WEBVTT 00:00:00.000 --> 00:00:03.240 Welcome back to another deep dive. Today we're 00:00:03.240 --> 00:00:06.000 doing something a bit different for you. Usually 00:00:06.000 --> 00:00:08.859 when you picture a mathematical genius, you know, 00:00:08.900 --> 00:00:11.400 you think of that classic goodwill hunting movie 00:00:11.400 --> 00:00:13.759 trope. Oh, right. The solitary guy just standing 00:00:13.759 --> 00:00:16.379 in front of a giant chalkboard. Exactly. Chalk 00:00:16.379 --> 00:00:19.579 flying everywhere, furiously writing out equations 00:00:19.579 --> 00:00:23.039 at like lightning speed. It's an image that really 00:00:23.039 --> 00:00:27.329 suggests true genius is manic. instantaneous 00:00:27.329 --> 00:00:29.969 and just totally disconnected from the physical 00:00:29.969 --> 00:00:32.250 world. It's a compelling image for Hollywood, 00:00:32.570 --> 00:00:35.409 sure. Yeah. But it creates this massive misconception 00:00:35.409 --> 00:00:38.109 about how high level mathematics is actually 00:00:38.109 --> 00:00:40.390 done. I mean, it's just not realistic. Right. 00:00:40.590 --> 00:00:42.609 And that brings us to today's deep dive into 00:00:42.609 --> 00:00:45.630 our source material, which is a really comprehensive 00:00:45.630 --> 00:00:48.689 Wikipedia biography of Miriam Mirzakhani. One 00:00:48.689 --> 00:00:51.229 of the absolute greatest minds of the 21st century. 00:00:51.289 --> 00:00:52.829 Without a doubt, because she didn't work like 00:00:52.829 --> 00:00:55.130 that movie trope at all. Instead of speed writing 00:00:55.130 --> 00:00:57.390 on a whiteboard, she spent her time sitting on 00:00:57.390 --> 00:00:59.549 the floor, surrounded by these massive sheets 00:00:59.549 --> 00:01:03.149 of parchment paper, just drawing intricate doodles. 00:01:03.509 --> 00:01:05.510 Yeah, her young daughter actually thought her 00:01:05.510 --> 00:01:08.269 mother's job was simply painting. Which is just 00:01:08.269 --> 00:01:11.099 so great. So our mission today for you listening 00:01:11.099 --> 00:01:14.540 is to really explore how that visual patient 00:01:14.540 --> 00:01:17.700 approach allowed her to map the universe of all 00:01:17.700 --> 00:01:20.120 possible geometric shapes. Right. We want to 00:01:20.120 --> 00:01:22.200 do that without getting bogged down by all the 00:01:22.200 --> 00:01:25.040 academic jargon. Exactly. We want you to walk 00:01:25.040 --> 00:01:28.739 away truly understanding her aha moments and 00:01:28.739 --> 00:01:31.599 also how her life ultimately shifted the entire 00:01:31.599 --> 00:01:34.180 cultural landscape of her home country. So, OK, 00:01:34.260 --> 00:01:37.000 let's unpack this. Well. To really grasp the 00:01:37.000 --> 00:01:38.780 magnitude of what she achieved mathematically, 00:01:39.200 --> 00:01:41.159 we need to understand the crucible in which her 00:01:41.159 --> 00:01:43.519 mind was forged. I mean, her story doesn't begin 00:01:43.519 --> 00:01:46.099 in some quiet, isolated ivory tower. No, not 00:01:46.099 --> 00:01:49.700 at all. It begins in Tehran, Iran, in 1977. Right. 00:01:49.959 --> 00:01:51.719 Just a couple of years before the Iranian Revolution, 00:01:51.859 --> 00:01:53.560 and then she grew up during the Iran -Iraq War. 00:01:53.920 --> 00:01:56.239 So her early childhood was set against a backdrop 00:01:56.239 --> 00:01:58.700 of immense national upheaval. Which is a lot 00:01:58.700 --> 00:02:01.799 for a kid to process. It is. But as the country 00:02:01.799 --> 00:02:04.959 began to stabilize, there was this massive push 00:02:04.959 --> 00:02:07.900 to rebuild and foster a completely new generation 00:02:07.900 --> 00:02:10.919 of intellectuals. And she ended up testing into 00:02:10.919 --> 00:02:14.099 the Tehran Farzanaigan School. Which the sources 00:02:14.099 --> 00:02:16.759 say was part of the National Organization for 00:02:16.759 --> 00:02:18.860 Development of Exceptional Talents. Yeah, this 00:02:18.860 --> 00:02:21.020 wasn't just a slightly advanced high school. 00:02:21.360 --> 00:02:24.159 It was a rigorous, highly competitive environment 00:02:24.159 --> 00:02:27.379 designed specifically to locate and cultivate 00:02:27.379 --> 00:02:30.729 the absolute brightest young minds in the country. 