1 00:00:18,240 --> 00:00:22,170 OK, well, thanks very much, Martin. Welcome everyone to Oxford. 2 00:00:22,170 --> 00:00:31,470 I'm going to be as Martin said, talking about knots and knots are pretty much exactly what you think they are. 3 00:00:31,470 --> 00:00:40,700 So you you take a piece of string or in this case, a skipping rope and you tie it up some way. 4 00:00:40,700 --> 00:00:48,470 And then you glue the two ends together, you glue the twins together to form a closed curve. 5 00:00:48,470 --> 00:00:55,190 And that is a knot, and you allow yourself to say, well, like wiggle the knot around. 6 00:00:55,190 --> 00:01:02,330 That doesn't change the knot, but you're not allowed, obviously, to cut it or make it pass through itself. 7 00:01:02,330 --> 00:01:14,330 So here are some knots as the simplest possible not the UN, not which is just the round circle. 8 00:01:14,330 --> 00:01:21,640 Then after that, the first non-trivial knot is is the trefoil and then the Figure eight. 9 00:01:21,640 --> 00:01:35,060 And so surprisingly. Mathematics has something to say about knots, but you may ask yourself why, why should we study knots? 10 00:01:35,060 --> 00:01:44,520 Well, they come up, it turns out all over the place, they come up in the physical and biological sciences. 11 00:01:44,520 --> 00:01:49,200 So let me just give you one example of that which is in. 12 00:01:49,200 --> 00:02:02,440 DNA. OK, so DNA is, as we all know, it's shaped famously like a double helix, so it has these two strands which curl around each other. 13 00:02:02,440 --> 00:02:12,610 And as you run along the the strands, there are various chemicals which encode our complete genetic makeup. 14 00:02:12,610 --> 00:02:20,120 They give the instructions for for life. And so. 15 00:02:20,120 --> 00:02:28,700 Clearly. Not only do these two strings tangle around each other, but it turns out the DNA ends up by being extremely knotted. 16 00:02:28,700 --> 00:02:35,100 And this is the particular relevance in the replication of DNA. 17 00:02:35,100 --> 00:02:39,230 OK, so sitting inside each of our cells in our body, 18 00:02:39,230 --> 00:02:48,800 there is a complete copy of our genetic code and every now and again, the cells divide divide into two. 19 00:02:48,800 --> 00:02:57,790 In fact, this happens incredibly often. It happens a thousand so a million times a second in each of our bodies. 20 00:02:57,790 --> 00:03:07,390 And what happens when the cell divides is that this genetic code or DNA has to and it has to 21 00:03:07,390 --> 00:03:13,990 be copied so that one bit of DNA ends up in one of our cells and one ends up in the other. 22 00:03:13,990 --> 00:03:18,520 So the way it actually works is that the two strands of this double helix get 23 00:03:18,520 --> 00:03:24,730 separated out from one strand ends up in one of the cells after division, 24 00:03:24,730 --> 00:03:35,320 and the other strand ends up in the other cell. And then the cells can rebuild the entire DNA from this process. 25 00:03:35,320 --> 00:03:42,460 Because as you run along the double helix, there are these different chemicals which specify the code that called a C, 26 00:03:42,460 --> 00:03:48,550 G and T and G only binds to C and A only binds to T. 27 00:03:48,550 --> 00:03:58,070 So if I have just one half of this double helix, then I can use that to recreate a complete copy of the double helix. 28 00:03:58,070 --> 00:04:06,450 So. These two strands, they have to pull apart, but this is actually a quite a complicated and difficult procedure. 29 00:04:06,450 --> 00:04:12,510 And the reason is that DNA is incredibly convoluted within our cells. 30 00:04:12,510 --> 00:04:17,950 So each cell nucleus is typically about five or six microns across. 31 00:04:17,950 --> 00:04:24,690 And if you were to take the DNA that was just sitting in that cell nucleus and stretch it out into one long line. 32 00:04:24,690 --> 00:04:26,520 It would be two metres long. 33 00:04:26,520 --> 00:04:35,710 So something two metres long has to end up like getting scrunched up into something just five microns across, so it's incredibly knotted. 34 00:04:35,710 --> 00:04:44,740 And when the cell replicates, these two bits have to somehow get from one one has to end up in one cell and the other one has to end up in the other. 35 00:04:44,740 --> 00:04:52,190 So clearly the not the two strings have to pass through themselves, pass through each other. 36 00:04:52,190 --> 00:05:06,890 And this is done by various enzymes and importantly, the enzymes that work for bacterial DNA, a different from the enzymes that work for human DNA. 37 00:05:06,890 --> 00:05:19,970 So in fact, lots of the antibiotics that we use, they they specifically block this untangling procedure in bacterial DNA. 38 00:05:19,970 --> 00:05:29,480 And so that means that the bacterial innate DNA cannot unlocked itself and so cannot replicate and so dies, 39 00:05:29,480 --> 00:05:34,580 whereas that has no effect on us because we use a different type of enzyme to do this. 40 00:05:34,580 --> 00:05:41,900 So clearly, understanding this unlocking procedure is important, but it's not completely well understood. 41 00:05:41,900 --> 00:05:50,730 And there's been a very productive collaboration between biologists and not theorists to be able to understand that in more detail. 42 00:05:50,730 --> 00:06:00,320 So that's just one example of the occurrence of knots and the importance of knots in nature. 43 00:06:00,320 --> 00:06:10,230 I'm a pure mathematician, and one of the reasons that I find not so interesting is because they are somehow intrinsic to three dimensional space. 44 00:06:10,230 --> 00:06:16,590 If we lived in two dimensions, then there wouldn't be enough space to be able to form nuts, 45 00:06:16,590 --> 00:06:22,050 whereas if we lived in four dimensions, there would be enough space so that every knot could be undone. 46 00:06:22,050 --> 00:06:29,030 And maybe if this time at the very end of this lecture, I'll show you how you do that, how you are not any, not in four dimensional space. 47 00:06:29,030 --> 00:06:38,000 So somehow, the existence of knots is one of the intrinsic features of the three dimensional space in which we live. 48 00:06:38,000 --> 00:06:42,410 So not theory is now part of topology, 49 00:06:42,410 --> 00:06:51,080 so part of the aim of this talk is to introduce you to topology and explain a little bit about what it's what it's found. 50 00:06:51,080 --> 00:06:59,780 So topology is a bit like geometry. So in geometry, you look at spheres and cubes and triangles and squares. 