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From Wikipedia, the free encyclopedia

Nina Claire Snaith is a British mathematician at the University of Bristol working in random matrix theory and quantum chaos.

In 1998, she and her then adviser Jonathan Keating conjectured a value for the leading coefficient of the asymptotics of the moments of the Riemann zeta function. Keating and Snaith's guessed value for the constant was based on random-matrix theory, following a trend that started with Montgomery's pair correlation conjecture. Keating's and Snaith's work extended works[1] by Conrey, Ghosh and Gonek, also conjectural, based on number theoretic heuristics; Conrey, Farmer, Keating, Rubinstein, and Snaith later conjectured the lower terms in the asymptotics of the moments. Snaith's work appeared in her doctoral thesis Random Matrix Theory and zeta functions.[2]

Nina Snaith is the sister of mathematician and musician Dan Snaith.

In 2014 Snaith delivered the 2014 Hanna Neuman Lecture [3] to honour the achievements of women in mathematics.

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  • ✪ Symmetry - what does it mean and why does it matter?
  • ✪ Random Matrix Theory and Zeta Functions - Peter Sarnak
  • ✪ Taming Riemann's Tiger

Transcription

I'm Nina Snaith. I'm a mathematician. So there are many practical applications of mathematics: to other sciences, to engineering, there's mathematics behind almost anything we refer to as technology. But for many mathematicians, it's not so much these applications that motivate us, but rather a love of the mathematics itself. So what does that mean? Well, symmetry is a characteristic that we humans appear to find very appealing, both aesthetically, for example in art, in architecture, in other human beings, but also we find a high degree of symmetry in many objects that are very functional. If you think about your phone, or your umbrella, or your car, or your house, and the same is really true in mathematics. So if you listen to the words that people who like mathematics use to describe it - words like elegant, structure, pattern, completeness - these are manifestations of symmetry. And whereas in the real world objects can never have exact symmetry in mathematics symmetry is perfect. So it's not hard to understand why some people can see mathematics as truly beautiful. But symmetry is also useful in mathematics. Symmetry can give us an idea whether a new theory is complete and it can point the way to a shortcut to a solution of a problem. In my lecture I want to illustrate what symmetry is in mathematics and to try to show how symmetry has led to breakthroughs that have revolutionised mathematics over the centuries. I'm a mathematician and so I'm here to talk to you about symmetry in mathematics, what it means and why it's useful and why it deserves to be a big idea in science. We see symmetry around us almost everywhere: In nature lots of things are almost symmetric, but none are really perfectly symmetric. We see a lot of symmetry in this flower, but you can see the petals are slightly different sizes, they're slightly different shapes. and so the symmetry isn't perfect. As humans we seem to value symmetry. We find it attractive, we unconsciously attribute to people who look symmetric valuable characteristics and so in man-made objects we find a lot of symmetry. Architects use symmetry to convey some characteristics like grandeur but even this object, which looks very symmetric, isn't perfectly symmetric because a millimeter off here and there breaks the symmetry. Architects obviously use asymmetry sometimes to make a point, so they deliberately break symmetry to express some value like energy or innovation. In mathematics symmetry is perfect, so these graphical representations of abstract objects which have absolute perfect symmetry. They may be shapes, or a function, an equation, an array of numbers, but in mathematics we can have absolute and perfect symmetry. So then the question is: how is symmetry useful in mathematics? Here's a question. Sometimes symmetry can help you solve a question, it can make a hard problem much easier by seeing the symmetry in the problem. So here's a problem: we have a cow, who is situated a this point. The cow is tired and wants to go home into the barn at the end of the day, but she's also thirsty. This here is a river. So the cow wants to go to the river and then to the barn to go home for the night. So she wants to do this: she wants to walk to the river and then go home to the barn. But this cow is lazy and she wants the shortest possible distance to walk to get a drink and then to go to the barn. Some of you may have seen this problem before. I want you to take two sheets of paper, the blue and the yellow one. Wave the blue one if you know the answer to this already, and the yellow one if you don't know the answer. Okay, so that's good - most people haven't seen this before. I want you to take a few minutes and think about this problem. You're trying to find the shortest option for the cow to go to the river and then to the barn. Feel free to talk to your neighbours to figure out how you might solve this problem. Audience discussion Okay, so how are we doing? Wave your blue paper