In Our Time - Emmy Noether

Episode Date: January 24, 2019

Melvyn Bragg and guests discuss the ideas and life of one of the greatest mathematicians of the 20th century, Emmy Noether. Noether’s Theorem is regarded as one of the most important mathematical theorems, influencing the evolution of modern physics. Born in 1882 in Bavaria, Noether studied mathematics at a time when women were generally denied the chance to pursue academic careers and, to get round objections, she spent four years lecturing under a male colleague’s name. In the 1930s she faced further objections to her teaching, as she was Jewish, and she left for the USA when the Nazis came to power. Her innovative ideas were to become widely recognised and she is now considered to be one of the founders of modern algebra.With Colva Roney Dougal Professor of Pure Mathematics at the University of St AndrewsDavid Berman Professor in Theoretical Physics at Queen Mary, University of LondonElizabeth Mansfield Professor of Mathematics at the University of Kent Producer: Simon Tillotson

BBC Sounds, music, radio, podcasts. Thanks for downloading this episode of In Our Time. There's a reading list to go with it on our website, and you can get news about our programs if you follow us on Twitter at BBC In Our Time. I hope you enjoyed the programs. Hello, Emmy Nurtur was one of the great innovative mathematicians of the 20th century, and her ideas have underpin much in modern physics and algebra. She has been greatly undervalued.

She was born in Germany in 1882, and with Nurta's serum, she showed some. scientists how to think about nature in a new way, to build ideas such as those that led to the search for the Higgs-Bosin. She helped Einstein understand some of the issues in general relativity. The narrative theorem has been described as the cornerstone of modern subatomic physics. She achieved so much, yet for years she was barred from teaching at universities as she was a woman. And when she finally had a paid post, she was sacked from this under the Nazis as she was Jewish. She died in exile in America in 1935 before her work was fully recognised.

With me to discuss Amy Nurtr, our Elizabeth Mansfield, Professor of Mathematics at the University of Kent, David Berman, Professor of Theoretical Physics at Queen Mary University of London, and Colver Roney-Dougall, Professor in Pure Mathematics at the University of St Andrews. Colver, what signs were there early on that she had this genius, as it turned out to be, for mathematics? Well, contrary to the standard ideas about genius, surprisingly few. So Emmy was born to a very prosperous Jewish family in the university town of Erlangen, just north of Nernberg. Her father was a professor of mathematics. She was the eldest of four children. Her education was standard for a middle class girl at the time.

So she went to girls' school and was taught German and English and French and arithmetic and how to play the piano. She apparently wasn't very good at playing the piano. And the only real early hint of any kind that she was going to go on to achieve as much as she did was that she was apparently good at solving logic problems posed at parties. But otherwise her early childhood was completely unremarkable. When she finished at school, she went to do the only kind of further education which was open to women at the time and trained to become a high school teacher in modern languages. So she trained to become a teacher in English and French and graduated with her. that in 1900, having got marks of very good. So she'd done well at that, apparently slightly let down by her large group classroom teaching. But in general, we can say that there's really

nothing pointing at that point to the fact that she's going to be such an influence later on, except for growing up in a very academic environment. Yes, and her brothers, one got graduated in maths, the other in chemistry and the other was very ill. So what was this revelation that she was going to switch at quite a late stage? Quite a late stage. So as late as 1898, the University of Erlangon had been declaring that the admission of women would overthrow all the academic order. But in 1900, the law changed and it became possible for women to at least audit classes at university with the permission of the lecturer. So she had to get permission from each individual professor to sit in at that class. At that point, she joined the University of Erlangen. She was one of the permission. She was one of the permission. of only two women amongst a thousand students. And initially, even then, was studying modern languages and history as well as mathematics, but slowly started switching to just studying maths. It seems to be, it seems to become soon obvious how very good she was going to be at it.

She caught the attention of David Hilbert and his group at the World Centre of Mathematics as another German city. Can you explain that? Absolutely. So Hilbert was arguably the the most famous mathematician in the world at that point. And he had initially become famous by solving a problem in an area called invariant theory. I won't go into too much details about what invariant theory is, but it's all about trying to find sets of polynomials, which can be used to describe infinite sets of polynomials. Now, the person who had posed this problem in invariant theory

with somebody called Paul Gordan, who was one of Emmy Nurtas' father's colleagues at the University of Erlangen. And he agreed to take on Emmy as his PhD student at the point when finally one was allowed to start studying for a PhD. And so she started working with Paul Gordan on this topic of invariant theory, which was very close to Hilbert's heart.