51 00:06:59,780 --> 00:07:04,850 But to a janitor, a sphere is different from a cube. 52 00:07:04,850 --> 00:07:13,820 Whereas to its apologist, they look the same. So in topology, you're allowed to deform objects with as long as you don't tear them apart. 53 00:07:13,820 --> 00:07:19,100 So you can deform this sphere, you just squish it down into this cube. 54 00:07:19,100 --> 00:07:23,510 So to it's apologists, these two objects are the same. 55 00:07:23,510 --> 00:07:31,340 So famously so its apologists, his coffee cup is the same as his doughnut. 56 00:07:31,340 --> 00:07:36,410 So there you can see the two deforming one into the other. 57 00:07:36,410 --> 00:07:44,150 So it kind of then makes sense that because when we're looking at knots, we allow them to deform but not pass through each other through them selves. 58 00:07:44,150 --> 00:07:52,440 That that is exactly the right sort of subject. Topology is exactly the right tool to be able to study knots. 59 00:07:52,440 --> 00:08:02,910 So the first person to realise that knots were actually part of mathematics was the great Carl Friedrich Gauss. 60 00:08:02,910 --> 00:08:09,060 So Gauss is widely viewed as one of the finest mathematicians who ever lived. 61 00:08:09,060 --> 00:08:14,010 He worked at the end of the 18th century and the beginning of the 19th century, 62 00:08:14,010 --> 00:08:19,110 and his mathematical notebooks contain lots and lots of pictures of nuts. 63 00:08:19,110 --> 00:08:23,100 He realised that they were mathematical objects. 64 00:08:23,100 --> 00:08:34,840 And what he did was he developed a way of encoding knots, using a sequence of letters and numbers called a Gauss code. 65 00:08:34,840 --> 00:08:40,270 So although he realised that knots were part of mathematics, he didn't really make much headway with them. 66 00:08:40,270 --> 00:08:47,180 He did it a little bit, but topology did not exist when he was alive. 67 00:08:47,180 --> 00:08:52,840 It was created at the end of the 19th century, the beginning of the 20th century. 68 00:08:52,840 --> 00:09:01,330 And so the real headway, the real, real progress and not theory had to be made by people after Gauss. 69 00:09:01,330 --> 00:09:12,460 In fact, the first person to really make any progress with knots was was not a mathematician, but a physicist. 70 00:09:12,460 --> 00:09:20,620 So here he is. Peter Guthrie Tate looking very stern and Victorian. 71 00:09:20,620 --> 00:09:32,140 He worked. He was a Scottish physicist working in the 19th century, and he got into knots via smoke rings. 72 00:09:32,140 --> 00:09:42,040 So I think you all know what a smoke ring is. Here is an example of a smoke ring that actually was belched out of Mount Etna. 73 00:09:42,040 --> 00:09:48,310 So coming out of the side of Mount Etna, there are various vents and every now and again, one of them. 74 00:09:48,310 --> 00:09:52,620 And look, if, if, if you're lucky, it'll form this. 75 00:09:52,620 --> 00:10:01,870 This smoke ring so tight it created these experiments to to create smoke rings. 76 00:10:01,870 --> 00:10:06,490 And one of the things about smoke rings is that they are surprisingly long lived. 77 00:10:06,490 --> 00:10:14,360 They fizzle through the air and don't break up instantaneously, as you might expect. 78 00:10:14,360 --> 00:10:18,980 So his experiments on smoke rings caught the eye of this man, 79 00:10:18,980 --> 00:10:28,430 Lord Kelvin Lord Kelvin is an extremely famous physicist for which the Kelvin temperature scale is named. 80 00:10:28,430 --> 00:10:40,850 And he was quite taken by Tate's experiments on smoke rings, and so he came up with a brilliant leap of imagination. 81 00:10:40,850 --> 00:10:46,550 He came up with a theory which was brilliant, but completely wrong. 82 00:10:46,550 --> 00:10:56,240 So his his his his theory was as follows was he thought that perhaps nots could be used to explain atoms? 83 00:10:56,240 --> 00:11:00,440 So at that point, atoms were known to exist. 84 00:11:00,440 --> 00:11:07,640 It was known that matter is made up of different atoms, but it had people had no idea what was inside an atom. 85 00:11:07,640 --> 00:11:15,560 So we now know, obviously, that it's protons and neutrons forming a nucleus surrounded by a cloud of electrons. 86 00:11:15,560 --> 00:11:22,610 But at this stage when when Kelvin was working, they had no idea what the structure of atoms was. 87 00:11:22,610 --> 00:11:29,570 So he thought the perhaps atoms were knotted vortices in the Aether, 88 00:11:29,570 --> 00:11:35,000 and that the long lasting nature of the smoke ring would explain the long lasting nature 89 00:11:35,000 --> 00:11:40,040 of the atom and the different types of knots would explain the different types of atoms. 90 00:11:40,040 --> 00:11:51,360 And maybe, for example, some of the structure of knots could explain some of the different properties of atoms, for example their absorption lines. 91 00:11:51,360 --> 00:12:05,040 Wonderful idea. Completely false. But what it did was it sparked Tate to investigate knots and he kind of got hooked on them and he created it, 92 00:12:05,040 --> 00:12:10,640 started creating tables of knots, which is a sort of very Victorian thing to do. 93 00:12:10,640 --> 00:12:20,280 So biologists would create tables of plants, and he was creating tables of knots ordered by then, not Guinness. 94 00:12:20,280 --> 00:12:28,320 So here's an example of he got a bit carried away, to be honest. 95 00:12:28,320 --> 00:12:33,090 Here is he's just one of the paper pages from one of his papers. 96 00:12:33,090 --> 00:12:37,800 Let me just zoom in a little bit. You can see this is just the top left of that page, 97 00:12:37,800 --> 00:12:48,090 and you can see there is the the truffle oil and the Figure eight and more and more complicated knots, and you can see the title of the page. 98 00:12:48,090 --> 00:12:53,490 The first seven orders of nuttiness. So what is nuttiness? 99 00:12:53,490 --> 00:13:00,150 So what we what what he called nuttiness is now called the crossing number of a nut. 100 00:13:00,150 --> 00:13:02,060 OK, so the crossing number of a knot, 101 00:13:02,060 --> 00:13:10,590 what you do is you look at the different possible projections of the knot and when you project the knot onto the plane, 102 00:13:10,590 --> 00:13:16,580 some bits of the knot have to cross over themselves, and the number of those is called the number of crossings. 103 00:13:16,580 --> 00:13:20,960 Different projections of the NOT will have a different number of crossings. 