 Rwm safon, dydych chi'n ddigon, am y cwylio'r un hearing buswll. Dwy'n meddwl, rynodd eich hunau i Mathematicio Ni, a rynodd eich hunau? Rwy'n meddwl, rwy'n meddwl... ..on yr hunau? Bydd hi'n growl! Rwy'n meddwl, rydw i'n mynd i. Dwy yma llawer? Rwy'n meddwl. Rwy'n meddwl! Ond llawer ac... ... newid ychydig a'i wneud hyn. Rwy'n meddwl, rydw i'n meddwl. Great, brilliant. Thank you. I knew if I didn't do both, someone would call me up in it. Right, okay. So Park Run. Park Run is a weekly event that happens all over the UK, simultaneously, probably, at 9am. Every Saturday morning, Friday today, so if you really want to, you can go tomorrow 9am to Ledbrie, I believe this is closest. It's on the EMF site somewhere. To quote Park Run directly, Park Runs are free, weekly community events all around the world. And what I'm trying to do is I'm trying to run the alphabet and it's exactly what it sounds like. I want to go, oh God, not yet. I'm not making that joke yet. I want to go to 26 Park Runs. Each of them start with a different letter of the alphabet. I'm doing very well at the moment. I've been to three. So I'm very dyslexic. I don't really do letters. So if there is any sort of mistakes in here, blame me because it was me. I'm a mathematician. I really like collecting things. So this follows up. So I want to collect all of my least favourite things. I'm a masochist. So I mentioned in the brief of this talk that I had realised something miraculous. As I was running around Heartlands Park Run, I got into the second lap and I was just going round the corner. I had a thought. Aren't all Park Runs the same? Surely they are. You're just going round in loops. And if they were, how can we explain this? So to the modern mathematician, they all are the same. You go round how many times you get to the end. You feel tired. Your legs give in and you need a drink. So how many Park Runs are possible? So another quick question for you. Who thinks there's only one Park Run? None of you. Great. Good. How many think two? How many think there are lots and lots that I can't really name on a slide and I couldn't really go through in the whole of this presentation? Great. All of you are wrong. So here's a mathematical idea. We're going to name it Homotopy. If there's any computer scientist in the audience, yes, this is like Homotopy type theory, but not the same. Okay. Here's a squiggle. They're going to squish it. Give it a squish and it turns into a line. That's the whole idea. We're going to take curves, closed curves, give them a squish, turn them into nicer things. Okay. So a Homotopy to the mathematicians is a continuous map between two spaces. Spaces are just kind of shapes and that's how we have to think about them. So on the right, we want for a squiggle to a line. They're both spaces. We call them topological spaces. Okay. So to the real mathematicians in the audience, a Homotopy is a continuous function between... We don't really care about this. So we now ask under what metric are two things similar? So if I want to say park runs are the same, how am I going to do this? So we're going to call them the same if they have the same fundamental group. Okay. So let's take some closed paths. So imagine I'm going for a run. Go down, down, down, down. And what I've done now is a closed loop because I started in the same place and I ended in the same place. So that's what we're going to do. We're going to start in the same place, go for a run, come back to the same place. That is a closed loop. And the fundamental group is all of those things you can do in your space. All of the runs that you can start at the same place and end at the same place and potentially maybe go somewhere. So this is what we call the fundamental group. And it's considered up to Homotopy. So squish them if I go all the way round. I'm not going to do it. But if I go all the way round outside, off the stage, back to here, that's the same as me going all the way down to the end here and back. They're the same because I can squish them in on each other. And the fundamental group has what we call a group structure. So what's a group structure? Well, to mathematicians, we need an addition. So we need to consider how we can add two things together. And as you see in my little animation over there, what we do is we take one end and we take the other and move them together and bang, we have one path. So we go from two to one by just going together. Right. So a group to a mathematician has the following structure, closure. So as we've seen, if we take any two paths, add them together, we get another path. That's pretty simple, right? So inverse, so we need to undo a path. And obviously, if I'm going to run all the way back, out there and all the way back, what I'm going to do if I want to do the inverse of that is I run all the way in the opposite direction. So the inverse is running in the opposite direction. Great. So if I'm going to do an identity, identity mathematics means that if you add it to something else, you get the same thing you started with. So adding zero, think, or multiplying by one, something like that. So the identity loop or the identity run is doing nothing. It's like adding zero. Great. So this is the complicated bit. Associativity, very, very complicated word. So this means basically if you add three things together, it doesn't matter if you add the first two first to the third or if you add the last two to the first. It doesn't matter. So I have gotten a screen. Ew, I know a sum takes right to school. But it doesn't matter. It's like that. OK. Good. So here we have a circle. Right. Circle, can we go over here? Right. Can we come with me? Over here. Right. It's going all the way round. So if we start up here and we go on a run all the way round, that is a loop. And we can do that twice, three