at me if you've solved the problem, yellow if you're still working on it. Okay, we have a few solutions. For those of you who are yellow, let me give you a hint. What if the cow was not at this point, but at this point? So now the cow is here, she wants a drink and she wants to go home to the barn. Think about how you'd solve that problem and then see if that helps you to solve the original problem. Again talk to your friends and see if you can figure that out. Audience discussion Okay, so now wave at me with the blue card if you've got it now, the yellow card if you need another hint. Okay, so well done we've got a lot of blue cards now. Of course if the cow was here, she'd walk in a straight line to the barn and would get a drink at the river. But now if we compare walking this distance and that distance, we see that the purple bit and the red bit are the same length so she walks the same distance. and if you chose to move that point, if you dragged it to any other point in the river, you'd easily tell that this distance would be longer, to go over here say, and since this is the same, that would also make the red distance longer. You can see by using symmetry, the question which you couldn't solve initially was made quite easy and that's true in mathematics in general. Often if you spot the symmetry that's at the heart of the mathematical problem it can make your life a lot easier. Here's another problem. We go through it right here, but I'll leave it as an exercise for you to do if you want. This question is: How do you draw a triangle with the shortest possible perimeter inside this angle here, one corner on the red dot and two corners on the blue lines? I've drawn the blue lines like the river as a sort of hint. You can think about this. The solution is at the end of these notes which will be put on the Big Ideas website. If you want to have a go at them and look at the solution then you can do that. These were two small examples of where symmetry can help us solve a particular, individual problem in mathematics and this happens in mathematical research. Sometimes you find a solution by using symmetry. But also, even more importantly, symmetry sometimes leads us to the most revolutionary breakthroughs in mathematics. If we think about imaginary numbers. So let's start with an equation: y = x^2. So if you plot this, say you choose the value 2 for x, then 2^2 = 4, so we have the point here, (2, 4). That's a point on this curve and if we choose various other x values, we can plot the curve y = x^2, which looks like a parabola. Then what if we ask the question: where is the height of this curve 0? This axis [Y-axis] measures the height of the curve and 0 is right here. Where is the height of this blue curve 0? It's the same as asking when y = 0, which is the same as saying when is x^2 = 0? So in this picture obviously the height is 0 when y = 0 and x = 0. Now what if we drew the picture of this curve? So now we've taken the picture of y = x^2 and just shifted it all down by 1 and so now the parabola crosses the X-axis at two points. If we asked the same question: where is the height of the blue curve 0? Or, where is y = 0? which is equivalent to saying where is x^2 - 1 = 0? If you solve it mathematically you get x = 1 and x= -1 solves this equation. If you look at it graphically, you can see that the height is 0 at -1 and that x is 1. Now let's look at this picture. Now we have y = x^2 + 1 and that's shifted that curve upwards so now it looks like this. We can ask the same question: when is the height of the blue curve 0? Or otherwise when is x^2 + 1 = 0? A lot of these quadratic equations have two solutions like we saw on the previous slide, two places where the curve cuts the real axis. Really by symmetry what we'd like is for all quadratic equations to have two solutions. But in the 1500s mathematicians only knew about real numbers, that is the numbers that lie on this number line here and so they knew that even if you square a negative number, so even if you square -1, you get a positive number [1] and so they couldn't even imagine a number that you would square to get a negative number. So if we rearrange this one, we have x^ 2 = -1 and so they couldn't even imagine what kind of number would you square to get -1. So they had a problem and they could solve it in one of two ways, they could say that this equation has no solutions, because they knew of no number that solved this equation. Or they could invent new numbers that would give you the desired two solutions to this equation. If you just do some manipulation with this equation here, to solve for x, you get that x = +- sq(-1). [or x = +- (-1)^(1/2)], whatever that means. Mathematicians were confronted with this. To see what they did let's consider this problem here. We want two numbers that when you add them together give you 10 and when you multiply them together give you 40. With our modern mathematics that we know, we know how to solve this. We can solve y, we can say y = 10- x by taking the x over there, so in the second equation we can have x times y (where y = 10 - x) = 40 and then we can rearrange the equation so we have a quadratic in x over here and by the quadratic equation we can solve that to get two