104 00:13:20,960 --> 00:13:30,650 So what we do is we consider the minimum possible number of projections and that then is the the crossing number of the not. 105 00:13:30,650 --> 00:13:35,560 OK, so just so that we're all on the same page. Let's just. 106 00:13:35,560 --> 00:13:44,540 Ask what is. What's the crossing number of that not? 107 00:13:44,540 --> 00:13:50,150 To do I hate her. You have no zero, right, you. 108 00:13:50,150 --> 00:13:57,410 You mustn't consider just this diagram. You have to consider all possible diagrams of this not and look at the minimal 109 00:13:57,410 --> 00:14:02,630 number of crossings so I can take this not I can just undo this and undo this. 110 00:14:02,630 --> 00:14:12,040 And I get. Just the UN not on this projection of the UN has zero number of crossings, and that's the minimum possible, obviously. 111 00:14:12,040 --> 00:14:19,150 So this has crossing No. Zero. 112 00:14:19,150 --> 00:14:23,140 OK, so that's the nuttiness of the knot or crossing? 113 00:14:23,140 --> 00:14:29,230 No. So it actually did a remarkable job. 114 00:14:29,230 --> 00:14:38,500 He has tables are amazingly accurate. So he didn't miss any knots out or at least within the type of knots that he was looking at. 115 00:14:38,500 --> 00:14:44,410 And I have very few duplications, so he or she did a remarkable job on it. 116 00:14:44,410 --> 00:14:52,330 By looking at this data, he actually came up with some conjectures which actually weren't finally resolved until the 1990s. 117 00:14:52,330 --> 00:14:59,810 So he really is, you could say, the founder of Of Of Not Theory. 118 00:14:59,810 --> 00:15:11,870 OK, so what I want to do now is get things a little bit more mathematical and I want to introduce the notion of addition of two knots. 119 00:15:11,870 --> 00:15:21,890 And this is going to be we're going to do this to sort of develop an analogy and analogy between knots and and positive whole numbers. 120 00:15:21,890 --> 00:15:29,360 OK. So just as we can add two whole numbers together and get another whole number, we're going to add knots together. 121 00:15:29,360 --> 00:15:32,710 So how might you add two knots together? 122 00:15:32,710 --> 00:15:44,590 So given two knots, K1 and K2, we're going to add them to form this new not called the connected sum, so K1 plus K2, and that's defined this way. 123 00:15:44,590 --> 00:15:48,670 So here's K1, for example, the Figure eight, his K2, 124 00:15:48,670 --> 00:15:57,130 the trefoil and the way we form the sum is you just cut the two knots and then splice them together. 125 00:15:57,130 --> 00:16:05,480 And it turns out that's a well-defined operation, and this is the then the sum of those two knots. 126 00:16:05,480 --> 00:16:14,840 And so I want to develop this analogy further between knots and and whole numbers, 127 00:16:14,840 --> 00:16:21,190 because whole numbers, as we all know, have this decomposition into primes. 128 00:16:21,190 --> 00:16:25,460 So the special hole numbers, these prime numbers and any hole number, 129 00:16:25,460 --> 00:16:32,330 any positive whole number can be written as a product of prime numbers and this can be done. 130 00:16:32,330 --> 00:16:44,030 This is unique, right? So if I say fifty seven and work out, it's prime de composition, well, that's three times 19, I think. 131 00:16:44,030 --> 00:16:49,100 And if I were to tell you to go away and do the same, you get the same answer, hopefully. 132 00:16:49,100 --> 00:16:56,330 So we want to come up with the same sort of result for lots. 133 00:16:56,330 --> 00:17:06,980 So what? First of all, what does it mean for not to be prime? So not as cool prime if it cannot be written as a non-trivial connected sum? 134 00:17:06,980 --> 00:17:13,970 So just as a number is prime, if it can't be written as a product of two smaller hole numbers, 135 00:17:13,970 --> 00:17:23,360 so not his prime if it can't be written as a connected sum of two two smaller knots. 136 00:17:23,360 --> 00:17:32,120 And in fact, just as there's a prime decomposition theorem for numbers, there's a prime decomposition theorem for knots, 137 00:17:32,120 --> 00:17:42,560 so any knot can be written as a connected sum of prime knots in an essentially unique way, just like crown numbers. 138 00:17:42,560 --> 00:17:49,760 So this is a theorem it was proved by Fields medallist John Milner in some of his early work. 139 00:17:49,760 --> 00:17:58,490 And what I want to do today is to explain an intermediate step in his theorem. 140 00:17:58,490 --> 00:18:01,310 So I'm going to explain this theorem here. 141 00:18:01,310 --> 00:18:07,610 I'm going to write it down so that because it's going to be something I'm going to refer to on and off throughout. 142 00:18:07,610 --> 00:18:18,060 So the theorem is. That if I take two knots together. 143 00:18:18,060 --> 00:18:27,730 And I add them. And suppose I get the simplest possible, not. 144 00:18:27,730 --> 00:18:36,960 Then, in fact, K1 and K2 must have been the simplest possible not. 145 00:18:36,960 --> 00:18:46,490 Namely. They're both. They are not. 146 00:18:46,490 --> 00:18:53,750 OK, so something like this has to be true if there's going to be a prime decomposition theorem for knots. 147 00:18:53,750 --> 00:19:03,230 So suppose it were false. Suppose that there were two non-trivial knots that when you added them together, gave you the andnot. 148 00:19:03,230 --> 00:19:11,240 So suppose that you could take two non-trivial knots K one plus K two and get the andnot. 149 00:19:11,240 --> 00:19:21,790 Well, now take any other prime, not say K three. And addicts take one plus K2. 150 00:19:21,790 --> 00:19:30,670 Well, then that gives me K3, some the are not, and if I add they are not to anything, I just get three. 151 00:19:30,670 --> 00:19:38,320 Well, maybe K1 and K2 weren't prime, but if this theory is going to be true, they'll have a decomposition into prime summons. 152 00:19:38,320 --> 00:19:45,280 And so that'll mean that I have this not here, which is prime written as in a different way as a sum of prime knots. 153 00:19:45,280 --> 00:19:50,440 And I would have them for a failure of uniqueness of prime decomposition. 154 00:19:50,440 --> 00:19:59,620 OK, so if I'm going to have this uniqueness theorem for prime decomposition, then I'm certainly going to have this results here. 155 00:19:59,620 --> 00:20:05,940 And in fact, this is an intermediate step in the proof of of this theorem. 156 00:20:05,940 --> 00:20:10,890 So I call this this, this this theorem, the garden hose theorem, 157 00:20:10,890 --> 00:20:18,150 and the reason is as follows so suppose I'm out watering the garden with my hose and being somewhat absent minded. 