":2266.935,"endTime":2270.291,"body":"So the point is that a Richter nine earthquake is 10 to the power of"},{"startTime":2270.291,"endTime":2270.515,"body":"seven."},{"startTime":2270.865,"endTime":2273.475,"body":"It's 10 million times more powerful than a Richter two."},{"startTime":2274.985,"endTime":2275.275,"body":"Okay?"},{"startTime":2275.275,"endTime":2279.955,"body":"So that's why people like the Richter scale turns numbers into things that are understandable."},{"startTime":2280.535,"endTime":2283.755,"body":"Now, the obvious question is social media followers."},{"startTime":2285.045,"endTime":2289.195,"body":"Christiana Ronaldo has 600 million people follow him."},{"startTime":2289.275,"endTime":2292.308,"body":"I mean, I, I wrote this talk a little while ago, but he, he's probably"},{"startTime":2292.308,"endTime":2292.915,"body":"got more now."},{"startTime":2293.135,"endTime":2295.955,"body":"He, he's got 600 million people follow him on Instagram."},{"startTime":2297.355,"endTime":2300.793,"body":"A lot of us, you know, a lots of people don't have 60 Instagram followers,"},{"startTime":2300.793,"endTime":2303.315,"body":"you know, so again, there's a factor of 10 million there."},{"startTime":2303.855,"endTime":2307.835,"body":"So why do these numbers vary on such a wide scale?"},{"startTime":2307.855,"endTime":2312.538,"body":"You know, why is Christiana Ronaldo 10 million times more popular than a lot of"},{"startTime":2312.538,"endTime":2313.475,"body":"us on Instagram?"},{"startTime":2313.535,"endTime":2319.6,"body":"You know, can we understand the way that social networks behave to understand how this"},{"startTime":2319.6,"endTime":2320.005,"body":"happens?"},{"startTime":2321.785,"endTime":2324.805,"body":"And if you are mathematicians, then you can try."},{"startTime":2324.905,"endTime":2327.81,"body":"So if you're a mathematician, the first thing that you might do is you might"},{"startTime":2327.81,"endTime":2330.716,"body":"draw a picture like this that you might have in your mind a picture that"},{"startTime":2330.716,"endTime":2331.685,"body":"sort of something like this."},{"startTime":2331.685,"endTime":2335.685,"body":"This is a sort of toy social network with six people in the network."},{"startTime":2336.245,"endTime":2339.599,"body":"I mean, maybe there's, you can think about this as a sort of class at"},{"startTime":2339.599,"endTime":2341.165,"body":"school that this represents people being friends."},{"startTime":2341.385,"endTime":2344.525,"body":"So there's a line between these dots."},{"startTime":2344.525,"endTime":2348.085,"body":"The dots represent, the circles represent people, the lines represent the idea."},{"startTime":2348.095,"endTime":2349.365,"body":"These people are friends with each other."},{"startTime":2349.425,"endTime":2353.606,"body":"So with six people, you could map out the sort of friendship relations within the"},{"startTime":2353.606,"endTime":2353.885,"body":"group."},{"startTime":2353.905,"endTime":2355.085,"body":"It might look something like this."},{"startTime":2356.305,"endTime":2359.285,"body":"If it's a social network, maybe you have arrows, not lines."},{"startTime":2359.565,"endTime":2361.685,"body":"I mean, I'm gonna kind of gloss over that really."},{"startTime":2361.685,"endTime":2364.947,"body":"Maybe you have arrows, not lines, because some people follow somebody else, it doesn't mean"},{"startTime":2364.947,"endTime":2368.21,"body":"they follow back, but you know, you could think about a graph that looks something"},{"startTime":2368.21,"endTime":2368.645,"body":"like this."},{"startTime":2369.825,"endTime":2374.893,"body":"Now the simplest model of this goes back to Erish and Renny who were Hungarian"},{"startTime":2374.893,"endTime":2376.245,"body":"mathematicians in the 1950s."},{"startTime":2376.745,"endTime":2380.374,"body":"So what they defined was they defined a very simple model that they defined a"},{"startTime":2380.374,"endTime":2384.003,"body":"kind of symmetric model that their model, what they do is you, you have n"},{"startTime":2384.003,"endTime":2387.633,"body":"people who could be in the network, some number of people who could be in"},{"startTime":2387.633,"endTime":2389.085,"body":"the network, you fix a probability."