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Emmy Noether

Meet the female mathematician that Einstein called a genius! Aarati takes a stab at explaining Noether's theorem and the underlying mathematical symmetries of our universe.

Episode Transcript Arpita: 0:10 Hi everyone. And welcome back to the Smart Tea podcast, where we talk about the lives of scientists and innovators who shape our world. How are you Aarati? Aarati: 0:18 I'm great Arpita. How are you? Arpita: 0:20 I am so good. I have a new mic and a brand new laptop, and I've just had... Aarati: 0:28 Hooray! Arpita: 0:28 So many technical difficulties over the last couple of weeks. So I'm moving up in the world. Hopefully the audio quality is better and editing is faster and I'm really excited to finally have a computer that works. Aarati: 0:42 Yeah, having new technology, it just opens up a whole new world. I feel like whenever I get a new computer, it's just like Oh my gosh, this is the world I've been missing, like where everything runs perfectly and I don't have to fight with my keyboard Arpita: 0:55 you realize like all the things you've been putting up with when your computer doesn't work, they just become normal to you. And then you're like, well, life could be really nice. Aarati: 1:04 Yeah, exactly. It's like life is a dream now. Everything is running so smoothly. It's beautiful. Arpita: 1:09 Totally, totally agree. How are you? What's going on with you? Aarati: 1:13 Okay, so I found another pop culture scientist reference. So I don't know if I told you, but my New Year's resolution for this year was to read more for pleasure. Arpita: 1:26 Nice. Aarati: 1:26 Because I used to love, love reading as a kid. I would get lost in books for hours. And then, I don't know, it just kind of like, petered off, life got busy. YouTube happened. TikTok, like Instagram, all those things. And then I was like, I need to really start spending less time on my phone and go back to reading books. But I'd been out of the game for so long. Like I don't even know which authors I like anymore. And so it's been fun so far, last few months, I've just been kind of like picking up books at random at the library and bringing them home and being like, all right, let's see, you know, if I like this and I saw a lot of recommendations for an author named Brandon Sanderson, who's a fantasy writer. Arpita: 2:11 Cool! Aarati: 2:12 Apparently he's huge and everyone's like in love with his books. So I was like, okay, great. I'm like 15 years late to this bandwagon, but let me see if I can climb aboard, you know? So I started reading one of his first books called The Way of Kings. Um, and, I am 200 pages into this 1, 000 page book. Arpita: 2:36 Oh my god. You really picked a heavy hitter for your intro back into reading. 1, 000 pages. Is this book, like, the size of my head? Aarati: 2:46 It's... yes. And it's part one of a trilogy, apparently. So.... Arpita: 2:52 So you're, you're like, I haven't read in a while. Let me read 3000 pages. Aarati: 2:58 Yeah, pretty much. Well, we'll see. Like, the nice thing about books is that I'm not committed to it. So if I hate it, I can always give up on it. But you know, so far it's really good. I'm really enjoying it. Um, but what really made me excited was, you know, it's a fantasy book. So there's a lot of like, you know, soldiers going into battle and all that sort of thing. But there's also a lot of talk of like, cleaning wounds and stuff. And one of the ways that they prevent infection is spreading the wounds with something called Lister's oil. Arpita: 3:31 Wow, even in a fantasy book? Aarati: 3:33 In a fantasy book, yeah. Like, it's just like someone did their research into surgery and antiseptic practice. I am so impressed. And it's just so random. It's just in there, like, you would never know if you didn't know who Joseph Lister was. It just kind of sounds like another, Oh, this is another fantasy potion. Something called Lister's oil that doesn't actually exist in the real world. Arpita: 3:58 Yeah. It's like subtle. Aarati: 3:59 Yeah. And that would have flown right over my head if we hadn't done that episode, but it didn't. And I'm so excited. So. Arpita: 4:07 That's amazing. I love that. Aarati: 4:09 Do you read a lot? Arpita: 4:11 I do. Um, I mostly just read on Libby on my Kindle so I can borrow books from the library onto my Kindle, which is great. Aarati: 4:18 Oh, how is your Kindle? I was wondering about that because part of the reason I wanted to get into reading books also was to cut down my screen time because I think it was really affecting my sleep. The books don't have that blue light that shines in your eyes and mess with your, you know. With your brain and prevent you from going to sleep. Does a Kindle like help with that? Arpita: 4:39 Yeah, I was a huge Kindle naysayer. I was like, I love physical books. Physical books are the best and I still love physical books, but the Kindle I don't know what magic it is but it's definitely not a screen. Like it is muted. It definitely is much more like a book, um, and I just love how portable it is. And I know that's the whole point and that's so silly, but I find myself, rather than going on social media, if I'm, you know, on Muni or waiting in line somewhere, I end up reading my Kindle as opposed to being on my phone just because it's so portable. Aarati: 5:16 That's so nice. Yeah, like think about lugging around a thousand page book, right? Which it's like at least five pounds. Arpita: 5:21 Yeah. I mean most of my books are not a thousand pages I will just say that but at least in Kindle form you Don't really notice it as much and I read my Kindle every day before bed for like an hour and it's definitely not blue light which is nice. Aarati: 5:34 Yeah, because I have noticed that i'm like sleeping better when I read I still haven't completely kicked the habit of, like, looking at my phone right before I go to sleep. So sometimes I'll read, I'll get tired and I'll look at my phone and mess up my sleep anyway. So yeah, it's still, it's still a work in progress. Arpita: 5:52 For sure. I do think the Kindle was really helpful though. Um, there's these TikToks of people who have this like, completely sloth Kindle setup, which I find aspirational where they have a clamp to hold the Kindle like in front of their face so they can lay down and the Kindle is just propped in front of them. And they use a Bluetooth remote to turn the page so that your arms are tucked in your bed. Aarati: 6:16 They have a bluetooth remote? Arpita: 6:17 And then you just click it and go to the next page. I actually got this for my sister for Christmas and she loves it. She doesn't have to pull her hands out and have her hands be cold. And she said it's the best thing. So I think I... Aarati: 6:29 Best. Gift. Ever. Arpita: 6:30 So I think I might have to get one for myself. Aarati: 6:32 That sounds amazing. There is nothing worse than like trying to figure out how to sit in bed and stay warm, but also have to turn the page. Arpita: 6:41 And then your fingers are cold. Aarati: 6:42 It's like, It's a struggle. Arpita: 6:43 I understand. Aarati: 6:44 Yeah. It's a struggle. Yeah. Great gift. That's a great gift. Arpita: 6:48 Um, great. should we dive into our story today? Aarati: 6:53 Yes. So today's story is actually also a listener request. We got a message on our website from Rachel, and she wrote that she would love to hear about female mathematicians. And we are so not math people. We are biology and health people. Um, I think the last time I took a math class was like, oh, well over a decade ago. Um, so I don't know about you, but... Arpita: 7:23 It's probably even been longer. I honestly, this