solutions: x = 5 + sqrt(-15) or x = 5 - sqrt(-15). But in the 1500s, the mathematicians didn't know what to do with roots of negative numbers, so let's just imagine that we accept that there is something, we'll call it "blob", that there is something which is this red blob which when you square it, it gives you -15. So we don't know what it is, we don't understand this thing, but let's say there is some thing which when you square it (when you multiply it by itself) it gives -15. Then we can see that it solves this little problem here because this solution, let's call this one x, this one y, so if we have x + y, then we have 5 + 'blob' + 5 - 'blob'. The 'blobs' cancel, so we get 10. [x + y = 10] If we multiply these together, you have to do all the cross terms: you get 25 (5 x 5) you get 5 x --blob and 5 x blob (they cancel) and you get blob x --blob, but we've been told that all we need to know is that when we multiply these two blobs together, we get -15. With the extra minus sign, you get 25 + 15 = 40 [xy = 40] In the 1500s, Cardano, a mathematician, realised that if you could just accept that there was a quantity which had these properties, then you could do algebra and he could do these problems just fine. It took another 300 years for complex numbers (imaginary numbers) to become well understood and well used but this was the beginning. Now we know that if we have any quadratic equation like this we'll always find two solutions if we allow ourselves to use imaginary numbers. You've probably met the quadratic formula which gives you those two solutions. So for example if we have x^2 = --1 we solve this and we have x= sqrt(--1) or x = --sqrt(--1). So now we have symmetry in the problem it has two solutions. In the process, we've invented a new kind of number. Obviously as the years passed, people became more familiar with imaginary numbers. It turns out you only need to invent one new number, that is the square root of -1, which we denote as 'i', because the square root of -15 can be written as sqrt(-1) x sqrt(-15), or i sqrt(15). We've developed ways to graphically represent imaginary numbers. Imaginary (or 'complex') numbers can have a real bit and an imaginary bit and we can plot them with a real bit on this axis and an imaginary bit on this axis so we can visualise a bit better what a complex number is. Now complex numbers are absolutely essential in modern science. This is a picture of a quantum wave function and you wouldn't have quantum mechanics if you didn't have imaginary numbers. You wouldn't have electrical engineering or digital signal processing or any of the things that technology relies on today. Imaginary numbers are absolutely fundamental in mathematics and other areas of science and we have them because mathematicians were trying to complete the symmetry, to solve the problem and make every quadratic have two solutions. Now let's talk about another kind of symmetry in mathematics. I want you to take your green piece of paper and I want you to make yourself an equilateral triangle, the instructions are on here. If you get it, you can help your neighbour so we all have a triangle. Okay so hold up your triangles. It looks good. If you don't ask your neighbour who has one to help you with it. When you have your triangle put it on your desk and label the corners 1, 2 and 3. Flip it over and label the other corners 4, 5 and 6. It doesn't matter how, just 1, 2, 3 on one side, 4, 5, 6 on the other. So now on your desk you want to imagine you have the outline of a triangle like this. I want you to figure out how many ways can you place the triangle into that outline so that it's in a different position. I want you to wave your blue card at me if you think there's more than one way to put the triangle down, and wave your yellow card if you think there's one or fewer ways to put the triangle down. Keep waving blue card if you think there are more than two ways to put the triangle down in different positions. Keep waving the blue card if you think there are more than four ways. Yellow card if you don't. More than five ways? More than six ways? I don't see any cards. Are there more than six ways to put the triangle down? Blue if yes, yellow if no. So a few people think they have more than six ways. Are there more than seven ways? Let's see. I could put my triangle down with 1 at the top like this. I could put it down with 2 at the top. I could put it down with a 3 at the top. Or I could turn it over. I have backwards numbers where you have 4, 5 and 6. I could have backwards 2, 3 or 1 at the top. So I think there are 6 distinct ways to put your triangle down into that outline. But now it's actually more helpful to think of it like this: Let's pick a reference position for our triangle. So I'm going to pick 1 at the top. If we think of moving the triangle into each one of those six positions, I could pick it up and put it down as it was. That's an action, which results it being in its reference position. I could pick it up and rotate it clockwise and put it back down, or, returning to my reference position, I could pick it up and rotate it the other way and I'm achieving one of the other six positions by doing this. Or, I could decide to flip it over. I could take one corner