158 00:20:18,150 --> 00:20:28,710 I managed to get this hose knotted up, save with a Knot K1, and I'm busy thinking about theorems and not really focussing on what I'm doing. 159 00:20:28,710 --> 00:20:35,700 And I get over here and I notice over there there's some annoying, not one which has got all tangled up. 160 00:20:35,700 --> 00:20:45,480 And I think, OK, well, I want to get rid of that knot and now being extremely lazy, I can't be bothered to go back and untie it. 161 00:20:45,480 --> 00:20:57,870 Instead, I think maybe if I just stand here and tie another knot K2 and then push that down there and hope that K1 will cancel out with K2. 162 00:20:57,870 --> 00:21:00,690 And this theorem says that's doomed. 163 00:21:00,690 --> 00:21:10,350 The connected sum of K1 and K2, which is what I would have just formed, would never be the andnot unless both K1 and K2 were the UN not to begin with. 164 00:21:10,350 --> 00:21:18,740 OK, so this is the garden hose theorem, and this is what I'm going to show you how to prove today. 165 00:21:18,740 --> 00:21:29,810 OK, so if you're going to try and prove a theorem like this, the first stage is to think, well, OK, let's let's to think why it might be true. 166 00:21:29,810 --> 00:21:35,660 OK, so what we want to try and do is we just want to think, think of a strategy. 167 00:21:35,660 --> 00:21:43,070 So why might this be true? Well, it seems just like let's just go back a bit. 168 00:21:43,070 --> 00:21:54,870 But when I formed this connected sum that somehow. This new note I've created is somehow more complicated than each of the original ones. 169 00:21:54,870 --> 00:22:06,100 So suppose. That were true, but somehow K1 and K2 was at least as complicated as each of the original knots. 170 00:22:06,100 --> 00:22:15,460 And that what I've created was the UN, not the simplest possible thing that must mean the K1 and K2 must therefore have been even simpler. 171 00:22:15,460 --> 00:22:21,130 No more complicated. Therefore, they are not. OK, so that's the idea. 172 00:22:21,130 --> 00:22:22,690 But how are we going to formalise that? 173 00:22:22,690 --> 00:22:35,710 What we want to do is we want to have a notion of complexity that behaves well when you add two knots together and you think we have such a notion, 174 00:22:35,710 --> 00:22:46,840 the nuttiness, the crossing number of the not so you're naturally led to think about the crossing number of connected sums and you think, well, well, 175 00:22:46,840 --> 00:22:56,590 maybe the crossing number of a connected sum behaves like this, that the crossing number of the sum of two knots is the sum of the crossing numbers. 176 00:22:56,590 --> 00:23:02,410 So this statement in red implies the golden host theorem. 177 00:23:02,410 --> 00:23:07,960 Right? Because suppose I take two notes and I add them together and I get the other note. 178 00:23:07,960 --> 00:23:14,040 That means the crossing number of the left hand side is zero. 179 00:23:14,040 --> 00:23:20,670 But if this were true, then that would mean that the crossing number of K one plus the crossing number of K2 would have to be zero. 180 00:23:20,670 --> 00:23:28,200 They're both not negative numbers and so therefore they must each have been zero and hence K1 and K2 must therefore have been both. 181 00:23:28,200 --> 00:23:36,380 They are not. OK. So this statement here implies the garden host there, unfortunately. 182 00:23:36,380 --> 00:23:39,290 This is a famous unsolved problem. 183 00:23:39,290 --> 00:23:46,910 OK, so this is this is I think this is bizarre and amazing that such a simple thing crossing number of the summer to Nazis, 184 00:23:46,910 --> 00:23:51,680 to some of the crossing numbers that should be unknown. 185 00:23:51,680 --> 00:23:58,520 But it shows you that mathematics is full of these simply stated conjectures, which seem. 186 00:23:58,520 --> 00:24:07,830 Beyond our reach. So you start to think about this and think, well, look, how could this not be true, right? 187 00:24:07,830 --> 00:24:11,070 I mean, it just looks like it has to be true. 188 00:24:11,070 --> 00:24:20,670 And you realise that actually your intuition is playing tricks with you, that actually your intuition actually only gives this inequality. 189 00:24:20,670 --> 00:24:27,030 So let me explain how you get this inequality. So you take K1 and take a minimal crossing. 190 00:24:27,030 --> 00:24:35,190 No diagram for K1 and K2 in a minimal crossing, no diagram for K2 and splice them together. 191 00:24:35,190 --> 00:24:43,320 That gives you a diagram for the sum, and the number of crossings in that diagram is the right hand side, 192 00:24:43,320 --> 00:24:49,020 and therefore the crossing number is of this connected sum. 193 00:24:49,020 --> 00:24:57,400 Is that most that. But maybe there is some other diagram with a few a number of crossings. 194 00:24:57,400 --> 00:25:04,940 Seems unlikely no one's ever found an example of such a thing, but no one has been able to rule it out. 195 00:25:04,940 --> 00:25:14,270 But all is not completely lost, so I have a theorem which is relevant to this, which is a slightly bit of a comedy theorem, really. 196 00:25:14,270 --> 00:25:17,060 What does it say I approved about 10 years ago? 197 00:25:17,060 --> 00:25:24,140 It says the following that the crossing number of the connected sum, it's a most assumes the crossing numbers, 198 00:25:24,140 --> 00:25:29,840 and it's at least the sums are crossing numbers divided by a hundred and fifty two. 199 00:25:29,840 --> 00:25:38,000 I know it's yeah. How can a theorem be funny? But it is unfortunately so. 200 00:25:38,000 --> 00:25:45,710 So, but this actually is powerful enough to prove the garden hose theorem, right? 201 00:25:45,710 --> 00:25:52,970 That so if K1 plus K2 is the andnot and this thing in the middle is zero. 202 00:25:52,970 --> 00:26:00,920 And therefore, this thing on the left hand side is less than or equal to zero, but therefore multiplying three by one hundred and fifty two. 203 00:26:00,920 --> 00:26:05,450 The crossing number of K1 plus across the number of K2 is therefore less than equal to zero. 204 00:26:05,450 --> 00:26:11,730 And therefore, each of the individual things is zero and therefore K1 and K2 are both the not. 205 00:26:11,730 --> 00:26:16,320 OK, great. It looks like the theorems proved, but that is a cheat. 206 00:26:16,320 --> 00:26:23,580 And the reason that's a cheat is because I use this theorem in the proof of that. 207 00:26:23,580 --> 00:26:27,810 So I'm not allowed to use that in the proof of that. That was. That is not allowed. 208 00:26:27,810 --> 00:26:35,510 OK, so we need a different strategy. And the different strategy is using topology. 