},{"startTime":2389.185,"endTime":2392.8,"body":"So there zen people in the network, there's a number P, which is between Norton"},{"startTime":2392.8,"endTime":2394.005,"body":"one, that that's a probability."},{"startTime":2394.105,"endTime":2397.839,"body":"And all you do is for each pair of people, you connect them with a"},{"startTime":2397.839,"endTime":2401.325,"body":"line or you don't, and you connect them with a line with probability P."},{"startTime":2401.325,"endTime":2406.085,"body":"So there's a similar chance of every potential pair of having the line between them,"},{"startTime":2406.085,"endTime":2410.845,"body":"each potential pair of people you have, um, a line between them with probability p."},{"startTime":2412.505,"endTime":2415.125,"body":"Now from a mathematical point of view, this is quite fun."},{"startTime":2415.705,"endTime":2418.565,"body":"Um, it turns out there's some, some quite interesting behavior with this."},{"startTime":2418.565,"endTime":2422.511,"body":"So for example, if n times P is less than one, it turns out that"},{"startTime":2422.511,"endTime":2426.458,"body":"the network, it isn't really a very good network that what you have is you"},{"startTime":2426.458,"endTime":2430.405,"body":"have very small little isolated islands that don't really kind of communicate with one another."},{"startTime":2430.825,"endTime":2434.528,"body":"If NP is bigger than one, then a fixed fraction of the network is joined"},{"startTime":2434.528,"endTime":2437.245,"body":"together, all of a sudden this whole thing starts to work."},{"startTime":2437.745,"endTime":2441.402,"body":"So if you are Elon Musk and you own the network, you want to get"},{"startTime":2441.402,"endTime":2445.059,"body":"over this threshold, you want to get over this level of, of connectivity so that"},{"startTime":2445.059,"endTime":2448.717,"body":"the whole network is, is, is joined together and so that things can, can pass"},{"startTime":2448.717,"endTime":2449.205,"body":"around it."},{"startTime":2451.025,"endTime":2453.965,"body":"But the problem is this doesn't capture what we see."},{"startTime":2454.245,"endTime":2458.49,"body":"I mean, what we can do is, this is a sort of simulation here that"},{"startTime":2458.49,"endTime":2462.735,"body":"if I took a model that we had say a hundred thousand people on the"},{"startTime":2462.735,"endTime":2466.981,"body":"network, we had, the probability is 0.0, 0 3, 0 and five, then this is"},{"startTime":2466.981,"endTime":2471.226,"body":"the kind of histogram that we see that what it turns out is that the,"},{"startTime":2471.226,"endTime":2472.925,"body":"the everybody has the same distribution."},{"startTime":2472.925,"endTime":2474.245,"body":"There's a lot of sort of symmetry here."},{"startTime":2474.545,"endTime":2478.802,"body":"And so what we see is that the, the distribution of, of friends that people"},{"startTime":2478.802,"endTime":2481.925,"body":"have, it's centered somewhere around 50, but it's, it's tightly constrained."},{"startTime":2482.795,"endTime":2485.245,"body":"Basically nobody has fewer than 20 friends."},{"startTime":2485.245,"endTime":2487.725,"body":"Almost nobody has bigger than more than 80 friends."},{"startTime":2487.955,"endTime":2490.165,"body":"Certainly nobody's got 600 million friends."},{"startTime":2490.945,"endTime":2496.428,"body":"So it, it, it's fine as a model, but it's not a model that is"},{"startTime":2496.428,"endTime":2497.525,"body":"anything like Instagram."},{"startTime":2497.625,"endTime":2501.565,"body":"So if we're trying to understand Instagram like dynamics, this is, this is too simple."},{"startTime":2503.545,"endTime":2507.463,"body":"So what we can think about is we can think about how social networks actually"},{"startTime":2507.463,"endTime":2507.725,"body":"evolve."},{"startTime":2508.505,"endTime":2511.885,"body":"And so what we can think about is the sort of dynamics of friendship."},{"startTime":2511.985,"endTime":2516.947,"body":"So for example, if Anna is friends with Becky and Charlotte, it's quite likely that"},{"startTime":2516.947,"endTime":2519.925,"body":"Anna will invite Becky and Charlotte to her party."