and I could flip it and get one of the six positions. Returning to my reference position I could again flip any one of the corners. Now instead of thinking of the position of the triangle, we're thinking of doing an action to take the triangle from its reference position into one of those six positions. The benefit of doing this, of thinking of this as an action instead of just a position, is that we've introduced time. Now we can think of a sequence of moves, one after the other in order, to the triangle. I want you to think about a series of questions: What happens when you make two of these moves in sequence? So maybe I take my triangle and from the reference position I make one move, and then without returning it to the reference position I do another move - maybe I rotate it again. Or maybe I start from this position and I decide first I'm going to flip it and then I'm going to rotate it anticlockwise. I want you to think. Take your triangle and see what you can do by doing two moves in sequence, so without returning to the reference position. You do one move, a flip or a rotation, and then you do another one. The questions you can think about are: When you perform two moves, one after another, is there any way to make the result different from one of the original single moves? If you've thought about that one you can move on to the other questions. Take your triangle and have a think about those. Qu. 2: Is the result the same if you do two moves or do them in reverse order? Qu. 3: Can you find any number of moves smaller than 6 that always give you back something in that smaller set? Okay, so what do we think? The blue card is an answer of yes, the yellow card is an answer of no. When you perform two moves in sequence, do you always get back to one of your original six positions? A lot of blues - that's good. You've just discovered group theory. This is an area of mathematics, which can be abstract, but we're going to use the rotations of this triangle to figure out some of its properties. What about the second question? If you do two moves in one order (1 then 2), then you do them in another order, do you always get the same result? Blue card if it's yes and yellow card it it's no. I'm seeing some yellow cards. Let's work this out. Let's make ourselves a table. You'll have to help me with this. First we're going to do a move down here and then we're going to do a move listed up here. 'i' is the identity, so we don't do anything. It's the move where we pick it up and put it down exactly where it was. 'C' is rotate clockwise, 'A' is rotate anticlockwise, 'T' is flip around the top corner, 'R' is flip around the right corner, 'L' is flip around the left corner. I want to fill out this table and we want to know the result of doing first this one and then this one. It's equivalent to doing one single move, and what is that single move that it's equivalent to? You have to help me. If I do first the identity (do nothing) and then the identity (do nothing), what do I get? Audience response: I [Identity/no move] Okay. So what if I do the identity then I do a clockwise rotation? What do I get? Audience response: C [Clockwise rotation] If I do the identity then I do anticlockwise rotation? Audience response: A [Anticlockwise rotation] If I do the identity and I do a flip on the top? I get... ? Audience response: T [Top corner rotation] If I do the identity and then a flip at the right corner? Audience response: R [Right corner rotation] If I do the identity and a flip on the left corner? I get L [Left corner rotation]. Let's do the second row. You might have to get your triangles out to do this. What if I first do a clockwise rotation then I do nothing? Audience response: C [Clockwise rotation] That's just C. What if I do a clockwise rotation then a clockwise rotation? What's that equivalent to? Audience response: A [Anticlockwise rotation] If I do a clockwise rotation and an anticlockwise rotation? Audience response: I [Identity/no move] What if I do a clockwise rotation then a flip around the top? What do I get in this box? Audience response: R [Right corner rotation] I get R? Does everyone agree that I get R? What if I do the clockwise rotation and then a flip on the right corner? Audience response: L [Left corner rotation] What if I do a clockwise rotation and a flip at the left corner? Audience response: C [Clockwise rotation] Is that right? Something's wrong here. If I do a clockwise rotation and then I flip on the right corner, that is the same as doing... What have I done? I'm doing a clockwise rotation and then I'm doing a flip on the right corner. So that's the same as doing a flip on the left corner. If I do clockwise rotation and a flip on the left corner, that's the same as doing T. Now for the nest line. If I do anticlockwise then nothing, I get anticlockwise [A]. If I do anticlockwise rotation then clockwise, I get back to identity [I] Anticlockwise then anticlockwise Anticlockwise then anticlockwise? Audience response: C [Clockwise rotation] Yeah, it's the same as doing one the other way. Now here are the trickier ones so you'll have to help me. If I do anticlockwise and then a flip at the top, what do I get? Audience response: L [Left corner rotation] Anticlockwise