00:02:31.169 --> 00:02:33.550 And Meerdakani obviously thrived there. But the 00:02:33.550 --> 00:02:35.550 funny thing is the sources note she wasn't actually 00:02:35.550 --> 00:02:37.409 interested in math initially. I mean she wanted 00:02:37.409 --> 00:02:39.789 to be a writer. She spent all her time just reading 00:02:39.789 --> 00:02:42.150 novels. Yeah reading novels. It wasn't until 00:02:42.150 --> 00:02:44.050 her older brother came home and told her about 00:02:44.050 --> 00:02:46.710 a famous math problem, the one involving adding 00:02:46.710 --> 00:02:49.389 numbers from 1 to 100, that she started to see 00:02:49.389 --> 00:02:52.490 the puzzle -solving beauty of mathematics. And 00:02:52.490 --> 00:02:54.590 from there her trajectory is just staggering. 00:02:54.729 --> 00:02:57.729 It really is. In 1994, she becomes the first 00:02:57.729 --> 00:03:00.669 Iranian woman to win a gold medal at the International 00:03:00.669 --> 00:03:03.770 Mathematical Olympiad in Hong Kong. Scoring 41 00:03:03.770 --> 00:03:06.270 out of 42 points. And then she follows that up 00:03:06.270 --> 00:03:09.289 the very next year in Toronto by achieving a 00:03:09.289 --> 00:03:12.629 flawless score, 42 out of 42, winning two gold 00:03:12.629 --> 00:03:15.550 medals. Which is incredible. Yeah. She even co 00:03:15.550 --> 00:03:18.110 -published a book on number theory in 1999 with 00:03:18.110 --> 00:03:20.629 her close friend and fellow Olympiad medalist, 00:03:20.830 --> 00:03:23.389 Roya Beheshti -Zavara. They were the first women 00:03:23.389 --> 00:03:26.099 to ever compete in the Iranian National Mathematical 00:03:26.099 --> 00:03:28.379 Olympiad. Right. But this is where her story 00:03:28.379 --> 00:03:31.280 takes a deeply tragic turn. And it drastically 00:03:31.280 --> 00:03:34.840 changes how you view her drive in her later work. 00:03:35.020 --> 00:03:37.379 Right. It's easy to look at her perfect Olympiad 00:03:37.379 --> 00:03:40.199 scores and assume she just had a computer for 00:03:40.199 --> 00:03:42.740 her brain. But how much of her early success 00:03:42.740 --> 00:03:45.060 was actually rooted in her specific environment 00:03:45.060 --> 00:03:47.560 and, well, her grit? If we connect this to the 00:03:47.560 --> 00:03:50.879 bigger picture, her... Genius wasn't just individual. 00:03:51.400 --> 00:03:54.379 In March 1998, Mirza Khani, Zavara, and several 00:03:54.379 --> 00:03:56.659 other exceptionally gifted students from Sharif 00:03:56.659 --> 00:03:59.020 University attended a mathematics conference. 00:03:59.259 --> 00:04:01.219 And on the way back? Yeah. On their way back 00:04:01.219 --> 00:04:04.199 to Tehran, their bus slipped on a wet road and 00:04:04.199 --> 00:04:06.500 fell off a cliff. The sources describe it as 00:04:06.500 --> 00:04:09.840 a devastating national tragedy in Iran. Seven 00:04:09.840 --> 00:04:12.680 passengers, literally some of the most brilliant 00:04:12.680 --> 00:04:15.060 young mathematical minds in the country, were 00:04:15.060 --> 00:04:18.740 killed. And Muzikhani and her friend Zavara were 00:04:18.740 --> 00:04:22.480 among the few survivors. Wow! Yeah. When you 00:04:22.480 --> 00:04:24.639 look at her life from that point forward, you 00:04:24.639 --> 00:04:26.779 have to factor in the psychological weight of 00:04:26.779 --> 00:04:29.980 that survival. I mean, she lost peers who shared 00:04:29.980 --> 00:04:33.310 her incredibly rare gifts. So continuing to pursue 00:04:33.310 --> 00:04:35.689 math with that kind of intensity wasn't just 00:04:35.689 --> 00:04:37.889 about raw talent anymore. Exactly. It was an 00:04:37.889 --> 00:04:40.769 act of profound resilience. She carried the legacy 00:04:40.769 --> 00:04:43.529 of an entire generation of Iranian mathematicians 00:04:43.529 --> 00:04:45.790 who didn't get the chance to finish their work. 00:04:46.490 --> 00:04:48.529 You know, you would think that surviving something 00:04:48.529 --> 00:04:50.810 like that might make a person want to rush through 00:04:50.810 --> 00:04:52.870 life, like to publish as fast as