makes a lot of sense because Aarati texted me earlier this week and said that she was thinking a lot about the science and content because it was so complicated and wanting to get it right. And now that you say that it's math, this makes so much more sense. Kudos to you. You're so brave for taking this on and we're doing it for you, Rachel, wherever you are. We're doing this for you. Aarati: 7:46 Yes. And if I get it wrong, I'm so sorry. I'm going to, I'm going to try and do justice to this because I, I went into this very naively. I have to say, like, I was like, my first instinct was like,"Oh, we're not math people. I don't know." But then I was like,"Oh, but it's a listener request. I really want to do it." And so I talked to my brother who is a total nerd and loves math and physics. Like he, that's his fun time. Arpita: 8:12 You know, I'm just really glad that we have him as a resource. You know, it's like friends of the pod. That's really what we need. Aarati: 8:17 Yeah, it is. He is. Yeah. And so I figured like, if anyone knew a good mathematician for me to do, it would be him. So I asked him and he's like,"You should do Emmy Noether". And I'm like, okay, great. And I looked up her story, just, really briefly and I was like,"Oh, wow, she has a really interesting story. This is great. Awesome." So I started researching her and like researching her story, writing it down, figuring it out. And then I actually started trying to get into what exactly it was that she was known for. And that's my world kind of like... Arpita: 8:49 That's when things took a left turn. Aarati: 8:51 Yes. I was like, what have I gotten myself into? And I was like, trying to do test runs with my family and ask them if they were understanding what I was talking about. And it led to some very like energetic conversations with everyone saying like, no, don't explain it that way. And you have to explain it this way. And, you know, I even talked to my cousin who is a physicist and she helped me. So shout out to my cousin Vasu who helped me. But basically there's a lot of people who are counting on me to get this right. Arpita: 9:24 I am so ex... I'm so invested, and I'm so excited, and I can't wait to see how the story unfolds. I'm so ready. Aarati: 9:32 Awesome. Her story really is amazing, so that's also part of the reason why I didn't want to give up. So, today, we're going to be talking about Amelie Emmy Noether, who very early on just ended up ditching her first name and going by Emmy to those around her and on all of her publications. She was born on March 23rd, 1882 to a fairly well off Jewish family in the small town of Erlangen, Germany. She was the eldest child of Max Noerder and Ida Amalia Kaufman. And her father was a well known mathematician and lectured at the University of Erlangen. And so, naturally, because of this, Emmy and her three younger brothers all developed a strong interest in science and math. Arpita: 10:20 That is so cute. I imagine her with like three little ducklings behind her. Aarati: 10:25 Yeah, basically. And they're all just smart little geniuses, all of them. But Emmy was a girl, so unlike her brothers who are able to just freely pursue their interests in science and math, it wasn't proper for her to do that because she was a lady. So instead she was taught more quote unquote womanly things like cooking, cleaning, playing music and dancing. But other than dancing, which she actually really did enjoy, the rest of what she was being taught wasn't very fulfilling to her. Overall, she was a happy child, very cheerful and friendly, but because she didn't really care about anything that she was being taught at school, she didn't really stand out as a particularly bright student or anything. Her friends and teachers did notice how logical her thinking was and how quickly she was able to solve puzzles and brain teasers, but they didn't really take that to be anything serious. That was all just like, Oh, she's good at solving, you know, this little brain teaser that we have, but that's nothing practical. Arpita: 11:25 Yeah. Like not actually useful to life. Like almost like she's good at games basically. Aarati: 11:30 Yeah, exactly. So, after graduating from high school in the year 1900, she took the exam to be a language teacher for French and English at girls' schools, which I guess would have been a respectable career for a woman, and she passed with one of the highest possible scores. However, that same year, she decided instead to screw gender norms, and she was going to pursue a career in math, no matter what anyone else said. Arpita: 12:00 Love that. Aarati: 12:01 So she decided to attend classes at the University of Erlangen where her father taught. At the time, there were almost 1, 000 students at the university and only two of them were women. Arpita: 12:13 Oh my god. Aarati: 12:14 Yeah, I know. But, As a woman, Emmy was not allowed to properly enroll and participate in the classes. She could only audit them and only if she got permission from the professor who was teaching the class first. Arpita: 12:31 So she wasn't actually even taking the classes for credit or having them count towards a degree or anything like that. She was just Just, just watching, basically, like sitting in. Aarati: 12:41 Well, I'm not 100 percent sure because, um, she was able to take the graduation exam, so I think she did end up getting some sort of degree, but she wasn't able to, like, participate, so I'm not sure where they drew that line. Arpita: 12:57 Okay. Somewhere in the middle. Aarati: 12:59 Yeah. But she fully leveraged the fact that her father was a well respected math teacher at that university, so, like, getting permission to sit in on classes and, uh, participate wasn't really too much of an issue for her. Arpita: 13:12 Yeah, she had an in a little bit. Aarati: 13:14 Yeah, exactly. Like, my dad works here. And he's well respected, too, so. So, after three years, she did take the graduation exam and passed with flying colors, but that didn't really mean much for her, as I said, because she's a woman, so it doesn't open any doors for her in terms of higher education or jobs or anything. She couldn't really do anything with the fact that she passed this graduation exam. So she spent the winter semester between 1903 and 1904 attending more math lectures at the University of Göttingen. Here she met and attended lectures from two very important mathematicians, Max Klein and David Hilbert, and they'll pop up in a minute. So I just wanted to mention them here. In the spring of 1904, the German government issued a decree that women would be allowed to enroll properly at any university within the German empire. So Emmy went straight back to the University of Erlangen to pursue her PhD in math. She worked under the supervision of Paul Gordon, and in 1907 she graduated and published her thesis work entitled, On Complete Systems of Invariance for Ternary Biquadratic Forms. Arpita: 14:25 Okay, pause, pause, pause. Aarati: 14:27 I, I don't know. Don't ask. Arpita: 14:28 Okay. Aarati: 14:29 I honestly, I honestly have no idea what that means. I did not even try to figure it out. Arpita: 14:35 I legitimately, don't know what a single word that you just said was. Aarati: 14:40 So all I know is that it has something to do with invariant theory, which is a branch of abstract algebra that she specialized in. And later on, one of those professors, David Hilbert, that she met apparently worked out a better approach for what she had done during her PhD. And when she was asked a question about it many years later, Emmy basically was like, Oh, that work that I did back then for my PhD was total crap. Um, I've forgotten