},{"startTime":2520.195,"endTime":2524.085,"body":"It's more likely than than average that Becky and Charlotte will meet at the party."},{"startTime":2524.085,"endTime":2525.845,"body":"They will talk to each other, they will become friends."},{"startTime":2526.115,"endTime":2527.845,"body":"They're not a random pair of people."},{"startTime":2528.705,"endTime":2532.62,"body":"So what you see is that within the network there is the sort of information"},{"startTime":2532.62,"endTime":2533.925,"body":"that builds up like this."},{"startTime":2534.745,"endTime":2538.685,"body":"And in particular real social networks, they have something that's called a Matthew effect."},{"startTime":2538.705,"endTime":2542.405,"body":"And this is named after Matthew 25, 29 in the Bible."},{"startTime":2542.545,"endTime":2546.165,"body":"So the verse is this, for whoever has will be given more."},{"startTime":2546.745,"endTime":2551.925,"body":"So people with lots of followers on social media, get more followers on social media."},{"startTime":2552.395,"endTime":2557.321,"body":"Once you are popular, then more people follow you, more people retweet your stuff, you"},{"startTime":2557.321,"endTime":2560.605,"body":"are more visible, more people then decide to follow you."},{"startTime":2560.605,"endTime":2564.805,"body":"So there is a sort of virtuous or vicious, depending on how you think about"},{"startTime":2564.805,"endTime":2567.045,"body":"it, circle as to how your followers grow."},{"startTime":2567.185,"endTime":2571.365,"body":"If you have an existing audience, then it helps build a further audience."},{"startTime":2572.265,"endTime":2576.725,"body":"Now, the thing that's interesting is that there is a mathematical model that captures this."},{"startTime":2576.725,"endTime":2580.575,"body":"There's something called the Barabasi Albert model, which was developed in, in, in the nineties"},{"startTime":2580.575,"endTime":2584.425,"body":"when people were starting to think about kind of the internet and things like this"},{"startTime":2584.425,"endTime":2585.965,"body":"that represents this kind of dynamics."},{"startTime":2585.965,"endTime":2591.405,"body":"It represents how these things grow and there's not enough time to get into it."},{"startTime":2591.665,"endTime":2595.477,"body":"But the thing that's interesting is that you can come up with these sort of"},{"startTime":2595.477,"endTime":2599.289,"body":"models where, for example, if I join the network, the chance that I follow somebody"},{"startTime":2599.289,"endTime":2602.085,"body":"is proportional to the number of followers that they have already."},{"startTime":2602.265,"endTime":2605.781,"body":"I'm more likely to follow the big hitters that I'm to follow the people with,"},{"startTime":2605.781,"endTime":2606.485,"body":"not many followers."},{"startTime":2606.865,"endTime":2610.548,"body":"If you do that, then it turns out that what you end up with is"},{"startTime":2610.548,"endTime":2611.285,"body":"something like this."},{"startTime":2611.285,"endTime":2615.294,"body":"You end up with something like the number of people with K followers behaves like"},{"startTime":2615.294,"endTime":2617.165,"body":"a times K to the minus gamma."},{"startTime":2617.785,"endTime":2620.045,"body":"So this isn't like the behavior that we saw before."},{"startTime":2620.065,"endTime":2625.365,"body":"The behavior that we saw before was this very fast drop off that this behavior"},{"startTime":2625.365,"endTime":2627.485,"body":"here is is slower drop off."},{"startTime":2627.485,"endTime":2632.426,"body":"It's, it's what's called a power law behavior and it it is dropping off, it"},{"startTime":2632.426,"endTime":2633.085,"body":"drops off."},{"startTime":2633.465,"endTime":2634.845,"body":"Um, much, much slower."},{"startTime":2635.105,"endTime":2638.565,"body":"Now this is something people claim to see."},{"startTime":2639.265,"endTime":2642.581,"body":"People claim that, well there is a mathematical model that says that we should see"},{"startTime":2642.581,"endTime":2643.245,"body":"a power law."},{"startTime":2643.385,"endTime":2646.205,"body":"People say there is a power law in real data."