possible just 00:04:52.870 --> 00:04:54.769 to outrun the clock. But when she moved to the 00:04:54.769 --> 00:04:57.009 United States to do her graduate work at Harvard, 00:04:57.259 --> 00:04:59.720 She does the exact opposite. It slows down. She 00:04:59.720 --> 00:05:02.720 really does. She began her PhD under the supervision 00:05:02.720 --> 00:05:05.399 of Curtis T. McMullen, who is a mathematician 00:05:05.399 --> 00:05:07.540 who had won the Fields Medal, which is the highest 00:05:07.540 --> 00:05:10.319 honor in math. Right. And at Harvard, she became 00:05:10.319 --> 00:05:13.360 known for her relentless questioning. She would 00:05:13.360 --> 00:05:16.199 sit in McMullen's office taking notes in Persian, 00:05:16.600 --> 00:05:19.399 just relentlessly drilling down into the foundations 00:05:19.399 --> 00:05:22.129 of an idea. She actually described herself as 00:05:22.129 --> 00:05:26.110 a slow mathematician. She explicitly said, you 00:05:26.110 --> 00:05:28.910 have to spend some energy and effort to see the 00:05:28.910 --> 00:05:31.250 beauty of math. She wasn't looking for shortcuts. 00:05:31.569 --> 00:05:34.490 No, not at all. And she had this incredibly vivid 00:05:34.490 --> 00:05:36.810 quote about her process. She said, discovering 00:05:36.810 --> 00:05:39.250 a mathematical proof is like being lost in a 00:05:39.250 --> 00:05:41.509 jungle and trying to use all the knowledge that 00:05:41.509 --> 00:05:43.509 you can gather to come up with some new tricks. 00:05:43.790 --> 00:05:45.750 And with some luck, you might find a way out. 00:05:45.959 --> 00:05:49.480 That visual of being lost in a jungle perfectly 00:05:49.480 --> 00:05:51.819 captures the nature of the specific mathematics 00:05:51.819 --> 00:05:54.160 she was studying. I mean, she worked in topology 00:05:54.160 --> 00:05:55.800 and geometry. Wait, wait, let's break that down 00:05:55.800 --> 00:05:57.579 for a second. When most people think of geometry, 00:05:57.759 --> 00:06:01.100 they think of rigid triangles or, you know, calculating 00:06:01.100 --> 00:06:04.100 the area of a circle. What does topology actually 00:06:04.100 --> 00:06:06.740 mean in this context? Right, so topology is often 00:06:06.740 --> 00:06:09.480 called rubber sheet geometry. Rubber sheet geometry. 00:06:09.720 --> 00:06:12.519 Yeah, it's the study of geometric properties 00:06:12.519 --> 00:06:15.000 that don't change even when a shape is stretched, 00:06:15.360 --> 00:06:18.060 twisted, or bent out of proportion. Okay. The 00:06:18.060 --> 00:06:20.459 only rules are you can't tear the surface and 00:06:20.459 --> 00:06:23.620 you can't glue bits together. So to a topologist, 00:06:24.040 --> 00:06:26.779 a coffee mug and a donut are the exact same shape. 00:06:26.980 --> 00:06:29.319 Because they both have exactly one hole. Exactly. 00:06:29.399 --> 00:06:32.060 You can continuously deform a mug into a donut 00:06:32.060 --> 00:06:34.040 in your mind without ripping it. Okay, so she's 00:06:34.040 --> 00:06:35.980 working with shapes that are just constantly 00:06:35.980 --> 00:06:38.319 morphing and bending. It makes sense why she 00:06:38.319 --> 00:06:40.519 needed those massive sheets of parchment paper 00:06:40.519 --> 00:06:43.459 on the floor. You can't just crunch a quick algorithm 00:06:43.459 --> 00:06:45.680 to understand a surface that is continuously 00:06:45.680 --> 00:06:48.279 stretching. No, you can't. You have to visualize 00:06:48.279 --> 00:06:51.000 it. You have to draw it out. It sounds like she 00:06:51.000 --> 00:06:54.199 wasn't trying to win a sprint. She was like a 00:06:54.199 --> 00:06:57.259 botanist wandering through a dense jungle mapping 00:06:57.259 --> 00:06:59.959 every single leaf. Which is honestly so comforting 00:06:59.959 --> 00:07:02.579 for you listening to hear that one of the world's 00:07:02.579 --> 00:07:05.319 smartest people embraced feeling lost. Precisely. 