all about it, so I can't really even try to explain it to you, because it sucked. Arpita: 15:13 So this isn't even the complicated math. We haven't even gotten there. Aarati: 15:17 No. Arpita: 15:17 Okay. Aarati: 15:20 Yeah. So yeah, cause I didn't, I like, I didn't even try. Like once I read that she was like, oh, that was total crap. I've forgotten all about it. I'm like, okay, so then it's not important. So the more I can spare my brain cells, the better. Arpita: 15:33 Speeding past that. So she did her PhD math, math, math. Aarati: 15:36 Yeah. So, but I just want to clarify that invariant theory. is not crap. Just the work that she did in that was. Arpita: 15:46 Her PhD wasn't crap. It was improved upon by another scientist. Aarati: 15:52 Yeah. And she became an expert in invariant theory. Okay. So now Emmy has her PhD, but once again, she's in this position where there's nothing she can do with it. Like, apparently the government, like, said it was okay for women to study at universities but not to hold a professorship or hold any kind of official position. Arpita: 16:14 Weird. Aarati: 16:14 So for the next seven years she taught at the University of Erlangen's Math Institute without pay. Arpita: 16:21 Seven years she taught without pay? I wonder what other people thought about this, like if she was trying to be a lady or she was taught to be a lady, and now she's gotten a PhD, she doesn't really have skills that would make her, you know, like, she doesn't have woman skills. I can't think of a better way to say that. It's like basically her skill set is not for a woman of society or whatever. Like, what's my Bridgerton vocabulary right now? Aarati: 16:50 Yeah, it's not proper. What are you doing? It's not proper ladylike behavior. Arpita: 16:54 I guess like, what was her position in the world? Aarati: 16:56 You know what? Like, a theme throughout this is she doesn't really care what other people thought about her. She just loved math and she was gonna do it. And she fully took advantage of the fact that her family was well off, so she didn't need the money. Yeah. So she didn't have to worry about that aspect of it. She could just continue doing what she wanted to do, basically. And this also meant that she was able to stay at Erlangen and continue to study math, work on invariant theory and stay close to her family. So around this time, her father's health was starting to decline. And so when he was too sick to lecture, she would fill in for him as a substitute. So during this time, her old professor, Paul Gordon, retired. And his successor, Ernst Fisher, became Emmy's friend, mentor, and a major influence on her life. They would spend hours discussing lectures that they had attended, and even though they both lived in Erlangen and they saw each other all the time at the university, there are still existing postcards between the two of them where they're just sending each other algebra equations and their thoughts on math. Like, they had just thought of something, and they couldn't wait to share it with the other person, so they just, like, wrote a postcard and dropped it in the mail and sent it to them. Arpita: 18:10 Oh my gosh, that's, like, actually adorable. Aarati: 18:12 And all it is is math. It's just, like, math scribblings on the back of this postcard. Arpita: 18:16 I love it. It's, like, kind of, like, old timey texting. It's, like, you know, I just, I thought of something and I wanted to share it with you. Aarati: 18:21 Yeah, and I was just, like, picturing, like, the postman's face, like, looking at the postcard, being like, what the heck is this? But she credited him with really shifting her way of thinking in a way that made her big breakthrough and what she's known for possible. So let's get into that. Now it's 1915, Emmy is 33, and something huge happens. Albert Einstein drops his theory of general relativity. So, basically, this was a brand new mathematical description of gravity. For nearly 200 years, scientists had been using Isaac Newton's laws of physics, which stated that gravity was a force exerted between two objects that was directly related to the mass or the amount of matter that those two objects had and the distance between them. Arpita: 19:12 Okay. Aarati: 19:13 And this was great for most things that people could observe and measure at the time, like we all know Newton's famous apple falling on his head, so the apple from a tree falls down to the ground because the earth has a lot more mass than the apple, and so it's pulling the apple down towards it, right? Arpita: 19:32 Right. Aarati: 19:33 So, Newton's laws more or less explained things like why the moon stays in orbit around the Earth, and why planets orbit around the sun, um, but there were things that his laws did not explain. For example, I didn't know this, but apparently the planet Mercury has kind of a funky orbit that no one was really able to explain. Arpita: 19:54 Is it an ellipse? Is that why? Aarati: 19:56 It is an ellipse, but then it also isn't like a constant ellipse. It kind of like wiggles around the sun. Arpita: 20:02 Oh, I actually didn't know that. Aarati: 20:03 Yeah. Arpita: 20:04 Yeah. Aarati: 20:04 It just kind of like wiggles around the sun. And yeah, no one could really explain it. And Newton's laws didn't explain it either. And so by the time Einstein rolled around 200 years later, physicists were starting to study super tiny, fast things like light particles and atoms. And Newton's laws weren't really applicable to these things that were moving faster than the speed of light or had no measurable mass. Arpita: 20:30 So you're like moving beyond things that you would consider visible or things that we like interact with like on a daily basis like the apple for example or the moon... Aarati: 20:39 Yeah. Arpita: 20:39 ...things that we are able to see and for the most part touch But now we're moving on into things that are moving so fast and are smaller than we're able to see with our eyes. Aarati: 20:49 Yeah, and if Newton is saying like gravity is this thing between two objects that have masses and you're like Well, what about light particles that have no mass? It's like, what do you do with that? So Newton's theories assumed that space and time were fixed constants. So physicists were working under the impression that you can't bend space and time will always move forward at a constant rate. And Albert Einstein comes in and basically says, not necessarily. Einstein instead hypothesized that there was this thing called spacetime. that we all exist in. So, just like a fish exists in water that's all around it, we all exist in spacetime. And everything exists in spacetime. Planets, stars, like everything. And spacetime can actually bend or contort based on the mass and energy that exists in the system. So, a really classic way that I found to picture this is the ball and trampoline analogy. So, You imagine you have a large trampoline, and that trampoline represents spacetime, and then you place a really big, heavy bowling ball, which represents the sun, in the middle of the trampoline, and that'll cause it to stretch and dip downwards, and that's basically how the sun causes gravity, right? So if you place something now at the edge of the trampoline, it'll kind of be pulled down towards the bowling ball, the same way the sun kind of pulls everything in towards it. Um, and the way that the trampoline is stretching and contorting down into the bowling ball is how spacetime contorts and bends in. Arpita: 22:29 That makes sense. So it's no longer a fixed constant, it's something that is able to bend and mold around something that also has mass. So that's interesting. Aarati: 22:40 Yeah, exactly. And