209 00:26:35,510 --> 00:26:39,980 OK, so topology is. 210 00:26:39,980 --> 00:26:43,620 So now the proof this term is going to be an extended tour through topology. 211 00:26:43,620 --> 00:26:47,120 So let me explain what topology is all about. 212 00:26:47,120 --> 00:26:59,810 So topology is, as I say, the study of spatial type objects like cube or the sphere rarely seen them or the Taurus. 213 00:26:59,810 --> 00:27:09,200 But there's a particular type of object which plays a big role in the subject, namely an end dimensional manifold. 214 00:27:09,200 --> 00:27:14,750 So in end, dimensional manifold is a topological space. 215 00:27:14,750 --> 00:27:21,490 Near each point just looks like ordinary Euclidean space. 216 00:27:21,490 --> 00:27:32,890 But globally need not. So your basic example of a of an end dimensional manifold in this case and equals two is is the surface of the Earth. 217 00:27:32,890 --> 00:27:45,220 So if I'm standing on the surface of the Earth and I just look around me say like a mile radius, then it just looks like a flat Euclidean plane. 218 00:27:45,220 --> 00:27:50,930 Maybe there's some hills and valleys, but politics don't care about that kind of thing. 219 00:27:50,930 --> 00:28:01,280 But that's only true to a certain scale. It's not a flat Euclidean playing the surface of the Earth curves round on itself in a non-trivial way. 220 00:28:01,280 --> 00:28:07,640 So globally, it's not a plane, but near each point it looks like a plane. 221 00:28:07,640 --> 00:28:14,370 So that's the defining property of a manifold, and manifolds are fascinating objects. 222 00:28:14,370 --> 00:28:18,440 And one reason there are many reasons why mathematicians are interested in them. 223 00:28:18,440 --> 00:28:22,870 But one of the main reasons is that we live in a manifold. 224 00:28:22,870 --> 00:28:30,980 OK, so according to Einstein's theory of general relativity, the universe is a manifold. 225 00:28:30,980 --> 00:28:36,290 So you look around you and it looks just like ordinary flat. 226 00:28:36,290 --> 00:28:43,130 You like Euclidean space. But that is only true locally. 227 00:28:43,130 --> 00:28:51,440 So what do we mean by locally? Well, in the relativistic world locally means maybe, I don't know, within 10 billion light years. 228 00:28:51,440 --> 00:29:01,460 So but if you allow yourself even larger scales, then maybe the universe curves around on itself in a non-trivial way? 229 00:29:01,460 --> 00:29:06,380 We don't know, actually. We don't know what the large scale topology of the universe is. 230 00:29:06,380 --> 00:29:15,930 We might never know. People have been trying to find out by looking at. 231 00:29:15,930 --> 00:29:25,140 Patterns in the microwave background radiation, but so far, no definitive answer as to what the topology of the universe is. 232 00:29:25,140 --> 00:29:31,620 So manifolds are interesting things. We're going to be focussing on things that we can visualise nice and easily, 233 00:29:31,620 --> 00:29:40,350 namely two dimensional manifolds such as the surface of the Earth, and they're called surfaces now surfaces actually a bad name. 234 00:29:40,350 --> 00:29:48,240 The surface of the Earth is clearly the surface of something, namely the the the the solid ball that forms the Earth, 235 00:29:48,240 --> 00:29:55,530 but which some of the things that we're going to look at won't actually be the boundaries of anything, but they're still called surfaces. 236 00:29:55,530 --> 00:30:00,090 OK, so here are some surfaces. 237 00:30:00,090 --> 00:30:04,680 The Taurus is a surface. This is a surface here. 238 00:30:04,680 --> 00:30:13,020 It's called pretzel or Jenness to surface. Here's another surface this is called the infinite Loch Ness Monster. 239 00:30:13,020 --> 00:30:22,350 So if you take a flat plain and you drill out little holes from it, there are those holes and then you attach onto each hole a little tube. 240 00:30:22,350 --> 00:30:31,410 Then the resulting thing is a surface. These guys go on forever and some infinite long. 241 00:30:31,410 --> 00:30:37,530 So we're going to focus on surfaces of the first two types rather than the third. 242 00:30:37,530 --> 00:30:42,060 The first two are what's called compact, which means they're nice and bouncy. 243 00:30:42,060 --> 00:30:48,690 They live inside. Some roughly means they live inside some finite portion of of of space, 244 00:30:48,690 --> 00:30:56,010 whereas this thing here goes on forever as non compact, and we're not going to attempt to try to understand this. 245 00:30:56,010 --> 00:31:01,290 So the reason why we focus on on compact ones is because there's a classification theorem. 246 00:31:01,290 --> 00:31:10,080 So one of the the early high points of of topology is the classification of compact surfaces. 247 00:31:10,080 --> 00:31:16,380 So it says that any compact surface is one of two infinite lists. 248 00:31:16,380 --> 00:31:22,230 So the first infinite list starts with a sphere and the tourists and the pretzel, et cetera. 249 00:31:22,230 --> 00:31:27,420 And each stage we just go from the previous one by adding on another handle. 250 00:31:27,420 --> 00:31:40,320 Or there's another infinite list, which I won't attempt to draw, but includes things like the Klein bottle, which I'm not going to focus on too much. 251 00:31:40,320 --> 00:31:45,240 This is what's called non oriented. All those are going to be bad, guys. 252 00:31:45,240 --> 00:31:50,310 I'll focus a little bit more on on orange surfaces in a little bit. 253 00:31:50,310 --> 00:31:54,450 OK. So those are surfaces. 254 00:31:54,450 --> 00:31:59,910 Now I want to generalise that to surfaces with boundary. 255 00:31:59,910 --> 00:32:06,240 So the surface with boundary is, by definition, a point on it. 256 00:32:06,240 --> 00:32:09,890 Either has a little portion around it that just looks like a desk, 257 00:32:09,890 --> 00:32:17,930 just like ordinary surfaces do or has a point on it where the disk just stops like a half disk. 258 00:32:17,930 --> 00:32:29,880 OK. So for example, this is a surface with boundary and the boundary curve is is is just this this round curve here? 259 00:32:29,880 --> 00:32:36,340 So this is this is a surface with boundary. It's just just a disk. 260 00:32:36,340 --> 00:32:44,290 So let me give you another surface with boundaries, so no talk on policy would be complete if I didn't include this example. 261 00:32:44,290 --> 00:32:57,500 So this I'm going to build you a Mobius Band. So Mobius Band, what you do is you take a long strip like this and you glue the two ends together. 