times, four times. I'm starting to name numbers here. And you may start to realize something. And the inverse is going in the opposite direction, negative one, negative two, negative three, and so on. And we've got zero. We can just stand at the top there and do nothing like a lazy bugger. OK. So as I said, you've got past on a circle. We can go plus one, which is forward, negative one, which is backwards, or zero, which is do nothing. And this forms our coveted group. And this group is kind of the integer. So all the whole numbers you know from school, I know school, we all hate school. But it is the integers. Now I've used some funny notation. There are pi one. That means the fundamental group because p is like th. And s one, which is just a sphere, is z, which is the integers. All the positive and negative whole numbers. Right. Great. So oh, God, it's a fancy name again. So what we're going to do now is to see Fert van Kampen theorem. That is a fancy name to mean that we can just cut things in essence. So we're going to consider wedge sum. And then we're going to do this see Fert van Kampen. Right. OK. So the wedge sum is we're going to take two circles, right? And I'm going to do what I've done here. I've taken one point and I'm going to move them together. Imagine their Play-Doh. And what happens is they just join together. And we use this fancy notation, which looks a bit like a V. And it is a V, really. OK. So we're going to take two circles, add them together at a point, and notate it using this fancy little V. So the see Fert van Kampen theorem says that if we take this fancy little V and we consider the fundamental groups or the different paths you can do on it, that's kind of the same as doing all the paths on one part of the circle times. Well, some sort of product, the other one. OK. Great. So if we consider the figure eight, which really is just the wedge sum of two circles, S1's a circle, we're going to add them together, join them together, and we get a figure of eight. See Fert van Kampen tells us that the fundamental group is just Z star Z. I'm being very careful not to use times here because it's not exactly the same as times. OK. So great. So here's another one. We can go around all the way around. There we are. That's a full loop. And then we see a blue one. That's a half loop. Carry on. It's a bit slow. Come on. Bit quicker. Thank you. OK. Great. So we have the full one, which we can call eight. And then we have A, which is the half bit, and B is the other half bit. And again, we can do the same thing. If we're going to run round A, we can run round in the opposite direction. And I'm going to call this A inverse. Doing A is forward, and going A inverse is backward, much like our one and negative one. So OK. What is this fundamental group of this one then? Well, what we have is what we call the free group. That's free, isn't it? I don't quite know the understanding of this, and I've changed the notation because I don't really like the notation. But the point is, when you go running round this park run route, what happens is you run round A, or in some way maybe forward or backwards, and you run round B. So we have this kind of this bit here. OK. I'm going to get to examples in a minute. I'm sorry. It's very, very kind of notation heavy. OK, great. So what is a path? Well, as I've just mentioned, we can run round A, we can run round B. So this one's running round A, then running round B, which is the whole space. We run round A, then we go round B in the reverse direction, then back round A. Remember, it's a figure of eight. So when we do A or B or A inverse or B inverse, we always end in the middle. So this one isn't complicated. I don't really know why I did this, but part of me decided that was fun to write out. And I mean here that when I've got A, B inverse A, you can't just multiply through to get like A squared B inverse. This is a non commutative algebra. And that's another funny word. And what it means is that, as I said, A times B is not equal to B times A. So remember when we were in school? I'm sorry again. This is not the best analogy. But we learned that one times two is two times one. And so we see here, again, I've given another example, that A, B, A inverse is not equal to B. Because if you think about it, running round A, and then going round B, and then going round A in the opposite direction while running, it's not the same thing as just running round B. That makes sense, right? Good. Oops, I've been through that bit. Okay, so that's a figure of eight. What about, we call it two rows, because rows are pretty. So if we consider rows with more petals, which part runs usually are, we'll see in a minute, what we're going to have is something like this. So we have all these wedge sums, fancy notation, use C for van Kampen several times, and then we get all these kind of integers multiplied together. And this is like the free grip on two letters, which we saw a minute ago for the figure of eight, but for N letters, where N is just a number. It doesn't really matter what it is. We'll use it in a minute. Great. Okay. How are we doing on time? So we're doing very well on time. So aside covering spaces, this is where I bring out my friend. I got a story to say about my friend. I went looking for him the other day. I went through, I live in Swansea at the moment, and I went into town. And kids don't play with slinkies anymore, apparently. I know I don't look old, but apparently the toys of my childhood aren't the toys of nowadays childhood. It is what it is. Okay. Great. So you may be asking me, okay, does it really matter if I'm going to go running round my park, run kind of loop, some sort of, go the whole loop twice? Does that really change the fundamental group? Well, no, not really, because if we think about going round circles