00:07:05.800 --> 00:07:08.240 And the specific objects she was studying were 00:07:08.240 --> 00:07:10.819 Riemann surfaces and their moduli spaces. Okay, 00:07:10.959 --> 00:07:12.920 here's where it gets really interesting. Let's 00:07:12.920 --> 00:07:16.100 talk about moduli spaces. Well, to understand 00:07:16.100 --> 00:07:18.699 her major breakthrough... we need to introduce 00:07:18.699 --> 00:07:21.420 a classic physics problem that mathematicians 00:07:21.420 --> 00:07:24.500 love, the billiards problem. Right, the sources 00:07:24.500 --> 00:07:26.779 actually mention mathematician Jordan Ellenberg 00:07:26.779 --> 00:07:29.660 using this exact analogy to explain her work. 00:07:29.959 --> 00:07:33.819 Yes, imagine a perfectly frictionless billiard 00:07:33.819 --> 00:07:36.759 table. You hit a ball and it bounces off the 00:07:36.759 --> 00:07:40.120 walls forever. In a standard rectangular table, 00:07:40.459 --> 00:07:42.519 the path of the ball is fairly easy to predict. 00:07:42.680 --> 00:07:44.540 But what if the table is shaped like a weird 00:07:44.540 --> 00:07:47.319 polygon with strange angles? Right. Tracking 00:07:47.319 --> 00:07:49.860 that ball as it bounces chaotically off the walls 00:07:49.860 --> 00:07:52.360 becomes a massive headache. The math just gets 00:07:52.360 --> 00:07:54.500 incredibly messy. Because every time it hits 00:07:54.500 --> 00:07:57.160 an edge, you have to calculate the angle of reflection 00:07:57.160 --> 00:08:01.420 and map the new trajectory. Exactly. So mathematicians 00:08:01.420 --> 00:08:03.720 use a brilliant trick. Instead of having the 00:08:03.720 --> 00:08:06.540 ball bounce off the wall, imagine you take the 00:08:06.540 --> 00:08:09.240 entire billiard table and reflect it across that 00:08:09.240 --> 00:08:11.730 edge like a mirror. Oh, wow. So the ball doesn't 00:08:11.730 --> 00:08:13.449 bounce at all. Right. It just rolls straight 00:08:13.449 --> 00:08:15.810 through the wall into a newly attached mirrored 00:08:15.810 --> 00:08:17.750 copy of the table. So every time it would have 00:08:17.750 --> 00:08:20.050 hit a wall, you just build another imaginary 00:08:20.050 --> 00:08:22.149 table next to it and let the ball keep rolling 00:08:22.149 --> 00:08:25.490 in a perfectly straight line. Yes. And if you 00:08:25.490 --> 00:08:28.629 keep doing this, reflecting the table over and 00:08:28.629 --> 00:08:31.310 over and gluing the corresponding edges together, 00:08:32.070 --> 00:08:35.960 that flat jagged polygon eventually folds up 00:08:35.960 --> 00:08:38.440 into a closed continuous surface. Which usually 00:08:38.440 --> 00:08:40.899 ends up looking like a multi -hole donut. Exactly. 00:08:41.080 --> 00:08:44.139 A Ryman surface. That is wild. So tracking a 00:08:44.139 --> 00:08:47.960 chaotic bouncing ball on a flat table is mathematically 00:08:47.960 --> 00:08:50.320 the exact same thing as tracking a perfectly 00:08:50.320 --> 00:08:52.480 straight line wrapping around a multi -hole donut. 00:08:52.620 --> 00:08:55.600 You got it. On a curved surface, the equivalent 00:08:55.600 --> 00:08:58.980 of a straight line is called a geodesic. If you 00:08:58.980 --> 00:09:01.379 want to walk from New York to London on the curved 00:09:01.379 --> 00:09:04.200 surface of the earth, the shortest path, the 00:09:04.200 --> 00:09:07.519 geodesic, is an arc. Great. Now, mathematicians 00:09:07.519 --> 00:09:09.519 were obsessed with finding a specific type of 00:09:09.519 --> 00:09:12.299 path on these multi -hole donuts, a simple closed 00:09:12.299 --> 00:09:14.580 geodesic. Simple closed meaning what, exactly? 00:09:14.639 --> 00:09:16.820 Let's make sure we're clear. Sure. Closed means 00:09:16.820 --> 00:09:18.799 the straight path eventually loops perfectly 00:09:18.799 --> 00:09:21.299 back onto its starting point, like a rubber band 00:09:21.299 --> 00:09:24.259 wrapped around the donut. And simple means the 00:09:24.259 --> 00:09:27.100 path never crosses over itself. Mm -hmm. So no 00:09:27.100 --> 00:09:29.809 figure eights. Just a clean, continuous loop. 00:09:30.049 --> 00:09:32.450 OK, I'm with you. And mathematicians wanted to 00:09:32.450 --> 00:09:34.590 know how many of these simple closed loops could 00:09:34.590 --> 00:09:37.049 exist on a given surface. Right. It was an incredibly 00:09:37.049 --> 00:09:39.950 difficult open problem. University of Chicago 00:09:39.950 --> 00:09:42.690 topologist Benson Farb actually called counting 00:09:42.690 --> 00:09:46.370 these loops the key object to unlocking the structure 00:09:46.370 --> 00:09:49.080 and geometry of the whole surface. Because as 00:09:49.080 --> 00:09:51.440 the loops get longer and longer, they wrap around 00:09:51.440 --> 00:09:53.860 the donut in more and more complex ways. How 00:09:53.860 --> 00:09:55.620 do you count something that is stretching into 00:09:55.620 --> 00:09:58.620 infinity? And this is where Mirzakhani's genius 00:09:58.620 --> 00:10:02.059 fully reveals itself in her 2004 PhD thesis. 