then each of our planets have their own smaller gravitational pull. So they would all form their own smaller dips or contortions in the spacetime trampoline. So that's kind of what's happening. That example's in 2D, obviously we're living in 3D, but in essence, Newton was sort of right ish, but when it came to studying light or tiny particles, Einstein's equations were a lot more reliable. But of course, it's new, and so universities all over the world are now scrambling to test it and see if it holds up to mathematical and physics experiments. And overall, it held up really well. And it also explained things that scientists had observed, but had not had answers to, like Mercury's weird orbit. It explains that. It also explained things like why astronomers had observed that light seemed to bend around stars. They had observed that phenomenon and they're like, wait, I thought light always moved in a straight line. How come it's seeming to bend? And Einstein's math equations showed why. Arpita: 23:44 What a crazy thing to discover though. Like you just, well, first of all, spacetime existing on its own... is nutso. And then also being able to be like, this thing bends. Like, what a mind trip. Aarati: 24:00 Yeah, like, space. Like, the space between me and the wall is actually bendable. Like, you can contort that. Arpita: 24:09 That is just a crazy train of thought to come up with in the, in the first place. And then to also then be able to have math to back it up is just... insane. And incredible. But, continue. So like, now it's, you're able to explain all these phenomena that were observed. Aarati: 24:26 Yeah. You're able to explain all of this, and you're able to explain where gravity comes from in the first place. Like, it's not this like, invisible nebulous force that just kind of exists between two objects. It's like, oh, it's coming because of the bend and contortions in spacetime, obviously. But Max Klein and David Hilbert at the University of Göttingen were looking at Einstein's theory and they hit a snag. They were like, wait a minute. So according to Einstein, mass can curve spacetime, but also according to Einstein, mass and energy are interchangeable. And that's the famous equation E equals MC squared. So that's energy equals mass times the speed of light squared. So those two are kind of, equivalent in a sense. And so they were like, wait a minute. So if mass can curve spacetime, that must mean energy can also curve spacetime. But that would break the laws of the conservation of energy. Because if energy can curve spacetime and spacetime contains energy, then spacetime would curve in upon itself. And that's weird. That shouldn't happen. That's problematic. Arpita: 25:40 Just to break that down. So if mass and energy are interchangeable, and we know that mass can bend spacetime because of the trampoline analogy, and now we are thinking about energy, but spacetime also has energy. So then, if we are saying that energy bends energy, now you don't have a finite amount of energy, which is, Aarati: 26:04 Well, so we're saying that if energy can bend spacetime, and we know that spacetime contains energy, then spacetime would bend spacetime, like spacetime would bend itself, which is clearly not happening, but according to Einstein's equations, it would be happening. And so they kind of hit this weird paradox. They're like, something's wrong here. Arpita: 26:29 It's almost like you're in a carnival with all the mirrors and they're all just like in and of themselves. It's like that, where it's like, well, this is bending itself, it's bending itself, it's bending itself. Aarati: 26:38 Yeah, exactly. They're like... Arpita: 26:40 They're like, that is weird. Aarati: 26:42 The math isn't mathing. Got it. Okay. Like something's... yeah. Something's... something's wrong here. And they asked Einstein about it and he was just kind of like, Yeah, that's weird. I don't know. Like, I don't know what to do. Arpita: 26:55 Keep that to yourself. Aarati: 26:56 He's like, I know I'm right, but I don't know what, how to explain that thing that you just found. Max Klein and David Hilbert are trying to figure this out, and they realized that what they really needed was a specialist in invariant theory. Arpita: 27:11 Nice. Aarati: 27:12 So they decide to try and recruit Emmy to join them as a faculty member at the University of Göttingen. However, the other faculty members were not happy about this. World War I was underway and one of the faculty members apparently said, what will our soldiers think when they return to the university and find that they're required to learn at the feet of a woman? Arpita: 27:33 That's the problem. Aarati: 27:34 Yeah, that's the problem. Arpita: 27:35 Also, What if they don't go to college? That doesn't even, that's such a stretch. Like they might come back for more and then they might want to go to college. And then there might be a woman. Aarati: 27:47 Where there's all this blood and destruction and death, and then they have a problem with the woman teaching them. Arpita: 27:53 Yeah, that feels like a stretch for sure. Aarati: 27:56 But David Hilbert had a pretty good response. He said,"I do not see that the sex of the candidate is an argument against her admission. After all, we are a university, not a bathhouse. So I was like, good job. Arpita: 28:08 We love an early feminist. Aarati: 28:12 Yes. So they really fought for her, but unfortunately they were not able to get her an official position. But Emmy didn't really care about that. She had her eyes on the bigger picture. She didn't care about titles and credentials and again, she didn't need the money. So, Emmy went to Göttingen anyway as an unpaid guest lecturer. And when she got there, she basically took one look at this paradox that Hilbert and Klein were struggling with and went, Oh, that's not a problem at all. You guys are just thinking too small. And she, like, solved it immediately. And.... Arpita: 28:45 What? This feels like a sitcom. She's just like, just shows up with her like little whiteboard and she just like scribbles all over it and she goes, done. Aarati: 28:56 Yeah, exactly. That's basically like what she did. And yeah, so like, she figured this out in probably less time than it's going to take me to explain it, but just immediately solved. Arpita: 29:09 She's like, got it. Too easy. Aarati: 29:10 This is not a problem. Arpita: 29:11 Yeah. Aarati: 29:11 Yeah. Arpita: 29:12 Next. Aarati: 29:12 Yeah. Too easy. Next. Yeah. Arpita: 29:18 Wait, sorry. Actually though, is this like hours or like days? What is the timeframe that we're talking? Aarati: 29:23 Well, I know she, she came to University of Gottingen in 1915 and she solved this in 1915. So we're definitely talking less than a year. But all the sources that I read were like, she pretty much immediately was like, this is, this is not a problem at all you guys. This is not, this is not a paradox. You guys aren't, you know, let me show you why you're wrong. Arpita: 29:47 This is just such good energy. Okay. Aarati: 29:49 So in solving this paradox, she came up with what she is actually most famous for, which is called Noether's Theorem. Okay. So We're gonna try to explain Noether's Theorem. So, Noether's Theorem, and this is, this is straight from Wikipedia, says that, every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. Got it? Arpita: 30:17 Totally, yeah, mm hmm, mm hmm, yeah. Aarati: 30:22 Yeah, now, okay, now you know why when I first started looking her up, I was like, crap, what did I get myself into? Arpita: 30:29 I think I had you in the first half, and then after that I was like, I think my eyes glazed over, so. Aarati: 30:34 So we're