Now, Hilbert, by this point, had been listening to ideas of Einstein and had become very interested in theoretical physics. And there was a problem which we will, I think, discuss more about later to do with general relativity. And Hilbert decided that what he needed was an expert in his old original field of invariant theory to come and work with him on that.

In her thesis, Nerta had calculated more than 300 invariants, so more than 300 of these very complicated polynomials, which had won her some renown. Her thesis had been published in one of the top. journals at the time. And so eventually Hilbert said, well, why don't we get this woman to come and work with us? Thank you. Elizabeth Mansfield. So can you refine or add to

the development that NERDA was showing in these early studies, particularly in this doctorate? She was really known for being able to do these vast calculations, which were extremely mysterious to everybody. Why were they so mysterious? Well, see, modern mathematicians think in terms of algorithms. You know, you have a starting point, you have a process and you have and you design your algorithm so you have a guaranteed output. But there doesn't seem to be any of these characteristics of this theory. They just seem to produce these invariants

almost out of nowhere, out of a rather mysterious theory which just goes by the name of transcendence, which tells you very little. Yes, tells you very little. And she was apparently, by all repute, absolutely incredibly remarkable at producing these things. But even later, her on. It's still really quite mysterious how she and Gordon did it. But Hilbert spotted this and he was as important as almost told us

he was and got her on side as soon as he could. That's right. They were very worried about a particular conservation law and general relativity. You know, where was the conservation of energy? Let's stick with her teaching there. Hilbert took her up. He thought she was that good, but she couldn't get a teaching

post. Well, she taught under other people's names. She taught under her father's name and she certainly supervised PhDs under other people's names. She was just probably, she was very social. It comes across that she really talked to people a lot, that she liked to be involved, she participated in seminars. And so she probably was just a person who was there who was just very good and very able. Was there a sense in which she felt thwarted and couldn't get on as she wanted to do because of that? The record shows that she actually seemed to be very mild in the face of what we would today consider a huge injustice

and that she was more active in pursuing the injustices against her students, her female students. And she wasn't paid for a lot of the work she did there, no, the family had money. When she went to America, she had a grant for a while and they were in the middle of trying to get her a salary when she died. So she was largely unpaid. And did, was it not just Hilbert, did the rest of them take her seriously as being? Oh, yes, yes, absolutely. Certainly Einstein and Vile and other people who... Certainly she always did have

a group of people who were very supportive. It was only the broader environment that was so misogynist. David Berman, what was Einstein working on that was to become important for NERTA? Einstein had already done the special theory of relativity and he was asked in 1907

to write a review about it. He was already famous then. And then in constructing that review he realized it's very limited extent. So he realized he needed a more general theory, which later we'd call general relativity. But to do so required a lot of mathematics that either he didn't understand or was not even yet developed. Einstein, after many years, started to correspond with Hilbert, who we've heard about. And he saw Hilbert's mathematical knowledge and his expertise in particular things, very useful.

So around the time of this correspondence was exactly when Hilbert realized what he needed. did was help himself and that's why he invited uh nerter to come to gertigan so there was a big push at the time to extend the ideas of relativity that Einstein had first come out with to understand relativity as a theory of gravitation and also i think in a bigger sense to understand these things of conservation laws conservation of energy that something we take for Granted, you know, in the theory of Newton, that's the theory of changes, of forces, of accelerations, of things altering and changing. But then people realized that there were these quantities that nature used that never changed,

energy, momentum, and other things. So rather than focus on the changing things in nature, this constant things in flux, instead science started to look at the things that did not change the conserve quantities that no matter what happened would remain the same but it was a mystery what these things were in general and why they even existed and then when they tried to construct the theory of generality the most basic thing why we can't destroy or create energy didn't seem to be part of that theory

and that bothered Hilbert that bothered Einstein and that was why nerter got involved So can you just explain what is this conservation of energy? It's something that we've all seen that we don't get energy from nowhere. We need to get it and just transform it from one form to another. So when I run, I convert the energy stored in my muscles or wherever into energy of motion. When I create a fire and create heat, I do so by releasing energy stored in the chemical bonds. So although there can be very complicated things going on,

in the end, nature has some bookkeeping where what it keeps constant is this thing called energy. And why had that been unspotted for so long and how did she arrive at it? People knew, of course, that energy was conserved, but it was more a case of why. Why did nature have these things that were kept the same no matter what you did?

So there was energy, there was momentum, something called angular momentum, which is like how when you spin, that you don't stop spinning, how gyroscopes work. And maybe other things, conservation of electric charges. We can't create or destroy those.