00:10:03.139 --> 00:10:05.259 She realized that trying to count the loops on 00:10:05.259 --> 00:10:08.159 just one single surface was completely the wrong 00:10:08.159 --> 00:10:10.879 approach. We had to zoom out. Yes. You had to 00:10:10.879 --> 00:10:13.019 look at the moduli space. This is a crucial concept. 00:10:13.200 --> 00:10:15.179 So a moduli space isn't a physical space, right? 00:10:15.379 --> 00:10:17.580 No. It is an abstract mathematical landscape 00:10:17.580 --> 00:10:20.779 where every single point in this space represents 00:10:20.779 --> 00:10:22.799 a different possible shape that your multi -hole 00:10:22.799 --> 00:10:25.000 donut could take. So if you stretch the donut 00:10:25.000 --> 00:10:27.340 to be longer, you move to a new point in the 00:10:27.340 --> 00:10:29.980 moduli space. If you pinch it so a hole gets 00:10:29.980 --> 00:10:31.799 smaller, you move to another point. Exactly. 00:10:31.899 --> 00:10:33.960 So she wasn't just studying one billiard table. 00:10:34.039 --> 00:10:36.759 She was studying the meta universe of all possible. 00:10:36.990 --> 00:10:39.870 billiard tables. And her big aha moment was realizing 00:10:39.870 --> 00:10:42.789 that counting the individual simple loops on 00:10:42.789 --> 00:10:46.450 a single surface is directly connected to calculating 00:10:46.450 --> 00:10:49.490 the total volume of that entire moduli space. 00:10:49.690 --> 00:10:52.509 Wow. By shifting her perspective from the micro, 00:10:52.909 --> 00:10:55.230 the single loop, to the macro, the universe of 00:10:55.230 --> 00:10:58.269 all possible shapes, she unlocked the whole problem. 00:10:58.539 --> 00:11:00.899 And the sources outline the actual formula she 00:11:00.899 --> 00:11:03.299 discovered. She proved that the number of simple 00:11:03.299 --> 00:11:06.000 closed loops of a certain length grows proportionally 00:11:06.000 --> 00:11:09.460 to a polynomial formula, the length L to the 00:11:09.460 --> 00:11:13.139 power of 6G minus 6. Let's translate those variables 00:11:13.139 --> 00:11:15.179 so we can see why it's so elegant. Yeah, please. 00:11:15.360 --> 00:11:17.299 So L is the maximum length of the loop you're 00:11:17.299 --> 00:11:20.019 trying to count. And g is the genus of the surface, 00:11:20.299 --> 00:11:22.200 which is just the mathematical term for the number 00:11:22.200 --> 00:11:24.500 of holes it has. So if our surface is a donut 00:11:24.500 --> 00:11:27.659 with two holes, g equals two, which means six 00:11:27.659 --> 00:11:30.799 times two minus six equals six. So for a two 00:11:30.799 --> 00:11:34.440 -hole shape, the number of simple loops grows 00:11:34.440 --> 00:11:37.000 proportionally to the length to the sixth power. 00:11:37.240 --> 00:11:40.080 Exactly. She didn't just guess this. She proved 00:11:40.080 --> 00:11:42.779 the exact underlying architecture governing how 00:11:42.779 --> 00:11:45.279 these loops behave. That's just brilliant. And 00:11:45.279 --> 00:11:48.009 she didn't stop there. She also provided a new 00:11:48.009 --> 00:11:51.129 proof for a long -standing conjecture by physicist 00:11:51.129 --> 00:11:53.929 Edward Witten regarding the mathematics of string 00:11:53.929 --> 00:11:56.710 theory. So she tied her geometric discoveries 00:11:56.710 --> 00:11:59.549 directly to theoretical physics. It's just astonishing, 00:12:00.129 --> 00:12:02.210 but she kept going deeper into that jungle, which 00:12:02.210 --> 00:12:04.970 brings us to another major concept she tackled, 00:12:05.289 --> 00:12:07.840 William Thurston's earthquake flow. Yeah, this 00:12:07.840 --> 00:12:10.259 sounds dramatic, but it's actually a very precise 00:12:10.259 --> 00:12:12.899 mechanical tool for mathematicians. Okay. Remember 00:12:12.899 --> 00:12:15.799 how we said moduli space is the map of all possible 00:12:15.799 --> 00:12:18.299 shapes a surface can take? Right, the meta universe. 