several times and then we turn it into a slinky, we do this, well, what we're really interested in is kind of looking down at it. We're interested at the actual shape, not kind of how many times we go round. And I give the analogy is if we're going to cut downwards on this slinky, which I'm not going to do because this cost me six quid, what happens is we got a load of circles and that they all have the same fundamental group and I don't really care about that. All I care about is the shape. Great. Okay. So here are the examples. Here's what you were waiting for. So we were waiting to find out how many park runs are there. I've left it as a surprise until the end and we've got N. N's just a number again. I'm a mathematician. We like putting variables everywhere. So we're going to consider some different park run courses up to Homotopy. I have run most of these. So let's see some of the park run courses. Well, firstly, here is Penrose Estate. This is one down in Cornwall where I'm usually based and we can see it's a nice line. It goes, don't go out and back, just out, right? So what happens here is we have an animation. Here's the root and it just obliterates itself under Homotopy and it goes to a point. This is no homotopic because it has fundamental group of zero because we can't really run around the point, can we? The only thing we can do when we're at a point is stay the same. So that is one type of park run. So at one, people who said one, you are wrong. That was none of you. So well done. So an overlapping loop. So just like this stuff. This is Gundersbury in London. I expect there are some people from London in New Orleans and may have even been to this park run. So this is Gundersbury. I ran this just before lockdown. I think this is my last park run before lockdown. I ran this one. Absolute nightmare. It was very rocky. I didn't like this one. I've done ones done in Cornwall. Okay, here's to park run. What's it going to do? Oh, it squishes into a circle. So this has fundamental group of the integers. So if we take any kind of squiggly park run that doesn't overlap, then we have a circle. And we have the integers because it's quite nice. You can run my once twice, three times, four times, five times. And so on. And you go in the opposite direction. Okay, great. So here's one crossing loop to any pedants in the audience. That was my Strava buggy act in that area up there. There's only actually one crossing loop. So this is Heartland. This is my home park run when I'm based down in Cornwall. That kind of hopefully doesn't dox me. The area is big enough. Okay, so, oh, I don't have an animation. Fine. What we can see here is we've got one crossing. So what's going to happen is it's going to turn into a figure of eight. So this, this out here expands, this expands and you have a figure of eight and we've seen that is the free group on two letters. So that's different. I said, I don't really care what the group is. All I care is it's different. Okay, two crossing. I've got two examples here because I quite like them. Here's Durstan. I haven't ran this one, but it looked quite fun. So it's got kind of all the way out. You do a loop. You come back. All right. So look what I'm saying. That one. Yeah, I don't like I've really got to learn blender, but I must note that this is continuous. So the animation shows it discontinuous, but it really is continuous. So what happens is we get a three rows. And how many letters is this free group on way three? Someone's learned something. Okay, great. What about this one? Forrest of Dean. Again, a bit out of my range, but let's have a look at this one. This one's quite fun because you go all around the place. I quite like this one. This one used to be one of my edge cases, but I realized that it wasn't an edge case because it does this, which is a circle. We've got three circles. And what you do is you take these two intersections, move them together and you get a three rows. And as our friend out in the audience told us, this has three letters. So it's different. Okay. This is where you come in. I couldn't find anymore. I spent two hours looking through the parkrun site. Have you ever tried to go and look at all of the parkruns in the UK on the parkrun site? Probably not because it's a pain in the ass. And nobody would really do it apart from a masochist. So I couldn't find something with three overlaps. So if there is anybody in the audience that has a home parkrun, you can prove me wrong and show me it's n equals four. But if not, it's n equals three. There are only three parkruns. You've heard it here first. Great conclusion. I should go to newspapers with that one, shouldn't I? I'm sure that one of them would buy it. So conclusion. Great. So we've quickly learned about algebraic topology. One of my favorite areas of maths. I'm an applied mathematician. Don't tell my university. We have seen the fundamental group and we've briefly seen the product and the free group. So here are a few conclusions that you should take away once you exit that door. Math is fun. That's my job. I'm paid to tell you this. So maths is fun. It's not just sums. The fundamental group is wacky and fun and it does several interesting things. James doesn't really like crunching data. I gave up after two hours. And I really, really need to learn blender. Did you see the state of those animations? I did those in PowerPoint and I am proud, I will tell you. But that is the end. And if you have any questions or you want to just come chat about topology, I am going to be based in the maths tense. So here we are in stage B. Go all the way round. You're going to go that way and you're going to go all the way round until you find workshop B where I will be standing outside smiling sweetly to come and get you to answer me questions. Thank you very much.