So people knew about the existence of these conservation laws, but why they existed at all was a bit of a mystery. So what did she bring to Hilbert that would help Einstein? What Hilbert needed was related to this old work of nurture,

which was to understand in the theory of relativity. We quit relativity, but what was important, what was left the same. So in other words, as you change things, and in particular, relativity was about changing things. We could change the coordinates that we used to describe science and physical systems. But what would stay the same as we changed the coordinates that we used? And what NERTA did was help Einstein and Hilbert answer that question.

What were the things that would remain the same? even if we change the coordinates that we used to describe that physical system. Can you give us some examples of that? Very simply. If we think about the laws of physics and we say it doesn't matter where you are, but the laws are the same, then we have something called conservation and momentum. And if it doesn't matter when you look at the laws of physics, they stay the same. Then we have conservation of energy.

the laws had existed before but why had not been explained and what NERTA did was give us a bridge between two things one these laws of conservation the things that have got to stay the same

and the other thing was this idea of symmetry that we could change something in the laws and the laws would remain the same and that bridge between symmetry in the laws of nature and the fact that there were quantities

is that we will measure and use, like energy, had not been done before. Do you want to take that up, symmetry, call it? Sure. So the thing with what NERTA did in some sense was when we think about the world around us and the laws that we use to describe the world around us, if I tell you how a ball is going to roll down a slope in London,

it would be exactly the same if the ball was rolling down the slope in Edinburgh, say. So it wouldn't be affected by where the experiment was taking place. Rolling a ball down a slope on Monday is the same as rolling a ball down a slope on Tuesday. So these are symmetries of the way in which we describe the situation. It doesn't care if you move the time forwards or backwards. So we say that it's time translation invariant. It doesn't care if we move where the experiment is happening by 200 yards. So we say that it's space translation invariant.

And it also doesn't care for this ball rolling down the slope example, whether the slope is currently pointing to the north or pointing to the south, or pointing to the south, so it doesn't care if we rotate where the experiment is happening. Now, what NERTA showed is that whenever you have a system describing the world, which has quite a lot of symmetry but not too much symmetry, and I'm hoping that one of the experts around me will pick up more on that later, but quite a lot but not too much, then to every single symmetry, something must be conserved.

And it was really radical her theorem because it applies all the way from physics at the scale of the universe, so galaxies moving around right down to physics at the scale of what's going on inside the atom. It's one of the very few theories that goes right the way from the very, very big, all the way to the very small. And it says that if your law has a symmetry and not too many symmetries, then something is conserved. Conversely, Einstein's theory of general relativity had too many symmetries. And that is exactly why they weren't able to get conservation of energy working properly. Is there any big reason why that hadn't been discovered before? I think the conservation laws that we knew about fell out so easily from the physics that was known

before that nobody had thought to ask quite why these ones and only these ones. Elizabeth. My understanding of physicists from the outside looking in as a mathematician is that they love these things to formulate a physical model in terms of something that minimises something. and minimises the amount of effort that the system has to undergo in order to move or to operate in some way. And so her theorem came out of this, almost this, what I view was a kind of physicist theology,

that the universe will minimize something. Sort of like a kind of physical Occam's razor. You know, the simplest thing, the particles will travel the least distance. The light will bend in such a way as to take the least time and so on and so forth. And if you formulate the physical models to satisfy this kind of principle, then you can prove that the symmetries give you the conservation laws. So it was the formulation of the physical model as a minimization problem, and then quite routine calculus operations give you the conservation laws

from exactly the same kind of invariances that they're talking about. So can you just try to explain, again, why symmetries give you these laws? The symmetries give you the tremendously important physical consequences. So the mathematics that goes into, you know, it doesn't matter whether I do it today or tomorrow, seems a little bit subtle. You can formulate that mathematically. And her proof of the conservation law is completely constructive

and uses quite elementary calculus. So this is to me a feature of absolute wonder that I can teach this theorem, the mechanics of the theorem, to undergraduates, even though the consequences are really quite profound. I think one of the things to realize is that physicists had to ask the question. Theoretical physics is very often

about realizing all of the things that we take for granted and never thought to ask about. And what was brilliant was that NERTA provided an answer to a question which everyone took for granted that there should be these conserved things like energy was something which people sort of were aware of but they never went why. And once you ask why, and then make a link to something as basic as symmetries,

again, symmetry is something every takes for granted. You expect the laws of physics to be the same tomorrow as today. You expect the laws of physics for me to be the same as you. It's the fact that those very simple rules that we can change something, like where is the origin of which we formulate our physical laws, is linked to this thing. Conserved energy, conserve momentum, which people have observed for years, that was the miracle.