gonna, we're gonna try to explain it. So the two big ideas that that are linked in Noether's theorem are symmetry and conservation laws. So let's first look at symmetry. So commonly we think of something as symmetrical if you change something about it and it looks the same. So for example, humans are symmetrical because if you take our left side and mirror it, it looks like our right side. And the same way like a flower that has six petals is symmetrical because you can rotate it in increments of 60 degrees and it'll still look the same. In physics, two systems are called symmetrical if you change something and they still act the same way. So, for example, if you imagine you're in outer space and there's absolutely nothing around you. There's no planets, no stars, no friction, no wind, no external forces of any kind. You're just floating around empty space all around you and you throw a ball. What happens to the ball? It'll leave your hand at a certain direction and speed, and it'll just keep on traveling that way forever. It'll just keep on going because there's no other external forces acting on it, right? I see, okay. So, it'll just keep going in that direction. So you can look at it one inch after it left your hand. And it'll be doing exactly the same thing one foot later, two feet later, a mile later, a thousand light years later, it'll still be going in the same direction at the same speed. no matter where you look at it along the trajectory. Arpita: 32:07 Okay. Aarati: 32:08 You can say that the ball is a system that has translational symmetry. That is, you can observe the ball in different places and all the forces acting on it will always be the same. Arpita: 32:21 So in this case, symmetry, can you say the definition of symmetry again? Aarati: 32:25 The definition of symmetry in physics is that if you change something, you can it will still be acting the same way. It'll still look the same. So in this case with the ball example, if you look at it in different places along its trajectory, I see the same forces are still working on it. And because of that, it's still moving in the same direction at the same speed. Arpita: 32:49 And then the thing that you're changing is where you're looking at it along the trajectory. Aarati: 32:55 Yeah, the different places. So, Noether's Theorem says that translational symmetry is linked to the conservation of linear momentum. So, when you look at the ball in different places, its linear momentum will be conserved. Okay, so now, what happens if you do the exact same thing on Earth, right? So, you throw the ball just like you did in space. Except now, when you look at the ball in different places along its trajectory, it's doing different things, right? So when the ball leaves your hand, like one inch after it's left your hand, it's going to be moving kind of horizontally away from you at a certain speed. But then if you look at it when it's one foot away from you, Now it's moving faster than it was before, and it's changed directions. It's getting pulled downwards. It's moving down. In space, like, it would have continued to move straight and keep moving away from you at the same speed, but now it's actually falling downwards. When it's two feet away from you, the ball is falling downwards even faster, and at some point it'll hit the ground and it'll stop moving entirely. So if you're only looking at the ball, you might be confused. And this is a dumb example, so like, bear with me, or a simple example. But in this case, it looks like linear momentum is not being conserved, right? The ball starts out at a certain speed, it gets faster, it changes direction, and then it stops. And, according to Noether's theorem, that's because the ball's motion is no longer translationally symmetric. You are missing something. You are missing some external factor that is acting on the ball to make it move that way. And, like I said, this is, uh, a really obvious, simple example, but the thing that you're missing is the earth, right? Once you take into account the earth, and you look at the ball and the earth together, and you see what's happening then, in that case, when you look at the ball, when it's one inch away from your hand, it's farther away from the earth. And then when you look at it one foot away from your hand, it's not. It's moved downwards, so now it's closer to the earth, right? And at some point it'll actually hit the earth. So the ball is no longer translationally symmetric because the distance between the ball and the earth is changing. And that's why the linear momentum of the ball is changing. Arpita: 35:20 I kind of get it. I think I kind of get it. So if you take into account the fact that the earth is acting on the ball. then linear momentum is conserved. Aarati: 35:31 Yes. Arpita: 35:32 Because basically what she's saying is it is conserved as long as you are taking everything into account. And in the version where you're just looking at the ball moving through the air on earth, you are not taking into account the earth. So it appears to not be conserved, but once you take into account. Aarati: 35:47 Yes. Arpita: 35:48 There's all the things that are acting on it, including the Earth, then you do see that it is indeed concerned. Aarati: 35:54 Yeah, so you're able to restore translational symmetry if you include the Earth and its gravitational forces on the ball in your system. So you have a system now that has two parts. It has the ball and the earth. And so now when you run your calculations again, linear momentum is indeed conserved because everything's accounted for. The ball is getting pulled towards the earth and the earth is actually moving just a teeny tiny little bit towards the ball. Arpita: 36:22 Right. Aarati: 36:23 And, um, that is the conservation of linear momentum. Arpita: 36:26 Okay. This makes sense. So when you're thinking about your system in physics, you would think about all the forces that are acting upon an object. And so you think about like a train or something and it's like, Oh, there's like the wheels on the track and there's friction there. And then there's wind and then there's like all these different things. And so you're thinking about a system and then in space it's vacuum because there are no forces acting upon an object, which is why you're able to have this symmetry. Right? Aarati: 36:55 Yes. Yeah. That's why linear momentum will be conserved because there's no external forces. Arpita: 37:01 Right. And then if you're thinking about it on earth, if your system does not include the Earth and its gravitational pull, it will not work because you're not taking into account all of the components of this system. Aarati: 37:12 It'll look like linear momentum is not being conserved. Yeah. Yeah. It'll look, it'll look a bit weird. Um, but now if you take into account the ball and the earth, you can move that entire system, the ball and the earth to different places and everything will work out fine. And that is actually what's happening, right? Our earth is orbiting the sun. So we are in different places in space all the time. And yet when we throw a ball, on Earth, we can predict that linear momentum will be conserved because we are taking into account the ball and the earth. So no matter what point we are in space around the sun, we can say that linear momentum will be conserved because we have established translational symmetry in our system. Arpita: 38:03 Got it. And the system is inclusive of all of these things that we've just discussed. Aarati: 38:07 Yes, exactly. That, that was a simple example. That was a simple example. Arpita: 38:14 I feel like I maybe 40 percent get it. Which is pretty good. Aarati: 38:20 Yeah, but