},{"startTime":2647.625,"endTime":2649.885,"body":"So the question is how would we tell?"},{"startTime":2651.385,"endTime":2655.041,"body":"And the answer is because this is to talk about logarithms, we will take the"},{"startTime":2655.041,"endTime":2655.285,"body":"log."},{"startTime":2655.905,"endTime":2660.373,"body":"So were it to be the case that PK was behaving like a times K"},{"startTime":2660.373,"endTime":2661.565,"body":"to the minus gamma."},{"startTime":2661.985,"endTime":2665.573,"body":"If we took the log of everything inside, it would be the case that log"},{"startTime":2665.573,"endTime":2668.445,"body":"of PK was log of a minus gamma times log of K."},{"startTime":2668.785,"endTime":2671.791,"body":"So in other words, what we'd have is we'd have a sort of straight line"},{"startTime":2671.791,"endTime":2674.798,"body":"relationship that what we could do is we could plot the log of the probability"},{"startTime":2674.798,"endTime":2677.804,"body":"on the y axis, we could plot the log of the K on the X"},{"startTime":2677.804,"endTime":2678.005,"body":"axis."},{"startTime":2678.235,"endTime":2683.525,"body":"This is a kind of y equals C plus MX type linear relationship."},{"startTime":2683.985,"endTime":2686.925,"body":"So if it's a power law, we should see a straight line."},{"startTime":2688.075,"endTime":2689.965,"body":"Like we saw a straight line with the Covid data."},{"startTime":2690.345,"endTime":2692.005,"body":"We should see a straight line when we plot this."},{"startTime":2692.995,"endTime":2693.285,"body":"Okay?"},{"startTime":2694.565,"endTime":2697.365,"body":"I have faith who thinks this is a straight line?"},{"startTime":2700.155,"endTime":2704.085,"body":"Okay, some people towards the back, this is Paul Krugman."},{"startTime":2704.085,"endTime":2705.445,"body":"Paul Krugman has a Nobel Prize."},{"startTime":2705.785,"endTime":2707.445,"body":"So this may, you know, this may alter your view."},{"startTime":2707.765,"endTime":2712.631,"body":"This is a blog that Paul Krugman wrote with Twitter followers and he's saying that's"},{"startTime":2712.631,"endTime":2713.605,"body":"a straight line."},{"startTime":2715.225,"endTime":2716.925,"body":"It sort of is, but it's not quite."},{"startTime":2717.865,"endTime":2722.085,"body":"But the point is that we at least know how to have this argument."},{"startTime":2722.105,"endTime":2723.605,"body":"You know, he's saying it's a power law."},{"startTime":2723.785,"endTime":2727.109,"body":"The way that we have this argument is that we, we plot a straight line,"},{"startTime":2727.109,"endTime":2729.325,"body":"uh, we plot it, it should be a straight line."},{"startTime":2729.325,"endTime":2731.365,"body":"If it's a power law, it will be a straight line."},{"startTime":2732.145,"endTime":2738.621,"body":"And so at least knowing about the logarithm gives us the way of, um, deciding"},{"startTime":2738.621,"endTime":2739.485,"body":"this question."},{"startTime":2741.785,"endTime":2743.405,"body":"It turns out this is pretty controversial."},{"startTime":2743.705,"endTime":2746.828,"body":"It this is the kind of thing that people who do networks yell at each"},{"startTime":2746.828,"endTime":2747.245,"body":"other about."},{"startTime":2747.865,"endTime":2751.725,"body":"Um, so health warning, sometimes it's not a power law."},{"startTime":2751.725,"endTime":2754.591,"body":"People will tell you it's a power law a lot of the time it's not"},{"startTime":2754.591,"endTime":2755.165,"body":"a power law."},{"startTime":2755.545,"endTime":2760.117,"body":"Um, Cosmo Shazi and his coauthors, for example, are very keen that it's not a"},{"startTime":2760.117,"endTime":2764.69,"body":"power law and they have statistical tests that come up with other ways of, of,"},{"startTime":2764.69,"endTime":2765.605,"body":"of showing things."},{"startTime":2765.865,"endTime":2770.393,"body":"So for example, here is a graph from Cosmo's talk where you can see, well,"},{"startTime":2770.393,"endTime":2774.921,"body":"no, this really isn't a straight line and this has one of my favorite talk"},{"startTime":2774.921,"endTime":2775.525,"body":"titles ever."