And can you describe, to listeners, how this idea of hers was received and broke through? Well, initially, it was a key part of Einstein's understanding of general relativity and Hilbert's understanding of general relativity. And both of those people were very supportive. They wrote about her contributions to others. There's a fantastic quote by Einstein saying, well, you know, if the easiest way to resolve this is just have NERTA to explain it to me. So there is plenty of evidence that once she had this insight,

it really helped the big guns of Einstein and Hilbert. When this sort of thing was happening, what effect did it have on her and her standing? Of course, when you've got the support of people like Hilbert and Einstein, it helps a lot. And very soon she was then allowed to teach in her own right. Yes. And go on and do more work.

I'd just like to push at this because obviously I know very little about this and I assume one or two of our listeners are in the same case as myself. In what way was it groundbreaking? I've spoken about how it linked and provided this bridge between the two worlds of things that remain the same and symmetries.

But I think what it really did and why it came up again, maybe 50 years later, was that it provided a vocabulary, a way of physicists to talk about the world, world. Not so much in terms of equations of change and all the things like that, but in terms of talking about when we describe the laws of physics, describe their symmetries. So it's saying, what should you talk about when you talk about nature? You talk about the symmetries that nature

have and then NERTA will do the rest because they'll give you the laws. And so now, even today, I was talking to a colleague and what do we talk about the symmetries that were present in the system we were thinking of, and then NERTA's result will do the rest for you. That's why it was groundbreaking. It gave us a different way to see the world, not of changes, but actually of things that were preserved and their symmetries. When was that acknowledged, Colba? So internally in Göttingen, she used this paper in which she'd proved these symmetry laws to submit for an exam that she was finally in 1919 after the end of World War I allowed to submit. which was the exam to get her habilitation, it's called in German,

which is the exam that enabled her to finally teach. So as soon as she'd done that, she got this permission to teach, and finally that meant that for the first time she became her own boss and she could actually start doing the research that she wanted to do. Which was? Pure mathematics. She didn't do any physics again after that. So she became her own boss and basically left all of that behind her in quite a radical move. But what was her mathematics? One presumes that was radical too.

Yeah, so she proceeded to change the language that we use in large quantities of pure mathematics. So let me try and describe a little bit about the area of mathematics that she moved into starting in 1919. So it's an area of mathematics which is basically due to her. The definition isn't due to her, but the fact that it's important is due to her, called ring theory. So before I try and tell you too much about what ring theory is, let me give you an example of a ring, because hopefully it's reasonably friendly. The example of a ring I want us to start with is the whole numbers, positive and negative. And what am I allowed to do with the

whole numbers? Well, I can add any two whole numbers and get another whole number. That's fine. I can subtract any two whole numbers. And since I've allowed myself the negative numbers as well, I stay within the whole numbers and that's fine. And I can multiply any two whole numbers and get a whole number. I can't necessarily divide. So we're not assuming we can do division. So two doesn't go into seven if I want to stay in the world of whole numbers because I'd get three and a half and that's not a whole number. So I can add, I can subtract and I can multiply.

As soon as I can do those things, subject to certain rules about things making sense, then I have something called a ring. Now, ring theory had come from two very, very different places of mathematics. On the one hand, it had come from geometry. The other place where ring theory had come from was attempts to prove Fermat's last theorem. So Fermat's last theorem had been jotted in a notebook by Pierre de Fermat back in 16 whatever, and it asserted that, for example, I can't solve the equation X cubed plus Y cubed equals Z cubed,

all in whole numbers. And that had given rise to some rings that looked very, very different. Now, what NERTA did was realize that these were both examples of the same structure and that she could prove theorems in far more generality with far simpler proofs than the people in these two separate fields had been doing before. Elizabeth, how did these developments affect the history of mathematics, the course of mathematics?

They were profound. I mean, if you talk to algebraists today, reading the original papers, they're really quite shocked at the originality and at the point of view took a long time to sort of ameliorate them and digest them, the originality was so striking and the way she wrote down her ideas was so original

that it really did, you know, even modern algebraes find it really quite profoundly, almost disturbing. Those are big words profound and disturbing. Yes. Is there any way, no, it's fascinating. Can you just give us some, put some meat on that? Well, mathematicians today are trained to be very rigorous to start off with an axiomatic system

and proceed in a very logical manner. and she really I think that she thought in a very original way and she had a way of talking about algebraic objects as though they were numbers and she had and she would manipulate them and do things to them

as though they were these whole numbers and this was really quite shocking so it took a long time Why do you use her a shocking? Just because they were used to thinking about things in a certain formulaic way and she was using a little bit of a different language She was doing things that were unexpected