00:12:18.600 --> 00:12:20.620 Mathematicians need a systematic way to travel 00:12:20.620 --> 00:12:23.000 through that space. They need a rule -bound way 00:12:23.000 --> 00:12:25.600 to deform one shape into another. The earthquake 00:12:25.600 --> 00:12:27.799 flow is one of those methods. How does it work? 00:12:28.019 --> 00:12:29.960 You take your multi -hole surface and you cut 00:12:29.960 --> 00:12:32.440 it along one of those closed geodesic loops we 00:12:32.440 --> 00:12:34.740 just talked about. Okay, so you physically slice 00:12:34.740 --> 00:12:37.139 the doughnut along a loop. Then you slide the 00:12:37.139 --> 00:12:40.059 two edges past each other by a specific distance, 00:12:40.559 --> 00:12:43.299 much like tectonic plates sliding past each other 00:12:43.299 --> 00:12:45.559 along a strike -slip fault line during an earthquake. 00:12:45.740 --> 00:12:48.879 Ah, hence the name. Exactly. Then you glue the 00:12:48.879 --> 00:12:52.299 edges back together. You have now created a mathematically 00:12:52.299 --> 00:12:55.399 distinct surface. You have moved to a new point 00:12:55.399 --> 00:12:58.409 in modulized space. But why do mathematicians 00:12:58.409 --> 00:13:00.870 actually want to do that? Like, what's the benefit 00:13:00.870 --> 00:13:03.429 of cutting and twisting these shapes? Because 00:13:03.429 --> 00:13:05.570 it allows them to map the dynamics of the space 00:13:05.570 --> 00:13:08.929 itself. Mirzakhani proved long -standing conjectures 00:13:08.929 --> 00:13:10.950 about how these theoretical earthquakes behave 00:13:10.950 --> 00:13:13.759 over time. OK. She showed that as you apply these 00:13:13.759 --> 00:13:16.100 earthquake flows, the transformations are not 00:13:16.100 --> 00:13:18.480 entirely random. They are measure preserving 00:13:18.480 --> 00:13:22.100 and ergodic, meaning they explore the space in 00:13:22.100 --> 00:13:24.779 a highly structured, predictable way. Which perfectly 00:13:24.779 --> 00:13:27.279 sets up her crowning achievement a decade later 00:13:27.279 --> 00:13:30.720 in 2014, the magic wand theorem. She developed 00:13:30.720 --> 00:13:33.320 this alongside mathematicians Alex Eskin and 00:13:33.320 --> 00:13:35.899 Amir Mohammadi. The magic wand theorem is arguably 00:13:35.899 --> 00:13:39.519 her most profound legacy. Imagine you are inside 00:13:39.519 --> 00:13:42.559 that moduli space, that universe of all possible 00:13:42.559 --> 00:13:45.620 shapes, and you decide to zoom off in a random 00:13:45.620 --> 00:13:48.899 direction, stretching and warping the shape continuously. 00:13:49.340 --> 00:13:51.740 It feels like that would eventually devolve into 00:13:51.740 --> 00:13:53.639 complete chaos, like you would just end up in 00:13:53.639 --> 00:13:56.340 a jagged fractal mess of unpredictable shapes. 00:13:56.460 --> 00:13:58.759 That's exactly what everyone assumed. But Mirza 00:13:58.759 --> 00:14:01.340 Khani, Eskin, and Mohammadi proved the opposite. 00:14:01.500 --> 00:14:04.039 They proved that the closure of these complex 00:14:04.039 --> 00:14:06.940 paths in moduli space are surprisingly regular. 00:14:07.139 --> 00:14:09.139 Wait, really? They don't just splatter randomly 00:14:09.139 --> 00:14:12.320 into chaos. No, they always land in neat, rigidly 00:14:12.320 --> 00:14:14.899 algebraic subspaces. The International Mathematical 00:14:14.899 --> 00:14:17.159 Union put in beautifully. They said it was astounding 00:14:17.159 --> 00:14:20.419 to find rigidity in an inhomogeneous world. That 00:14:20.419 --> 00:14:23.399 is the core of her genius. She proved that even 00:14:23.399 --> 00:14:25.940 in a system that allows for infinite mind -bending 00:14:25.940 --> 00:14:28.559 transformations, there are strict, beautiful 00:14:28.559 --> 00:14:31.039 underlying rules. She mapped the hidden order 00:14:31.039 --> 00:14:33.659 