262 00:32:57,500 --> 00:33:01,600 OK, so instead of doing them like this, this would form what's called an annulus. 263 00:33:01,600 --> 00:33:07,250 We're not going to do that to form a Mobius strip, you glue the two ends together. 264 00:33:07,250 --> 00:33:17,270 But with a half twist. OK, so that's a Mobius Band, and it has all sorts of interesting and paradoxical properties. 265 00:33:17,270 --> 00:33:22,250 So, for example, it has a single boundary curve. 266 00:33:22,250 --> 00:33:28,490 All right. So if I start here and want around, I come back to the other side and then keep on going. 267 00:33:28,490 --> 00:33:34,770 So that's just as a single boundary curve. So the boundary curve of this is actually a not. 268 00:33:34,770 --> 00:33:41,760 Any ideas, what not that might be? 269 00:33:41,760 --> 00:33:51,330 It's the unknown, it turns out I could have I could have made this a bit more complicated, so I could have instead of gluing it with one half twist, 270 00:33:51,330 --> 00:33:57,270 I could have made it with three half twists that would still be a Mobius Band 271 00:33:57,270 --> 00:34:01,050 because I would have glued the two parts together in exactly the same way. 272 00:34:01,050 --> 00:34:05,870 It just would be sitting inside three dimensional space in a different way. 273 00:34:05,870 --> 00:34:16,470 OK, and now the boundary of that is is the track. While it turns out OK, so Mobius Band is a black eye because it's non oriented. 274 00:34:16,470 --> 00:34:27,390 So what Oriental means is that so this is oriental because it has two sides which you can colour with different colours, 275 00:34:27,390 --> 00:34:35,010 whereas this it starts off before we do the gluing with two different colours red on one side and black on the other. 276 00:34:35,010 --> 00:34:40,050 But when I glue it up. You get a discontinuity like that. 277 00:34:40,050 --> 00:34:49,860 And that means that there's no basically this is a one sided surface if I start when the Reds come back and end up on the other side. 278 00:34:49,860 --> 00:34:55,290 And so this is a will surface. So it's a bad guy, right? 279 00:34:55,290 --> 00:35:02,610 So we're going to focus on the oriental ones because, well, the non-renewable ones are classified as well. 280 00:35:02,610 --> 00:35:12,660 But the oriental ones, they are very simply to state simply class simple classification theorem that any compact Oriental will surface with. 281 00:35:12,660 --> 00:35:16,890 Boundary is exactly one of these guys here. 282 00:35:16,890 --> 00:35:22,140 So you take a long strip and then you attach onto it to G. 283 00:35:22,140 --> 00:35:27,600 Alternating bands for some negative integer G. 284 00:35:27,600 --> 00:35:32,880 There may be an isolated bands like So. 285 00:35:32,880 --> 00:35:41,190 So actually, in our case, we're going to focus on the case where any zero because we're going to want this to be just a if. 286 00:35:41,190 --> 00:35:46,230 So, each time you put on an isolated band, you end up with a new boundary curve. 287 00:35:46,230 --> 00:35:51,060 And I want this to have connected boundary. In other words, I want this to have a boundary. 288 00:35:51,060 --> 00:36:02,540 I'm not. So at the no, when you have to try alternating pens, G is it's called the genesis of the surface. 289 00:36:02,540 --> 00:36:09,570 So. This surface could be embedded in three dimensional space if I wanted to in some complicated way, 290 00:36:09,570 --> 00:36:13,150 so I could have those bands and I could tie little knots in them and that kind of thing. 291 00:36:13,150 --> 00:36:25,770 Still be the same surface. OK, so we have this classification theorem for surfaces with boundary compact Oriental surfaces with boundary. 292 00:36:25,770 --> 00:36:33,270 So we want to use that classification theorem in making progress with knots. 293 00:36:33,270 --> 00:36:43,550 So surfaces with Boundary. They are related to knots via the notion of a cyphers surface. 294 00:36:43,550 --> 00:36:54,980 Sifford surface for a not OK. It's a compact oriental surface sitting inside three dimensional space whose boundary is equal to K. 295 00:36:54,980 --> 00:37:02,870 So for example, this is a flat surface for its boundary curve, which is the, um, not. 296 00:37:02,870 --> 00:37:12,500 Whereas on the other hand, this thing here where the Mobius Band with the three half twists in its boundary is the trefoil lot. 297 00:37:12,500 --> 00:37:18,300 But it's not a cipher surface because we want to focus on the Orient, but once. 298 00:37:18,300 --> 00:37:26,490 And so it turns out that any not nevertheless has some surface surface, so every every, every mathematical talk should have a proof. 299 00:37:26,490 --> 00:37:33,480 And so let me show you how to build a cipher surface for any not. 300 00:37:33,480 --> 00:37:39,270 So there we are. Any not has a siphoned surface, I'm going to build one for the travel. 301 00:37:39,270 --> 00:37:47,240 So here's the track oil and the way you build a surface is you start with any diagram for the not so like so. 302 00:37:47,240 --> 00:37:55,110 And then you give it an orientation, so I've put arrows on that, not. 303 00:37:55,110 --> 00:38:03,020 And then that orientation allows you gives you a way of getting rid of each of the crossings. 304 00:38:03,020 --> 00:38:10,490 So you resolve each of the crossings using that orientation, so so when if you like that, you just go like that. 305 00:38:10,490 --> 00:38:17,030 So this is a from a not theory point of view. This is a terrible thing to do because I've completely changed the knot type. 306 00:38:17,030 --> 00:38:23,430 In fact, I've now just got a collection of simple close curve sitting inside the plane. 307 00:38:23,430 --> 00:38:28,320 So I view that plane now as as lying inside three dimensional space, 308 00:38:28,320 --> 00:38:36,390 as just the horizontal x y plane, and there are my three well in general, my oriented curves. 309 00:38:36,390 --> 00:38:43,480 And now what I do is I now attach disks above the plane onto those curves. 310 00:38:43,480 --> 00:38:53,190 OK, so now I've got some disks sitting inside three dimensional space and the boundary is equal to those curves. 311 00:38:53,190 --> 00:39:03,090 And now that's not quite what I wanted, I wanted to have a surface whose boundary was the not so to what I end up by doing is I re-instate. 312 00:39:03,090 --> 00:39:10,170 I enlarged my surface as follows. I add in half twisted fans at each crossing. 