},{"startTime":2775.545,"endTime":2779.128,"body":"His talk is called, so you think you have a power law, do you Well,"},{"startTime":2779.128,"endTime":2779.845,"body":"isn't that special?"},{"startTime":2780.505,"endTime":2784.25,"body":"So you know, you can, you, you can see that that you know, at least"},{"startTime":2784.25,"endTime":2787.995,"body":"within the levels of mathematical discourse, this is, this is kind of gloves off type"},{"startTime":2787.995,"endTime":2788.245,"body":"stuff."},{"startTime":2788.245,"endTime":2791.285,"body":"People are fighting about is this a straight line?"},{"startTime":2792.025,"endTime":2793.085,"body":"Is this a straight line?"},{"startTime":2793.425,"endTime":2794.685,"body":"Should it be a straight line?"},{"startTime":2794.685,"endTime":2798.073,"body":"What's the model by which it's a straight line, but the point is it almost"},{"startTime":2798.073,"endTime":2798.525,"body":"doesn't matter."},{"startTime":2798.945,"endTime":2803.156,"body":"The point is that we at least know how to have this argument that the"},{"startTime":2803.156,"endTime":2806.245,"body":"way to have this argument is to work on log scales."},{"startTime":2806.425,"endTime":2810.571,"body":"So what you can see is that across the bottom, the number of followers, uh,"},{"startTime":2810.571,"endTime":2814.165,"body":"sorry, this is the, the in degree distribution of web logs in 2003."},{"startTime":2814.165,"endTime":2819.882,"body":"This is how many, um, the, the, the, where the ranking is of the most"},{"startTime":2819.882,"endTime":2820.645,"body":"popular logs."},{"startTime":2820.875,"endTime":2823.725,"body":"This is the proportion you can see these are both log scales."},{"startTime":2824.145,"endTime":2828.351,"body":"So the theory says, if you want to decide this question, the way to do"},{"startTime":2828.351,"endTime":2832.558,"body":"it is to go back to Gresham, to go back to Briggs in 1599 and"},{"startTime":2832.558,"endTime":2836.205,"body":"say to him, I've got this data, what can I do with it?"},{"startTime":2837.705,"endTime":2839.405,"body":"And that's the kind of key message here."},{"startTime":2839.905,"endTime":2845.607,"body":"So my claim is this, that logs are wonderful things, they're powerful things and the"},{"startTime":2845.607,"endTime":2849.79,"body":"more you start looking for them, the more you'll see them."},{"startTime":2850.065,"endTime":2854.181,"body":"So, uh, the, the more that you use them and become aware of them, the"},{"startTime":2854.181,"endTime":2856.925,"body":"more that you'll see them cropping up in different places."},{"startTime":2857.665,"endTime":2861.405,"body":"Now what we'd like to do is we'd like to understand why that is."},{"startTime":2861.985,"endTime":2868.227,"body":"So for example, with the epidemic data, we have these equations that says why this"},{"startTime":2868.227,"endTime":2870.725,"body":"should be true with Shannon's information."},{"startTime":2870.745,"endTime":2874.467,"body":"We have this argument based on independence that says that the log is the right"},{"startTime":2874.467,"endTime":2876.205,"body":"thing to do with the Instagram data."},{"startTime":2876.265,"endTime":2879.94,"body":"We have this, um, Baria Albert argument that people argue about that says that this"},{"startTime":2879.94,"endTime":2883.125,"body":"is why it should be true, but this is what we're looking for."},{"startTime":2883.125,"endTime":2884.765,"body":"We don't want it to just be a coincidence."},{"startTime":2884.765,"endTime":2888.845,"body":"We want structural reasons why the log is the right thing to appear."},{"startTime":2889.385,"endTime":2892.57,"body":"But if we can do that, then my claim is that the, the logs help"},{"startTime":2892.57,"endTime":2893.845,"body":"us make sense of the world."},{"startTime":2894.385,"endTime":2898.765,"body":"And so that's where I'd, I'd like to finish except one more slide."},{"startTime":2899.025,"endTime":2901.405,"body":"So if you like this, this is my book."},{"startTime":2901.405,"endTime":2902.165,"body":"This is number crunch."},{"startTime":2902.355,"endTime":2905.485,"body":"This is, this is out wherever books are sold."