it is, it's really good. Like, you don't, you don't understand the amount of arguments and like the amount of work that I put into making this sort of understandable and then my brother coming in and just ripping it apart and saying, no, you're completely wrong. Just like, what, what is happening? I still feel like at the end of this episode, he's going to listen to it and be like, you got it wrong. How, how did I get it? I'm really hoping it's not the case. The, this was, this was like the simplest way I could boil it down, I think, and what it really showed, the power of Noether's Theorem here, is that it helps you see if you're taking everything into account when you are talking about very complex systems. So we were talking about a ball, and then a ball, and then a ball and the Earth. But when things get more complicated than that, which they do very quickly in physics, um, Noether's theorem helps you see if you're taking everything into account. So in the example with the ball in space, as long as there was nothing around to influence the ball, the ball by itself was a translationally symmetric system. Arpita: 39:37 Uh huh. Aarati: 39:38 But once you move the ball to Earth, If you just look at the ball, you lose that symmetry. Arpita: 39:43 Yep. Aarati: 39:43 Right? So you have to include the Earth in your system and then everything is accounted for and the system is translationally symmetric again. Arpita: 39:53 Yep. Aarati: 39:53 And Noether's theorem lets us say with certainty that the laws of conservation linear momentum will hold true in that case. So it kind of like lets you know if you're missing something, basically. Arpita: 40:06 Got it. That is very useful. Aarati: 40:08 Yes. Um, and there are also different symmetries a system can have. So Noether's theorem not only proved that translational symmetry is linked to the conservation of linear momentum, but she also showed that rotational symmetry is linked to the conservation of angular momentum. And so that means that you can rotate your system and still see the same thing. So that's more akin to throwing the ball in different places on earth. Like if you throw the ball here in California, or you throw it in India, or you throw it in Antarctica, you can still predict the angular momentum of the system will be conserved. Arpita: 40:50 I see. So it doesn't matter where on the sphere you still have rotational.... I understand. Aarati: 40:56 Yeah, you have rotational symmetry, then we can say something about the angular momentum. We can say angular momentum is conserved. And she also showed that temporal symmetry, or symmetry over time, is linked to the conservation of energy. So that's if you throw the ball today, or tomorrow, or ten years from now, the energy of the system will still be the same. Arpita: 41:18 Do we not know that before? Aarati: 41:19 So we didn't, we didn't have a mathematical proof for that before. We couldn't say that with like complete certainty. Um, and so it came to Einstein's theory, Emmy basically said that if the conservation of energy laws appear to be broken, then you haven't included everything you need to in your system. Somewhere along the way in your calculations, David Hilbert and Max Klein had lost temporal symmetry. And once they figured out whatever it was that was causing them to lose temporal symmetry and added that back into their calculations, they were able to explain why it seemed like the law of energy of conservation was being violated and they were able to resolve the paradox and fix their own mistake, basically. Arpita: 42:09 Got it. So it's like, not like a sanity check, but she's basically creating a way to check your work, almost. So it's like you're able to account for all these things because you're using her principle to ensure that you have symmetry in all of these different dimensions. Aarati: 42:25 In, in the case of Einstein's paradox, yes. But physicists later were able to use this because if they're able to, um, establish an underlaying symmetry in whatever experiment or problem that they're trying to figure out, then they can use the laws of conservation of momentum or energy to figure out how their system should behave because they've established this underlaying symmetry. So it's useful in kind of studying other things too. So resolving the paradox that was seeming to arise in Einstein's theory, that was just kind of like a side effect of the actual power of her theorem. Arpita: 43:07 Okay. Aarati: 43:08 Yeah. Arpita: 43:08 But it wasn't necessarily the sole use. Aarati: 43:11 Oh, not at all. Not at all. So, Emmy proved her theorem in 1915, which was the same year she got into Göttingen, and she published it in 1918, and it fundamentally changed how physicists approached their research. So now, they could look for these types of symmetries in their experiments. And if they were able to establish symmetry, they could use the conservation laws freely to solve problems. So afterwards, scientists found symmetries in other branches of physics, like quantum field theory. And when you start to get into things like particle physics and quantum physics, the link between symmetries and conservation laws is huge. Because before that, physicists didn't have a generalizable rule that they could apply to any situation and predict whether the conservation laws would hold true or not. So when Emmy showed that these conservation laws are rooted in symmetries, it allowed kind of a better understanding in all branches of physics and led to developments of things like semiconductors and even the Higgs boson particle in 2012. Arpita: 44:23 That's amazing. Aarati: 44:23 So it's, it's huge. It's like this huge underlying theorem that physics and math people use all the time. Arpita: 44:32 So if we had to think about maybe like a very big picture way of thinking about it. So it's maybe not a sanity check or some kind of quote unquote checklist to make sure that symmetry exists in your system. That was almost a, like a byproduct of finding this because they were just thinking about that specific problem at the time. But then making this so much bigger, it's almost a different lens through which physics problems are viewed. Because it's taking into account all of these different things. So it's almost like she's like shifted perspectives through this theorem, as opposed to creating a checklist. What I, which, is what I originally said. Aarati: 45:12 Yeah, absolutely. Arpita: 45:13 Okay. Does that seem right? Aarati: 45:14 Yeah. Yeah, absolutely. I think that's much more accurate. It's really allowing you to predict if you can use these laws of conservation. And I actually watched some videos where people said, like, you know, it's almost like magic. Like if they could, if they could establish some underlying symmetry, it's like, everything would fall into place and they could do all these kinds of things that they wouldn't have been able to do otherwise. Arpita: 45:37 So establishing symmetry is a good thing to do when you're trying to start thinking about a physics problem. It's like trying to understand symmetry within all of these different dimensions, like temporal symmetry or translational symmetry, or. Angular. Aarati: 45:51 Yes. Arpita: 45:52 That would, that's like. Aarati: 45:53 Yeah, it makes things so much easier. Arpita: 45:56 Understand. Okay. I see. Aarati: 45:57 It's so important. Arpita: 45:58 I'm actually thinking about this also. Cause like, it feels like maybe this is just the world that I live in, but it feels like a lot of physics discoveries and things that happen in space and like quantum theory and all of these things. Like, I feel like we hear about them when they're shiny, like