and a flip on the right corner? Audience response: C [Clockwise rotation] Anticlockwise and a flip on the left corner? If I do a flip on the top corner and then identity, That's just T. If I do a flip on the top corner and then C (clockwise)? Audience response: R [Right corner rotation] Does everyone agree? If I do a flip at the top corner and then anticlockwise? Audience response: L [Left corner rotation] If I do a flip at the top corner and a flip at the top corner? Audience response: I [Identity/no move] A flip at the top corner and a right corner flip? Audience response: C [Clockwise rotation] C? Do we agree with C? Flip in the top corner, left corner rotation? Hopefully is 'A' If I do a flip on the right corner and then a clockwise rotation? Audience response: L [Left corner rotation] Flip on the right corner and then anticlockwise rotation? Should be T [top rotation]. Flip on the right corner and flip on the top corner? Audience response: C [Clockwise rotation] Flip on the right corner and flip on the right corner? That' should be 'I' [Identity] Flip at the right corner and flip at the left corner? There's something wrong with that. Can we check this one? If you do the right corner and then the top corner? Can someone do it for me? Audience response: A [Anticlockwise rotation] So we think this one's A. This one's C, let's see what we get in the last line. If we do left and identity that's left [L]. If we do left corner flip and clockwise rotation? What do we get? Someone said R, can we check that? Audience response: T [Top corner rotation] Someone says T, that sounds like it could be right. Left corner and then anticlockwise? Audience response: R [Right corner rotation] Left corner and then flip at the top? If we flip at the left corner and then flip at the top? Do we all agree on A? So we have an error somewhere else. Is this one A? Well something is wrong here. Let's do this one. If we do the left corner and the right corner what do we get? If we do the left corner and the right corner what do we get? What do you think? Try it with your triangle. We flip at the left corner. We flip at the right corner. Okay so there's a problem somewhere. Can we check this line here? The line with the right corner. So if we do a right corner flip and then we do a top flip. Then this one is C according to me. Do you guys want to confirm that? That hopefully means that this one is... So that means that this one has to be different. If we do a right corner and then we do a left corner, Audience report their findings What do we think? Those two? Okay, are we happy with that? Thank you - that at least looks right. Okay, so let's assume we've got that right now. So what have we discovered? First of all, we have indeed confirmed that if you make two moves, one after another, then you get back to one of the original six moves. You can see that eventually we managed to fill out this table. Is the result the same if you do two moves in one order or the reverse order? So say we do a T and then a C, or if we do a C and then a T. So those are the same. If we do (let's try find one that's different) a T and then an R, verses an R and then a T. So here if we do T then R, or R then T then we get different answers So doing the moves in different orders sometimes gives us different answers. So it's not like multiplying numbers where you can multiply them in any order, the order matters. This is important in areas like quantum mechanics, the uncertainty principle is due to this issue that you can't always multiply two objects in the same order and get the same thing. What about the last question: Can you find any number moves smaller than six, smaller than the full set, that always gives you something back within that set? Can you look at this picture and see if that happens? Audience response: I C A Exactly. So if we take that top corner, then we can see that if you perform in any order two operations on I, C or A and then a second one, you always get back I, C or A, you never get into the region where you have a transformation from a flip so the rotations form a group on their own that you always get back the elements from that same group. So like I say, we've discovered group theory, which means that when we perform these operations, you can perform any number of them and you'd get back something that's equivalent to one of the original operations. We have an operation which does nothing, and we have operations which sort of restore you back to where you were, like if you do a C then an A then you get back to the original position. So we've learnt what symmetry means in mathematics, and that symmetry can be useful, not only for solving individual problems, but also some of the biggest breakthroughs in mathematics have come about because mathematicians are looking for the symmetry in the problem and trying to complete it the symmetry in the problem. So that's all for symmetry in mathematics. Next week you'll hear from Mark about symmetry in Physics, and so thank you very much. Applause

References

  1. ^ [1]
  2. ^ Nina Claire Snaith, Mathematical Genealogy Project
  3. ^ "Hanna Neumann Lecture r".
This page was last edited on 28 April 2019, at 22:12
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