},{"startTime":2905.985,"endTime":2909.085,"body":"Um, and I talk about some of these things and other things as well."},{"startTime":2909.105,"endTime":2910.525,"body":"And thank you very much for listening."},{"startTime":2914.665,"endTime":2921.22,"body":"An online question, uh, is Shannon's coining of the term bit where the term bit"},{"startTime":2921.22,"endTime":2923.405,"body":"in computer science comes from,"},{"startTime":2924.025,"endTime":2929.095,"body":"Or, um, yes, e except I, I got picked up on this that somebody told"},{"startTime":2929.095,"endTime":2934.165,"body":"me that it was actually Qy who coined it and that Shannon pinched it off"},{"startTime":2934.595,"endTime":2934.885,"body":"Offy."},{"startTime":2935.305,"endTime":2936.445,"body":"But yes, it's, it's those people."},{"startTime":2936.505,"endTime":2940.757,"body":"So it, it's exactly the bit in, in, in, in computer science is, is exactly"},{"startTime":2940.757,"endTime":2941.325,"body":"Shannon's bit."},{"startTime":2941.745,"endTime":2946.681,"body":"So Shannon, uh, when he was 21, he wrote an MSC thesis that basically invented"},{"startTime":2946.681,"endTime":2951.617,"body":"the whole of this, that what he realized is that everything could be represented as"},{"startTime":2951.617,"endTime":2952.605,"body":"zeros and ones."},{"startTime":2952.825,"endTime":2956.042,"body":"So at the age of 21, he already, you know, had changed the world, but"},{"startTime":2956.042,"endTime":2959.045,"body":"then he came along and did, did this stuff to do with entropy afterwards."},{"startTime":2959.105,"endTime":2961.325,"body":"But, but yes, it's, it's exactly the same bit."},{"startTime":2961.715,"endTime":2962.245,"body":"Okay, cool."},{"startTime":2962.245,"endTime":2966.078,"body":"And, and while we're talking about words, someone else has asked, where does the word"},{"startTime":2966.078,"endTime":2966.845,"body":"logarithm come from?"},{"startTime":2967.145,"endTime":2967.685,"body":"Do we know?"},{"startTime":2967.985,"endTime":2968.205,"body":"Oh,"},{"startTime":2970.285,"endTime":2970.845,"body":"I ought to know"},{"startTime":2970.845,"endTime":2974.877,"body":"That I, you, you've all got phones, you've got Wikipedia, you \u003claugh\u003e, you look, the"},{"startTime":2974.877,"endTime":2977.565,"body":"only thing I do know is it is an anagram."},{"startTime":2977.565,"endTime":2978.685,"body":"It's an anagram of algorithm."},{"startTime":2978.705,"endTime":2980.925,"body":"But I think that's, that, that, that's a coincidence."},{"startTime":2980.945,"endTime":2981.685,"body":"But, uh, \u003claugh\u003e,"},{"startTime":2981.875,"endTime":2982.405,"body":"Yeah, yeah."},{"startTime":2982.505,"endTime":2984.445,"body":"Oh, that's a nice coincidence."},{"startTime":2984.985,"endTime":2988.885,"body":"So we've seen things based on, um, powers of two, based on powers of 10."},{"startTime":2989.305,"endTime":2993.245,"body":"But, um, what about this natural algorithm that some of us may remember from school?"},{"startTime":2993.245,"endTime":2994.045,"body":"What does that help us?"},{"startTime":2994.425,"endTime":2994.645,"body":"Yes."},{"startTime":2994.675,"endTime":2997.925,"body":"Okay, so, so I deliberately skirted over this."},{"startTime":2998.005,"endTime":3000.528,"body":"I sort of said, oh yes, this is log to base two, and I didn't"},{"startTime":3000.528,"endTime":3003.052,"body":"really say what I meant by that, but so, so yes, it's the thing that"},{"startTime":3003.052,"endTime":3003.725,"body":"you're taking the power."},{"startTime":3004.385,"endTime":3010.485,"body":"So the reason or a reason that the natural logarithm is natural is to do"},{"startTime":3010.485,"endTime":3012.925,"body":"with the, the slope of it."},{"startTime":3012.995,"endTime":3017.309,"body":"That the, the, the, if you take E to the X, where he is this"},{"startTime":3017.309,"endTime":3021.623,"body":"sort of magic number of, you know, this 2.7 moralities number, that if you take"},{"startTime":3021.623,"endTime":3025.938,"body":"EE of the X, that's the thing whose slope is equal to itself, that you"},{"startTime":3025.938,"endTime":3029.965,"body":"take the derivative of that the slope is the same as the function itself."