the James Webb telescope or something like that, but we don't actually get to hear all of the things and steps that come up leading up to it. And I listened to, I should find what it was, but I listened to a different podcast with a science communicator. And she is also a physicist and she talks about how it's very difficult to get the public to understand what they're doing because it is so difficult for even physicists to like conceptualize and to put things into words. And the reason they're using math is because explaining it in words is so difficult to do. And that's why they use math. Aarati: 46:48 It's so hard. Arpita: 46:49 And so it's not for a lack of, you know, understanding it or whatever, but it's difficult for even these people who have done this for a living for so long, put it into words, number one, and then put it into words that lay people can understand, which is why I think a lot of this stuff doesn't really get talked about that much. We just don't have like a context for it as much. Aarati: 47:11 Yeah, absolutely. Because even like, like I said, even the ball in space versus the ball on Earth. Like, I don't, I can't explain to you how many iterations of that scenario we went under and, like, even up till yesterday, I was still saying it wrong. And I'm like, how complicated can this be? It's a ball in space and then it's a ball on Earth. Okay, I think we're, we're almost, we're almost done with the math. So at the end of World War I, the attitude towards women's rights in general were starting to become more progressive, but despite this, and despite the major breakthrough that Emmy had just made, she still wasn't really getting the recognition that she deserved. Arpita: 47:52 Classic. Aarati: 47:53 To give lectures, she technically had to do so under David Hilbert's name. So to get around the system, they would advertise her lectures as David Hilbert with guest lecturer or assisted by Emmy Noether. And then David Hilbert would quote unquote forget to show up to his lecture so that Emmy could just give the lecture on her own. Arpita: 48:15 He really sounds like an ally. I love it. Aarati: 48:17 It took the university three years to allow her to take an exam for which she would become eligible for tenure. But even though she was successful, it took another full year for her to get a position that actually paid her a tiny salary. But, again, Emmy didn't really let any of this phase her. She continued to work in abstract algebra. And in 1921, she published a paper called Ideal Theory in Ringberechen, which gave rise to the term Noetherian Ring and was described as revolutionary by mathematics. And again, like, I did not go into this. I was just like, I can't. Arpita: 48:56 Yeah, let's cut, cut our losses here. Aarati: 48:59 I know it's like, but one interesting thing I did find about it is that actually today many universities call the clubs and support groups that they have on campus that support women and minorities studying math the Noetherian Ring. So it's like their club name. Emmy was very enthusiastic when she gave her lectures, but students were of mixed opinions as to how good they were because oftentimes her lectures weren't really planned out. So basically, if you. didn't already know mostly what she was talking about, they weren't very useful, and you just get lost, which I can totally understand. Arpita: 49:34 I totally understand. That sounds like a lot of my lectures in undergrad. Aarati: 49:39 Yeah. So like, one cute story I found was like, one time she had a new idea for a way to solve a mathematical proof, but she wasn't able to work it out before her lecture. So, during the class, she just started teaching it, and tried to work out the new proof in front of her students. Arpita: 49:57 In real time? Aarati: 49:58 Yeah, in real time. And then halfway through, she realized it actually wasn't going to work out the way that she had thought it would. And so, she threw down her piece of chalk, stomped on it, and yelled,"There! I'm forced to do it in a way I don't want to!" And then she taught the class the normal, traditional mathematical proof perfectly. Arpita: 50:17 Can you imagine being a student in that class? And then you're just like desperately taking notes. You're just like writing as fast as you can. Like your hand is cramping. And then she's like, Oh, this doesn't actually even work. And then starts over. I would lose my mind. That would be like, I'm done. Aarati: 50:30 It's like scrap the last like 45 minutes. That was all useless. That's actually all wrong. Arpita: 50:36 I would absolutely lose my mind if that was me as a student. I would be like, I'm done. We're trying again tomorrow. I'm going home. Aarati: 50:42 Yeah, absolutely. Um, but if you could get on the same level as her, her lectures were actually transformative and many of her students credit her for completely changing the way that they thought. She, if, if you hadn't noticed, uh, she also didn't really care about having good manners or seeming lady like at all. Um, one example of this is she would stuff her handkerchief down the front of her blouse and pull it out and stuff it back in repeatedly, making her dress wrinkled and untidy, which students found really funny. Arpita: 51:13 I feel like that just feels like such a mild offense. But then, like, In the early 20th century, they're like,"You're pulling it out from where? And you're wrinkling your dress." Like that just feels so funny. I feel like the archetype of a professor, male or female, is just the sort of crumpled person anyway, you know, like, they're just like, have like coffee on their shirt. They're just like, you know, trousers they wore yesterday. Like, it's like, not this person who's like, Aarati: 51:41 Yeah. Arpita: 51:42 Perfectly kept, you know what I mean? Like that's kind of the archetype of what you think. Aarati: 51:45 Yeah, but since she's a woman, it's like she's supposed to be. Arpita: 51:48 Exactly. Aarati: 51:49 But that's exactly what people would say. They would say like she would spill food on her dress because she was like talking while she ate and like gesticulating wildly. And so like food would fall onto her dress and she would just kind of wipe it away and like not care that it left a stain or it's like whatever who cares. And then. Like, I thought this was so funny. So like one time she was teaching and her hair was falling out of her clip. And two students really wanted to talk to her and tell her that her hair was falling out of her clip and she should maybe like go and tidy it up, but she was just too busy talking to some other students and so they couldn't even get a word in to talk to her about her hair. She's just like, I can't, I can't do, who cares about my hair? Who cares? Math. Math, math. Arpita: 52:36 Not important. Aarati: 52:36 Um, she was always ready to share new ideas and help her students. And she had a dedicated group of students called the Noether Boys who would promote her. I know, I was like, that's so cute. Arpita: 52:48 And she had her little brothers too. So I'm sure it was just like, yeah, exactly. Aarati: 52:52 Um, so her Noether Boys would promote her wherever and however they could. And her math colleagues who saw how brilliant she was helped shine the spotlight on her by inviting her to lecture and share her work. So throughout the 1920s and 30s, Noether really started making a name for herself. She was becoming a very prominent figure in mathematics and mentoring many young mathematicians during their PhDs, and they went on to make their own amazing breakthroughs. Um, And so it seemed like she had more or less broken through society's sexism, but now I unfortunately have