and obtaining results that were striking. So these innovations, mathematical innovations, do take a while to simmer out, particularly because of the demand for rigour and to have a very firm, logical foundation to it. So if somebody's being a bit original and they're putting things together in a very different way, it takes a while to make sure that those logical foundations are there,

but they were there. And how did she send them? these off to be these ideas of her to people in her group to be checked to be value? Oh she certainly was engaged in all of the normal processes so she was certainly talking to people and she was writing papers

and she was giving lectures and talking to students so all of the normal things that allow mathematicians to develop and mature their theories were there for her. So it was out on but it wasn't David it wasn't taken up very vividly very early was it

this idea and these ideas in this thing which is now called Nurtus theorem, which was this work with Hilbert and Einstein, then it became almost forgotten. It was a thing that some people knew about, but it took until the 1950s and 60s to people to rediscover their importance, which only came about

because they were trying to formulate the laws of fundamental physics for particles. What happened? Why was it delayed? To pick up a word from Elizabeth, why was the shock so delayed? It's a very good question. It's hard to know whether it was sociological or scientific. I think the models of particle physics that people had

didn't need Nertes theorem quite at that point. But then in the 1950s with the work of Younger Mills and then later in the 60s, when people started to try and understand subatomic physics in a certain way, They realized that language of symmetry that I mentioned before was crucial. And I'd like to have just one more thing. Even now, people think of there is this thing called Nurtus theorem, which is this idea of this conservation law I've mentioned.

But in the same paper, she had another theorem and actually it describes something which are now called gauge theories. And gauge theories are the foundation of all of the laws of physical. that we know in the standard model of particle physics that's tested at CERN, it is the way in which we think about the world. What's amazing is even today, most of my colleagues aren't aware that the first person to understand nature is a gaitre's eminerta. Well, some of the most influential writers in the West didn't cite her work at all.

So, for example, the Nobel laureate, Eugene Vigno, cited everybody around her, but not her. Why is that? Was it deliberate omission or was it because she didn't get published? It's very hard to say it was deliberate. It is the condition of being a female academic that you're a little bit invisible. The paper itself has no examples and it's expressed in extreme generality. So you can say that there's a variety of different reasons. And also she had turned her own attention to algebra and left the mathematical physics behind. So she didn't advertise it herself, even though it was her habilitation.

subject. So there's a variety of different reasons. And, you know, when the world was in chaos as well. Oh, well. Yes, so with her. Colman. Yeah, I'm feeling very proud as a mathematician at this point in the conversation because her work in mathematics was immediately recognized. So the pure mathematics that she started doing after proving Nurtus theorem in physics had a massive impact. Not so much directly by her own publications, but she became someone who drew many, many young mathematicians, so nowadays what we'd call postdoc phase, from all over the world to come and

work with her. So she had a school of algebraists working underneath her in Gertigan. She wasn't formally anybody's boss, but she was so inspirational in the way that she taught them that they would come and just spend semesters with her whenever they could. And one of them in particular, a Dutch mathematician called Vandevaden, based a book called Modern Algebra at that point, which came published in 1931, so not very long after Nerdy Vardin. had done the work on Nurtur's lectures and explicitly said that his book was based on work of Nurtr and of somebody else called Emil Artaan. And that book became an instant classic.

So writing 40 years later, the famous French mathematician Diodonne says he can remember the day in 1931 that he first got his hands on a copy of Vandevarden's modern algebra and how it had changed his life completely to start reading this book. And one of the fascinating things there, I was actually really, reading van der Varden's book on the train on the way down to London, the order in which he presents the topics, the choice of topics, the choice of what to teach and what to leave as an exercise,

are exactly how every single mathematics department teaches pure mathematics today. So in that approach to what we put in, what we leave out, what's basic, what we derive, NERTA through her students, transformed pure mathematics instantly. It informs what you've been doing intensity for the last 10 years.

Oh, I've been working on the mathematical physics theorem, the conservation laws for the last 10 years, trying to embed them into the computer simulations, yes. But you find the same influence as permeated to the people around you and the work you do? That theorem is just so key, her mathematical physics theorem, and to try to generalise that, to the kinds of mathematical objects that a computer can handle,

that's been what I've been very involved with. and it's not just engineers and scientists and weather forecasters who need her theorem, it's also video games because anybody, you don't need to be a scientist, look at a video game action and be able to tell if that looks like a reality or not. So, you know, those symmetries have to be embedded

into how the computer produces the animations. So it's actually kind of key. It seems to me, David, that she did so much and it's so important, why is it taking so very long for how to surface. I think there's one other aspect perhaps. And it's, you know,