inside the chaos. And the global mathematical 00:14:33.659 --> 00:14:36.940 community recognized that immediately. In August 00:14:36.940 --> 00:14:40.340 2014, Mariam Mirzakhani was awarded the Fields 00:14:40.340 --> 00:14:44.080 Medal in Seoul. The Fields Medal is widely considered 00:14:44.080 --> 00:14:47.139 the Nobel Prize of Mathematics. It is the absolute 00:14:47.139 --> 00:14:49.600 pinnacle of achievement in the field. Her win 00:14:49.600 --> 00:14:52.519 was historic on multiple fronts. I mean, she 00:14:52.519 --> 00:14:54.659 became the first woman ever to win the prize 00:14:54.659 --> 00:14:57.919 since it was established in 1936. Wow. and she 00:14:57.919 --> 00:15:00.320 also became the first Iranian to win. But the 00:15:00.320 --> 00:15:02.779 sources reveal a pretty heartbreaking reality 00:15:02.779 --> 00:15:04.879 playing out behind the scenes of this triumph. 00:15:05.500 --> 00:15:08.019 Just one year prior to receiving the medal, in 00:15:08.019 --> 00:15:10.600 2013, she had been diagnosed with breast cancer. 00:15:10.820 --> 00:15:12.799 So while she is stepping onto the world stage, 00:15:13.220 --> 00:15:15.059 being recognized for mapping the universe of 00:15:15.059 --> 00:15:18.440 shapes, she's quietly undergoing severe medical 00:15:18.440 --> 00:15:21.080 treatments. And despite her incredible resilience, 00:15:21.500 --> 00:15:23.639 the cancer eventually spread to her bones and 00:15:23.639 --> 00:15:26.519 liver. Mary Mirzakhani passed away on July 14, 00:15:26.639 --> 00:15:29.799 2017 at Stanford Hospital. She was only 40 years 00:15:29.799 --> 00:15:32.759 old. It is a profound and premature loss for 00:15:32.759 --> 00:15:35.679 science. But the impact she had in those 40 years 00:15:35.679 --> 00:15:38.460 didn't just stay in the realm of abstract geometry. 00:15:38.700 --> 00:15:41.879 No, not at all. Her life and her death triggered 00:15:41.879 --> 00:15:44.139 immediate and massive ripples in the physical 00:15:44.139 --> 00:15:46.960 world, particularly in Iran. So what does this 00:15:46.960 --> 00:15:49.639 all mean? She studied these abstract earthquakes 00:15:49.639 --> 00:15:51.980 in geometry, but it sounds like she caused a 00:15:51.980 --> 00:15:54.460 literal cultural earthquake in her home country. 00:15:54.879 --> 00:15:57.139 What's fascinating here is the cultural reaction 00:15:57.139 --> 00:15:59.840 detailed in the sources. The Iranian president 00:15:59.840 --> 00:16:02.620 at the time, Hassan Rouhani, publicly praised 00:16:02.620 --> 00:16:05.899 her, noting how her scientific achievements made 00:16:05.899 --> 00:16:08.539 Iran's name resonate globally. And demonstrated 00:16:08.539 --> 00:16:10.960 the immense capability of Iranian women. Yes. 00:16:11.419 --> 00:16:13.460 But the reaction in the press is what really 00:16:13.460 --> 00:16:16.600 highlighted her cultural impact. Iran has strict 00:16:16.600 --> 00:16:19.779 societal rules regarding the hijab, and media 00:16:19.779 --> 00:16:22.200 protocol generally dictates that women must be 00:16:22.200 --> 00:16:24.519 shown with their hair covered. Right. However, 00:16:24.720 --> 00:16:27.480 upon her passing, several major Iranian newspapers, 00:16:27.879 --> 00:16:30.480 as well as President Rouhani and his public tributes, 00:16:30.899 --> 00:16:33.419 broke those taboos. They published photographs 00:16:33.419 --> 00:16:35.820 of Mirza Khani with her hair uncovered. Now, 00:16:35.940 --> 00:16:38.159 to maintain impartiality, we should note the 00:16:38.159 --> 00:16:40.080 sources mentioned that some newspapers adhered 00:16:40.080 --> 00:16:42.480 to the strict rules by digitally manipulating 00:16:42.480 --> 00:16:45.659 images or using paintings to obscure her hair. 00:16:45.960 --> 00:16:48.620 But the fact that so many mainstream outlets 00:16:48.620 --> 00:16:52.440 chose to publish her true image was noted internationally 00:16:52.440 --> 00:16:55.019 as a really significant break from protocol. 