},{"startTime":3030.455,"endTime":3035.563,"body":"Other powers other, sorry, other, you know, A to the X for something else, there's"},{"startTime":3035.563,"endTime":3036.245,"body":"a factor."},{"startTime":3036.245,"endTime":3037.205,"body":"There'll be a multiple."},{"startTime":3037.705,"endTime":3041.976,"body":"And so that's sort of the reason that it's natural is that, that it has"},{"startTime":3041.976,"endTime":3043.685,"body":"that property That's a handy property."},{"startTime":3044.345,"endTime":3048.47,"body":"It it, it's annoying when you write papers in an information theory because in information"},{"startTime":3048.47,"endTime":3052.045,"body":"theory papers, the convention is that log always means log to base two."},{"startTime":3052.465,"endTime":3055.765,"body":"And then when you read anybody else's papers, the convention may be that log means"},{"startTime":3055.765,"endTime":3056.205,"body":"something else."},{"startTime":3056.545,"endTime":3061.045,"body":"So it, it's one of these things when we're all broadly speaking, speaking the same"},{"startTime":3061.045,"endTime":3063.445,"body":"language, but only up to a constant factor."},{"startTime":3063.945,"endTime":3067.605,"body":"So is is logarithms what you do, logarithms at university?"},{"startTime":3067.665,"endTime":3069.765,"body":"Is that much different from what you do at a level?"},{"startTime":3070.985,"endTime":3074.725,"body":"Um, I'm not sure."},{"startTime":3075.185,"endTime":3075.925,"body":"I'm not sure they are."},{"startTime":3076.045,"endTime":3077.645,"body":"I mean, you know, it's, it, it is the same function."},{"startTime":3077.685,"endTime":3081.39,"body":"I mean, it's the same, you know, this is the nice thing that you, Briggs"},{"startTime":3081.39,"endTime":3085.095,"body":"or whoever wrote down this function, it's still the same function Now that, that everybody"},{"startTime":3085.095,"endTime":3087.565,"body":"talks about the log, they're talking about the same thing."},{"startTime":3087.595,"endTime":3089.405,"body":"They're probably being used for different things."},{"startTime":3089.625,"endTime":3093.492,"body":"So, um, I mean the, the advert for doing a, a, a university math degree"},{"startTime":3093.492,"endTime":3096.845,"body":"is that what you can do is you can specialize in particular directions."},{"startTime":3096.845,"endTime":3099.405,"body":"So, for example, you can specialize in number theory."},{"startTime":3099.745,"endTime":3102.763,"body":"You can be interested in primes, you can count primes, and the log comes up"},{"startTime":3102.763,"endTime":3102.965,"body":"there."},{"startTime":3103.145,"endTime":3107.485,"body":"You're interested in statistics, you know, you do you see things to do with data."},{"startTime":3108.105,"endTime":3112.159,"body":"Um, I think in turbulence, in applied maths, they probably crop up to do with,"},{"startTime":3112.159,"endTime":3116.213,"body":"with, with things where I'm very much getting beyond my pay grade, but they come"},{"startTime":3116.213,"endTime":3117.565,"body":"up all over the place."},{"startTime":3117.565,"endTime":3121.165,"body":"It's the same function, but the same function is a tool for understanding all kinds"},{"startTime":3121.165,"endTime":3123.325,"body":"of different relationships and all kinds of different circumstances."},{"startTime":3124.555,"endTime":3124.845,"body":"Okay."},{"startTime":3125.185,"endTime":3126.685,"body":"Um, we're gonna need to finish there."},{"startTime":3126.705,"endTime":3129.967,"body":"So in the room, I'm sure all of will be willing to stay and answer"},{"startTime":3129.967,"endTime":3131.925,"body":"any questions you may have, uh, after we stop."},{"startTime":3131.945,"endTime":3134.765,"body":"But I just want to finish by saying thank you to everyone for coming."},{"startTime":3135.185,"endTime":3139.597,"body":"And I'm so delighted that the London Mathematical Society and Gresham College have this fantastic"},{"startTime":3139.597,"endTime":3142.245,"body":"partnership and brings these lectures to you each year."},{"startTime":3142.585,"endTime":3146.205,"body":"Um, let's finish by thanking the speaker Oliver Johnson, again for his wonderful talk."}]}