I was thinking about this when I was preparing for this program. It is hard to explain what the theorem is. When people even talk about something like relativity, trains get involved, watches are moving, light beams are shining, and there's all these marvellous thought experiments that one can do. But there's something so general and so deep about Nurtus theorem. just explaining what a symmetry is of a law of nature,

just explaining about these kinds of quantities is not something that's immediately appealing. So it's very deep and profound for the professionals, but very hard to explain to the general public because it's not intuitive and it's so general. And at this time, to come back to the life for a moment, she's battling away, first of all, as a woman, and then the Nazis come into power,

and she knows she's in trouble. people are leaving and she leaves and goes to America. Can you take that up a little? She went to Brinmore College where they were very welcoming to her. They seemed to appreciate her. They liked her a lot. I think she had a small stipend from a...

She had a small grant from a foundation and they were in the middle of trying to get her a permanent salary to be there because they really did like her a lot. But when she died, it's clear from her letters that she did feel that she was in exile and that she would have liked to have

gone back home. But the Americans at the time really did take to her and were very welcoming. They were very proud of her. That she was there. She died comparatively young at about 53, didn't she? Yes. Sorry. Yes, so she died very, very suddenly. She had developed an ovarian cyst. So it wasn't expected to be a major operation. She had even been joking to her female postdoc at the time that she was looking forward to losing a little bit of weight when the

operation was carried out and had arranged with the surgeon that whilst he was in there, he might as well take out her appendix as well. She must have had a little bit of fear that she was going to die because she had arranged for certain of her belongings to be given to friends if she did die. But she had the operation, it all seemed fine, and then four days later she suddenly developed a really high temperature and died. So this was really at the peak of her powers. One big recognition thing that we kind of skimmed over along the way was in 1932 she'd been invited to give a plenary address at the International Congress of Mathematicians. Now, this is the big conference for mathematics. It happens every four years. It moves around the world. She was the

only female plenary lecturer at that conference out of about 800 people present at the conference. That was 1932. The next female plenary speaker at an International Congress of Mathematicians was in 1990, to give you an idea of quite far, how far ahead of the rest of the world. was in her development at the time. So the Americans had been incredibly excited to have her, was struggling slightly with what to do with her, because in some sense she was too advanced for most of the mathematics that was going on at Bryn Maw.

She wasn't going to be giving undergraduate lectures there. There wasn't much chance you'd get into Princeton. Most of the princess where no female principals were allowed to exist. She was invited to give some lectures there, but she writes in her notes that she did not feel welcome, and they were not interested in anything female. So that was the great shame Because there were people there who were

That was where she should have been That was where the people on her level were David And I think one of the things to bear in mind Is that what made her move Was this law in 33 that came in Which said you couldn't teach at universities If you were Jewish

What's amazing is It took less than a year For her to find a position in America That allowed it to move There's something about her standing already In 1933 that allowed her to move, it's not something a very ordinary academic

would have been able to do and many were still trapped in Germany. So already by then you can see that the fact that she could go to America was an indication that she was already well received and well appreciated by the mathematics community. You mentioned that Brinmar wasn't a place really on her level.

So what did she manage to do while she was in America? I think that she was just so self-motivated and self-generating of mathematics. and she could talk to a lot of different people. So she found people to talk to. She certainly talked to people at Princeton. She just didn't feel welcome generally there. To me, she just comes across as being so happy to do mathematics

that she would have done it anywhere. She was busy building up a group around her at the point when she died. So she only had a year and a half in America before dying. She was continuing her work with Richard Brower, who went on to form a completely new area of mathematics called modular representation theory in 1937, so just two years after she died. She was beginning to build up. She had one post-doctoral student.

She had a PhD student who managed to finish in that year and a half whilst she was there. So she was certainly hitting the ground running. And she was being recognised in more and more different areas of mathematics for having injected algebra into bits of maths where it really wasn't expected at that point. So let me give an example from just before and when she was moving to, to America. There had been some visiting mathematicians, a Russian mathematician called Alexandrov

and a German one called Hopf, who came to Gertingen, not so long before she was forced out. Alexandrov also tried to get her a job in Moscow, but it didn't come through in time. And they were introducing NERTA to the discipline that we now call topology, which is the study of properties of shapes

which don't change when you're allowed to stretch the thing as much as you like and twist it as much as you like, but not cut it. And the big question for topologists is, how do I tell the difference between two things? So I'll give you a concrete example. If I consider the surface of the football, so I've got a sphere and just the surface of that, and contrast with the surface of a donut. So I've got just the surface of it, not the interior. How am I going to prove that those are different surfaces? So we have some numerical invariants called betty numbers that tell us how to tell them apart. The first betty number is just,