313 00:39:10,170 --> 00:39:16,050 And then when I do that, so I put one this way where the original crossing was there, one there, one there, 314 00:39:16,050 --> 00:39:25,770 so three half twisted bands in this case and I've created a surface with boundary and the it's exactly equal to the knot that I started with. 315 00:39:25,770 --> 00:39:30,600 And because I took with the orientations, the resulting surface really is oriented. 316 00:39:30,600 --> 00:39:37,170 You can trace that through and therefore really is a cipher surface with a knot. 317 00:39:37,170 --> 00:39:43,140 OK, so this is good because what we've got now is we starting to introduce some of the tools from topology, 318 00:39:43,140 --> 00:39:49,770 particular understanding of surfaces to provide some information about knots. 319 00:39:49,770 --> 00:39:54,270 And so that leads naturally to the notion of the genesis of a knot. 320 00:39:54,270 --> 00:40:02,890 So the Genesis G.K. of ANOC is the minimal possible genesis of a cipher surface for K. 321 00:40:02,890 --> 00:40:07,170 OK, so just as you know, so before when we're looking at crossing number, 322 00:40:07,170 --> 00:40:13,260 we looked at all possible diagrams of the knot and minimised the number of of crossings here. 323 00:40:13,260 --> 00:40:18,240 Instead, we look at all possible surfaces, cipher surfaces that the knot bounds. 324 00:40:18,240 --> 00:40:27,450 We look at each of their genesis and we take the minimum. And the resulting thing is the genesis of the note, and this has some nice properties. 325 00:40:27,450 --> 00:40:35,220 So in particular, like crossing number or nuttiness, it's the K is the unknown if and only if the Genesis zero. 326 00:40:35,220 --> 00:40:42,960 That's just because if case the are not a LB disk and that has Genesis zero. 327 00:40:42,960 --> 00:40:51,630 Conversely, if the knot has Genesis zero, will the only surface with Genesis zero and one boundary component is the disk. 328 00:40:51,630 --> 00:40:55,830 And if not bounce a disk, it's it's the output. 329 00:40:55,830 --> 00:40:59,070 So we have that property there. 330 00:40:59,070 --> 00:41:08,130 And now, because we're dealing with something more mathematical, something topological, this really does behave well with respect to connected. 331 00:41:08,130 --> 00:41:14,460 So it's possible to prove that the genesis of the sum is the sum of the genesis genesis of each of the knots. 332 00:41:14,460 --> 00:41:22,680 And that is enough to prove that God knows there because if K1 plus K2 is the andnot. 333 00:41:22,680 --> 00:41:32,620 Has Jenna Zero, and therefore each of those numbers must have been zero and therefore K1 and K2 must have been me, I'm not. 334 00:41:32,620 --> 00:41:44,340 And so we've proved that my strategy in the back garden of trying to undo mine not hit home is doomed to failure. 335 00:41:44,340 --> 00:41:53,040 OK, so that's an example of the use of topology in and not theory. 336 00:41:53,040 --> 00:42:10,920 And so you can say, well. What where does the subject go from here and some actually taped when he was writing down his tables had 337 00:42:10,920 --> 00:42:16,620 really quite an interesting perspective on on the subject before he even knew that the subject exists. 338 00:42:16,620 --> 00:42:21,900 So you can ask you, can you classify knots? If so, how many are there? 339 00:42:21,900 --> 00:42:29,460 And you immediately come to the problem, not how do we decide whether to not of the same or not? 340 00:42:29,460 --> 00:42:34,500 So, for example, we are at the very beginning. I gave you the the trefoil and the Figure eight. 341 00:42:34,500 --> 00:42:43,220 And I said, those are different knots. But but how do we know for a fact that the foil and the Figure eight are different? 342 00:42:43,220 --> 00:42:51,080 And Tate realised this, so he says, though I have grouped together many wildly different but equivalent forms, 343 00:42:51,080 --> 00:42:56,270 I cannot be certain that all those groups are essentially different from one another. 344 00:42:56,270 --> 00:43:04,000 So this is slightly convoluted Victorian way of saying that we don't really know whether the Figure eight and the Truffle really are different. 345 00:43:04,000 --> 00:43:10,450 For example. Now, it turns out that we can now proof that they're different. 346 00:43:10,450 --> 00:43:15,220 But you could ask yourself, like, is there a general procedure if I give you two different knots? 347 00:43:15,220 --> 00:43:21,700 Is there a general procedure for being able to decide whether they really are different? 348 00:43:21,700 --> 00:43:28,390 And interestingly, this particular question was taken up by Alan Turing. 349 00:43:28,390 --> 00:43:38,890 So Alan Turing is famous for his work at Bletchley Park, and he's also famous for his work in the foundations of computer science. 350 00:43:38,890 --> 00:43:49,400 And one of his key, his key contribution is that he established that there are fundamental limits to what computers can do. 351 00:43:49,400 --> 00:43:56,600 There are some questions that mathematical questions that computers will never be able to answer. 352 00:43:56,600 --> 00:44:01,730 So, for example, here's a system of polynomial equations. 353 00:44:01,730 --> 00:44:08,580 Does it have a solution in integers? Computer will never be able to answer that question. 354 00:44:08,580 --> 00:44:14,490 But Alan Turing didn't realise that at the time, so he he he knew that there was some fundamental limits to what computers can do. 355 00:44:14,490 --> 00:44:19,830 He could come up with problems that computers would never be able to solve. 356 00:44:19,830 --> 00:44:23,670 He did this by inventing the idea of what's now known as a a cheering machine, 357 00:44:23,670 --> 00:44:31,710 which is this super powerful computer powerful enough to emulate any other computer that we have, including quantum computers. 358 00:44:31,710 --> 00:44:39,840 And he showed that that's super powerful computer was not powerful enough to be able to answer certain questions. 359 00:44:39,840 --> 00:44:45,510 Unfortunately, the questions that it couldn't answer was somewhat slightly artificial at that stage. 360 00:44:45,510 --> 00:44:51,520 And so he was searching for natural problems that computers might not be able to solve. 361 00:44:51,520 --> 00:44:53,500 And so he came to nuts. 362 00:44:53,500 --> 00:45:02,050 So in his final paper that was published in the year of his death, nineteen fifty four called solvable and unsolvable problems. 