to remind you that she is a very prominent Jewish person living in 1930s Germany. Arpita: 53:34 I have a feeling this story doesn't have a happy ending. Aarati: 53:36 Well, it's, it's not in the way you would think, actually. So, At first, Emmy doesn't really take the rise of the Nazis and the growing antisemitism very seriously, but pretty soon it does start to affect her. In 1933, Hitler's administration passed a Law for the Restoration of Professional Civil Service which removed any Jewish person from having a government job. So Emmy was immediately let go from her position at the University of Göttingen along with several other of her colleagues. But, she wasn't really phased by it, probably because, you know, she'd faced this all her life. Arpita: 54:11 Right. It didn't feel that different. Aarati: 54:15 Yeah, and she had gone through years of not having a steady job anyway, so she's like, what else is new? Arpita: 54:21 It's like, it was originally because she's a woman and now it's because she's Jewish, but probably to her it didn't really feel any different. Aarati: 54:26 Exactly. Instead, she opened up the front room of her home and started teaching private lessons even to the students who showed up at her house in Nazi uniforms. Um, she basically just ignored it and taught them the same way as everyone else because she quote,"never doubted the brown shirt students integrity". Wow. So yeah, if you're interested in math, come learn. It doesn't matter what your politics are. Arpita: 54:54 I mean, it probably makes sense from her perspective too, right? Is that she was faced with so much discrimination and she was. you know, face so many roadblocks that her perspective, she's like, why would I put roadblocks up for someone else to learn who really wants to? Aarati: 55:07 Yeah. I just, I still found it kind of incredible though, that it's like, you're, you're teaching someone who is like, clearly a Nazi or clearly align themselves with that way of thinking. And is clearly against you, but you're able to put that aside and just be like, it doesn't matter. Um, but with the situation getting worse and worse in Germany, Emmy finally realized she had no choice but to leave the country. Many universities around the world were offering refuge to German scholars who might be targeted by the Nazi regime. So a grant was approved by the Rockefeller Foundation for Emmy to take a position at Bryn Mawr. So she immigrates to America and she starts her position at Bryn Mawr in late 1933. And these were some of her happiest and most fulfilling days. I think because she was at a college for women. And so she was respected and surrounded by support like she never had been before, and she was able to fully be herself and work on math. Arpita: 56:08 Totally. I'm really glad this story didn't end the way I thought it was going to. So, I'm, yes, she has a happy life. And she's in America. Aarati: 56:17 Yes. Um, she was invited to give lectures at the Institute for Advanced Study in Princeton, which is actually where Albert Einstein ended up, because he also left Germany. But she noted that she did not feel welcome at Princeton at all, which was a quote unquote men's university. But unfortunately, just 18 months after she arrived in America, doctors found a tumor on Emmy's pelvis, and then when they went in to operate, they discovered that she also had an ovarian cyst the size of a large cantaloupe, as well as two smaller tumors on her uterus. Arpita: 56:51 Oh my god. Aarati: 56:53 Yeah. So, she went under the knife, and three days after the operation, she appeared to recover normally, but on the fourth day, her temperature suddenly rose to 109 degrees Fahrenheit, and she died. So, physicians aren't really sure exactly what happened, but they do think she most likely succumbed to a viral infection of some sort due to the operation. Arpita: 57:14 That's unfortunate. Aarati: 57:15 Um, she was 53 when she died. Her friends and colleagues at Bryn Mawr held a small service for her. German mathematicians came out in force to pay her tribute, including Albert Einstein. He actually wrote an obituary for the New York Times. in which he wrote,"In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began." Arpita: 57:46 That's amazing. And to like, for someone who is already really respected like Einstein, to be able to say this about someone, it's like he's using his power to like, help lift her, which is amazing. Aarati: 57:56 I was just like, for Einstein to say that, like, Arpita: 58:01 It's interesting too that she does seem like she had a pretty privileged life, which gave her the space to, I mean, she was a woman in the early 20th century, but she had a family who supported her and, you know, they could have disowned her. They could have been like, you need to be a lady or you need to like do the things that ladies are supposed to do, but they supported her and allowed her to do this work. Aarati: 58:20 Yeah. Arpita: 58:20 And believed that math was something important because of her dad's position. And so she was able to like take these risks, like. It's interesting that even though she, you know, had all these obstacles, that from this perspective, she was able to make, take these risks because of the privilege of the position she had. So like, that's also kind of an interesting way to think about it. Aarati: 58:42 Yeah, absolutely. And I think that she fully took advantage of that. Arpita: 58:46 Yeah. Aarati: 58:47 But yeah, that is her story. And I actually found out that although many mathematicians and physicists know, of her name because they know Noether's Theorem because it's so fundamental. Many people just kind of assume Noether was a man. Arpita: 59:01 Yeah. It feels like a fair assumption given the landscape of math and physics, but that's amazing that it's a woman. Aarati: 59:09 I really hope I did justice. to her actual, like, work and her actual theorem and what she's known for. I'm so sorry if I got it wrong. Please write in and tell us what I got wrong. Arpita: 59:20 I'm not gonna lie, I definitely fell out of my depth even listening to that. So kudos to you for writing that story and taking it on. I think you did a great job and I really, really liked your story. Aarati: 59:32 This is all Rachel's fault, so you can thank her. Thank her for writing in. Yeah, yeah. It was, it was a fun story, but I think for the next couple episodes I'm gonna have to stick to something that I'm a little more familiar with. Arpita: 59:48 Thanks for listening. If you have a suggestion for a story we should cover or thoughts you want to share about an episode, reach out to us at smartteapodcast.com. You can follow us on Instagram and Twitter at@smartteapodcast and listen to us on Spotify, Apple Podcasts, or wherever you get your podcasts. And leave us a rating or comment. It really helps us grow. New episodes are released every other Wednesday. See you next time!

Sources for this Epsiode

1.Angier, Natalie. “The Mighty Mathematician You’ve Never Heard Of”. The New York Times. Published March 26, 2012.

2.  "Emmy Noether". Wikipedia.

3. “Emmy Noether: The Greatest Forgotten Mathematician in History”. BioGraphics. February 17, 2020. YouTube.

4. Taylor, Mandie. "Emmy Noether". Biographies of Women in Mathematics. February 21, 2023. 

5. Dick, Auguste (1981). Emmy Noether: 1882–1935. Translated by Blocher, H.I., Boston: Birkhäuser, doi:10.1007/978-1-4684-0535-4ISBN 978-3-7643-3019-4

 

6. "General Relativity explained simply & Visually." Arvin Ash. Jun 20, 2020. Youtube.

7. Einstein, Albert. "The Late Emmy Noether," letter to the Editor of the New York Times, May 1, 1935.

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