is this in one piece or two? In both of those instances, there's only one of them. So that doesn't tell you very much. It's one football and one donut. But then the next Betty number says, in how many ways can I cut a circle around it and leave it all in one piece? Now, if you imagine cutting a circle on a sphere,

wherever you do, you're going to be pulling a bit out. It can't stay in one piece if you cut a circle on your football. You're going to take a tile out. Whereas on the donut, you could not only score right around the top, leaving a floppy disk, you could also score a vertical circle, so chopping it into a cylinder. Now they explained this to Emmy Nurta,

and she thought about it for a bit, and she pointed out that there was some algebra going on there. And in describing to them how these numbers, which they were saying, so for the sphere we had one sphere and no ways I can cut a circle, for the donut we had one donut and two different ways I could cut a circle. Those numbers are actually telling you something about some algebra, and having got this algebra, there's now a field called algebraic topology, which basically was born out of a chance conversation that Emmy Nurseer had with two visitors,

immediately started having results. So one which was published about the time that Emmy moved to America, tells you the startling fact that at any given point in time, there are two antipidal points on the surface of the world, which are the same temperature and have the same air pressure, always, opposite points on the surface of the world. beautiful little theorem that came out basically from a chance observation of Emmy Nurta.

You look terrible pleased about that, Elizabeth. I do. I've studied this theory quite extensively, and I think it's absolutely beautiful, and I've... And I actually use this theory in all of my various different things that I do, so I'm very happy to... What gives you so much pleasure about it? I don't know. I just find it totally enrapturing,

and I just thoroughly enjoy it. I totally can understand Emmy Noda just being so happy in the activity of mathematics that she could transcend all of the problems of the world around it to do with misogyny and all of the rest. And I can really understand that because I feel

similarly. I'm sure Colbert does as well. David, you're all putting her on a very high level. Are we talking about how she lived a little more and had the world tilted a little bit towards a favouring women? She would have won the Nobel Prize and that sort of thing?

I think she certainly would be enough for something called the Fields Medal, which is the Nobel Prize of Maths. She was an amazing woman and I think perhaps you've got a flavour also from listening to us, the sort of emotional response that her work gives in the sense... Why do you think that is?

It's that, you know, very deep mathematics or deep theoretical physics is about opening your eyes to something that you could never have imagined before and that's what a lot of her work did, whether it's in this theory of topology or whether it's in

theoretical physics, so that you see connections that were just impossible to imagine. And that's and that gives us deep pleasure. And I think that's the emotional side to it. And what would you say a greatest legacy was Elizabeth? Oh, it's not just the fact that she was a genius. For me personally it was her resilience as well as the mathematics. So to me it's both a personal and a scientific legacy that's really right up there. And Colma? As a pure mathematician, it's teaching a

to generalise and look at the not worry too much about whether we're talking about numbers or geometric shapes or anything else, but instead to think about the relationships between things and the ways in which we can combine things. Every undergraduate mathematician in Britain now is taught to start by thinking about combinations and operations of objects,

not to think about the objects in themselves. So she's changed how we think about what pure mathematics is. And as I understand it, David, her work is becoming increasingly influential. Absolutely. So there is probably not a single paper in theoretical physics gets published without a discussion of symmetries. It's the absolute language that we all now use. And so, as I say, whether I'm doing work today, talking to colleagues about concert charges that come out of some hidden symmetry,

that's using NERTA's ideas. It's that thing where it provides us the way of thinking, that language of how. how we talk about the world and the universe. Well, thank you very much, David Berman, Elizabeth Mansfield, and Colver Roney Dougal. Next week, it's Owen Glendower and his revolt early in the 15th century against English rule over Wales. He was captured, he escaped, and he was never found again.

Thanks for listening. And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Melvin and his guests. Yeah, and I didn't get to explain what she actually did with rings. We can do it now. Well, I really wanted to explain. So as one moved from just the integers to these other rings, so I was talking about rings of polynomials

or the rings that have been introduced to solve Fermaz-Lar's theorem, there's various nice properties of the whole numbers that stop behaving. So, for example, in the whole numbers, if I take two numbers like, say, four and six, where there's the biggest number that divides both of them is two. so that's called their greatest common divisor. There's no number bigger than two that divides both four and six. Then it's a theorem going all the way back to the Greeks

that I can write this number two as a sum of multiples of six and multiples of four. So in this case it's just six take four, which is nice and easy. When you move to these other rings, that breaks. That breaks badly. So if I'm talking about polynomials over X and Y, the biggest thing that divides both X and Y is one. But I can't take a multiple of X and add some Y's to it.

made one. That's just not going to work. So what Nurta did was come up with a general theory of when properties like that were going to work in any ring you like. So she had these two families of examples, which were the ones from Fermas-Lars-Thirum, and these polynomial rings, but actually we now know many more than that. And she was able to prove in massive generality statements like, when can I factorize something into primes? When can I find greatest common devisers? And this this clarity of vision was just amazing. But she had the two good sets of examples, and it's amazing how much good mathematics

comes from having a decent example. Yeah. The right example to look at. I wanted to mention a bit about the different sorts of symmetry that are in her two theorems. Because everyone talks about the first theorem, and then there's this thing which is called global symmetry.