363 00:45:02,050 --> 00:45:12,550 He examined notes and wrote No systematic method is yet known by which one can tell whether two knots are the same. 364 00:45:12,550 --> 00:45:17,740 So this is basically the same statement that Tate said. 365 00:45:17,740 --> 00:45:22,930 There's no how do we know whether to not say the same or not? So he clearly was envisaging, 366 00:45:22,930 --> 00:45:30,220 and this is a picture from an excerpt from his paper that you what you might do is you might encode your not in some mathematical way. 367 00:45:30,220 --> 00:45:37,690 So, for example, as a union of straight arc sitting inside all three and you'd feed these so you take you to different knots 368 00:45:37,690 --> 00:45:43,990 and you'd feed this mathematical representation of them into a computer and the computer would say yes or no, 369 00:45:43,990 --> 00:45:54,820 or maybe. And it was pretty clear that he was thinking that maybe there would be no such way of doing this, that maybe this was an unsolvable problem. 370 00:45:54,820 --> 00:46:02,940 So it turns out. That it is a solvable problem, and this was proved by this guy here, Wolfgang Harken. 371 00:46:02,940 --> 00:46:14,950 So Harken in the early 60s showed that there actually is a systematic method for reliably determining whether two knots are the same. 372 00:46:14,950 --> 00:46:22,690 So he's he's a very famous mathematician, not just for his work on knots, but also for what he did afterwards. 373 00:46:22,690 --> 00:46:32,140 So after he proved this theorem, he then turned his attention to two plain graphs and in conjunction with Kenneth Apple, 374 00:46:32,140 --> 00:46:37,300 they proved the famous full-color theorem for plainer graphs. 375 00:46:37,300 --> 00:46:43,900 So but before that, he looked at knots and he showed that there was this. 376 00:46:43,900 --> 00:46:48,790 There is a systematic method of deciding whether to not of the same. 377 00:46:48,790 --> 00:46:58,300 And he did this using the theory of three dimensional manifolds, so spaces that locally looked like three dimensional space. 378 00:46:58,300 --> 00:47:02,890 But globally, like curved round on themselves in unexpected ways. 379 00:47:02,890 --> 00:47:07,060 So that's actually my area of research. I work in three dimensional manifolds theory. 380 00:47:07,060 --> 00:47:13,210 And so I find this this theorem very impressive. 381 00:47:13,210 --> 00:47:19,820 It's still highly relevant today. I should say that. 382 00:47:19,820 --> 00:47:27,920 This this method, it's very much a theoretical method, but if you if you if you apply it to two knots, then yes, 383 00:47:27,920 --> 00:47:34,640 there is a computer programme that will decide whether or not they're the same or not, but it'll take the lifetime of the universe to run it. 384 00:47:34,640 --> 00:47:43,030 I mean, even the simplest ones. So it's very, very much a mathematical mathematicians theorem rather than an actual practical result, although. 385 00:47:43,030 --> 00:47:49,630 The search for a practical algorithm to do this is still very much an open problem. 386 00:47:49,630 --> 00:47:56,860 OK, so let me just close by talking about one final question that we don't know the answer to, 387 00:47:56,860 --> 00:48:02,080 which is a really simple question, which is how many knots are the. 388 00:48:02,080 --> 00:48:10,370 OK, so we know, by the way, the infinitely many knots before I go any further, we've actually proved that implicitly today. 389 00:48:10,370 --> 00:48:19,040 So if I take the trefoil, it has Janice won because the non trivial knot in it bounds a cipher surface of Janice one. 390 00:48:19,040 --> 00:48:24,320 Take the sum of two trifles by the theorem about A.R.T. that has Genesis two. 391 00:48:24,320 --> 00:48:33,350 So this is different not and then add on another trefoil as Genesis three, etc. So there's an infinite collection of different notes, provably so. 392 00:48:33,350 --> 00:48:36,920 But how many are there as a function of their crossing them? 393 00:48:36,920 --> 00:48:42,800 They're not in us. So let can be the number of prime knots with crossing. 394 00:48:42,800 --> 00:48:49,530 No end. OK, so we know Cayenne for some small values of. 395 00:48:49,530 --> 00:48:57,080 Here it is. So for example, like an equal zero crossing no0, there's just one, namely we are not. 396 00:48:57,080 --> 00:48:59,190 Then there's none with crossing number one. 397 00:48:59,190 --> 00:49:08,300 There's none with crossing number two, we've already saw that crossing number two really isn't crossing number two at zero. 398 00:49:08,300 --> 00:49:16,590 First, non-trivial, not there is the trefoil, this one, and then by the time you reach 14, there's over a million of them. 399 00:49:16,590 --> 00:49:23,910 A lot of work went into that number, by the way. So how does this how does this sequence go? 400 00:49:23,910 --> 00:49:31,350 Well, pretty clearly, I mean, not provably so, but pretty clearly you're unlikely to get it like a closed formula for it. 401 00:49:31,350 --> 00:49:36,330 But what about how does it grow asymptotically? 402 00:49:36,330 --> 00:49:42,330 So how does Cayenne grow asymptotically? So we don't know, is the answer. 403 00:49:42,330 --> 00:49:48,480 The best we can do is due to Thistlethwaite Samberg and Dominic Welch. 404 00:49:48,480 --> 00:49:53,940 Dominic is an emeritus professor here at Oxford, and between the three of them, 405 00:49:53,940 --> 00:50:01,530 they proved the following very strange theorem that says the Cayenne lies between, well, the top end. 406 00:50:01,530 --> 00:50:10,930 It's twenty seven over two to the power and. The bottom end, it's this given by this very complicated formula, don't worry too much about this. 407 00:50:10,930 --> 00:50:16,630 This is really the dominant term. And this is growing exponentially with a function of it. 408 00:50:16,630 --> 00:50:19,810 So these numbers, they definitely blow up exponentially. 409 00:50:19,810 --> 00:50:29,950 But the precise exponential growth rate is not known, and to be honest, I think we have no prospects of knowing it in the near future. 410 00:50:29,950 --> 00:50:35,890 So that is just one example of some of the things that we don't know in the subject, 411 00:50:35,890 --> 00:50:42,010 which I think is remarkably interesting and intricate and fascinating. 412 00:50:42,010 --> 00:50:56,620 I'll leave it there. Thank you very much. So thank you all for coming. 413 00:50:56,620 --> 00:50:59,980 It's our tradition that we don't take public questions in these lectures, 414 00:50:59,980 --> 00:51:06,820 but the speaker case mark is more than happy to hang around and if you want to come forward and ask him any questions privately, 415 00:51:06,820 --> 00:51:31,922 but I'm sure you'll agree that really was a beautiful lecture, so let's thank Mark again.