So the easiest way to think about it is we spoke about you can change something. So you think about change the way in which we orient the laws of physics, which way we're facing. when we orient the laws of physics. And we all agree we're facing the same way, but we can change,

then that's what we call the global symmetry. And that was what the first theorem is about, and that symmetry is what gives us these conservation laws. But then if we say, well, we could all face a different way, and the laws of nature should still be invariant, there's what's called local symmetries. And that's what gives these things called gauge symmetries, which we base all the forces of nature on, just that idea.

Yeah, are those the ones that are like the people? prediction of the Higgs-Botome one we got into that level of data. Well, effectively, I would say it's about the standard model of physics, which is the thing that's tested at CERN and all those things, is built around the gauge theory principle. And there were probably five Nobel Prizes awarded for that because it took a while to say, what are the symmetries and all these things? But the idea that you could have a symmetry that's local, in other words, you can rotate, but we could all face different ways.

So we all make our own choice, and nature still shouldn't care, that's the gauge theory principle. And that structure, no matter what, is basically what led to how we structure the loss of physics. But it's not just particle physics, it's fluid mechanics and weather forecasting, those exact same local symmetries. And one of the symmetries is called potential vorticity. And if your numerical model doesn't get it right, you do not predict the path of the hurricane. So these things are, they're very professional. found significance for our day-to-day life because of the applications to fluid mechanics as well as particle physics. Oh, and I'd meant to talk about another small, tiny, cute bit of impact. So I'd

mentioned that one of the inspirations for ring theory had been trying to prove Fermaz Las Lest theorem and the facts that you can't divide things uniquely into prime factors like we can in the whole numbers had caused at least one false proof of Fermas-Lost theorem. Well, the very final paper published by Andrew Wiles and a collaborator in 1995, finishing his great proof of Fermat's last theorem and plugging the final gap was called Ring theoretic properties of certain hecker algebras and neither the study of rings nor arguably the study of Hecker Algebras

would have existed without Amy Nurtt. So she developed some of her theory beginning out of attempts to prove Fermat's last theorem. And in fact, when it finally was done, it pointed back to the work that she had done. takes your breath away, isn't it? And she sounds like such a nice person as well. I mean, Herman Valle described it was warm like a loaf of bread,

which just gives you a real image of... I think also having this big school of PhD students is a real measure, I think, of an academic. It's not just that she had her own theory and then went away. She saw that she was going to bring on the next generation. She was going to bring in a whole bunch of people to take the ideas forward. And they loved her. Yeah.

It's amazing when you look at the list of her students. If you look them up, say, on Wikipedia, you'll discover that 90% of them have their own Wikipedia entries because they went on to do amazing things. So there was something about the way that she taught people that really she attracted the best people and then she made amazing things out of them. And also the mathematical opportunities that she gave them

because of what she was doing. Yeah. Is there a sense in which her reputation is overtaking that of some of our contemporaries like Hilbert and so on? Or is that a nonsense question? I don't know I can answer that. Yeah. Because an awful lot of the stuff she did is now,

embedded in how we think and how we teach rather than via direct citation. So there are objects in maths that are called Natyrion after her. But otherwise it tends to go without direct attribution and I don't think that's a sexism thing. I think just she told us how to think and once we knew to think like that, then we didn't bother citing it. Yeah. I mean, it's her, I'm hoping her star has risen in mathematical physics. So it's approaching that of Einstein. But, you know, Einstein will always be that archetypal theoretical physics with crazy hair. And somehow it's not what people expect. But in terms of the people who work on the theories,

the realisation of the importance of gauge theories at that stage, yeah, it's incredible. Well, thank you all very much once more. The producers bursting through the door. Simple question. Tea or coffee? Anybody? I'm okay, actually.

I have tea. Tea, please. In our time with Melvin Bragg is produced by Simon Tillotson. Hello, I'm Tess I'm here to tell you about my podcast, Tess Talks. It'll make you laugh, cry and even question the cultural choices you have historically made. You can subscribe to Tess Talks on BBC Sounds.