00:16:55.320 --> 00:16:58.460 And her impact extended beyond media representation 00:16:58.460 --> 00:17:02.320 all the way to national legislation. Her death 00:17:02.320 --> 00:17:05.039 actually sparked a massive political debate in 00:17:05.039 --> 00:17:08.099 Iran regarding citizenship laws. Because under 00:17:08.099 --> 00:17:11.079 Iranian law at the time, citizenship was passed 00:17:11.079 --> 00:17:13.220 down patrilineally, meaning through the father. 00:17:13.559 --> 00:17:16.160 Exactly. But Mirza Khani's husband Jan Vondrak 00:17:16.160 --> 00:17:18.359 is a Czech mathematician whom she met in the 00:17:18.359 --> 00:17:20.420 U .S. And they had a daughter together named 00:17:20.420 --> 00:17:23.339 Anahita. Because Anahita's father wasn't Iranian, 00:17:23.500 --> 00:17:25.839 she could not easily get Iranian nationality 00:17:25.839 --> 00:17:28.500 to visit her mother's home country. So following 00:17:28.500 --> 00:17:31.240 Mirza Khani's death, The sources state that 60 00:17:31.240 --> 00:17:33.400 Iranian members of parliament immediately urged 00:17:33.400 --> 00:17:35.779 the speeding up of an amendment to the law. Wow. 00:17:36.200 --> 00:17:38.119 They pushed to allow the children of Iranian 00:17:38.119 --> 00:17:40.619 mothers married to foreigners to be granted Iranian 00:17:40.619 --> 00:17:43.339 nationality, specifically citing the desire to 00:17:43.339 --> 00:17:45.440 make it easier for Mirza Khani's daughter to 00:17:45.440 --> 00:17:47.690 visit Iran. When you step back and look at that 00:17:47.690 --> 00:17:51.029 trajectory, it is staggering. I mean, she dedicated 00:17:51.029 --> 00:17:53.329 her life to sitting on the floor, deep in the 00:17:53.329 --> 00:17:56.210 mathematical jungle, studying abstract earthquakes 00:17:56.210 --> 00:17:59.309 and the rules of imaginary morphing donuts. Yet 00:17:59.309 --> 00:18:02.289 her actual life caused a literal cultural earthquake. 00:18:02.769 --> 00:18:05.869 She inadvertently forced a reevaluation of strict, 00:18:06.089 --> 00:18:09.170 real -world societal rules simply through the 00:18:09.170 --> 00:18:11.950 undeniable magnitude of her existence. It's incredible. 00:18:12.170 --> 00:18:14.109 And her memory continues to be honored around 00:18:14.109 --> 00:18:16.680 the world today. International Council for Science 00:18:16.680 --> 00:18:19.200 declared her birthday, May 12th, as International 00:18:19.200 --> 00:18:21.950 Women in Mathematics Day. High schools and university 00:18:21.950 --> 00:18:24.349 libraries in Iran have been named after her. 00:18:24.809 --> 00:18:26.950 There is an Iranian microsatellite, named in 00:18:26.950 --> 00:18:31.549 her honor, and an asteroid. Asteroid 321357 bears 00:18:31.549 --> 00:18:34.750 her name. That's so cool. In 2024, the International 00:18:34.750 --> 00:18:37.529 Astronomical Union even named a lunar crater 00:18:37.529 --> 00:18:39.769 after her. And the Breakthrough Prize Foundation 00:18:39.769 --> 00:18:42.730 created the Maryam Mizikani New Frontiers Prize, 00:18:43.109 --> 00:18:46.130 which is a $50 ,000 award to support early career 00:18:46.130 --> 00:18:49.029 female mathematicians. She fundamentally altered 00:18:49.029 --> 00:18:51.259 what the world picked when they think of a mathematical 00:18:51.259 --> 00:18:53.720 genius. She really did. For you listening, if 00:18:53.720 --> 00:18:56.200 you are tackling a complex problem right now, 00:18:56.339 --> 00:18:58.440 whether you're trying to learn a new coding language, 00:18:59.160 --> 00:19:01.799 analyzing a massive data set, or just figuring 00:19:01.799 --> 00:19:04.660 out a complicated life decision, Mirza Khani's 00:19:04.660 --> 00:19:07.799 process is a master class. True mastery doesn't 00:19:07.799 --> 00:19:10.059 always look like furious speed. Sometimes it 00:19:10.059 --> 00:19:13.000 requires you to sit down. lay out the pieces, 00:19:13.279 --> 00:19:15.460 and just be willing to wander slowly through 00:19:15.460 --> 00:19:18.359 the jungle until the underlying structure reveals 00:19:18.359 --> 00:19:21.099 itself. That is the enduring lesson of her slow 00:19:21.099 --> 00:19:23.579 math. And if there is a final thought to mull 00:19:23.579 --> 00:19:25.700 over, building on the mathematics we've explored 00:19:25.700 --> 00:19:28.880 today, it is the profound contrast between her 00:19:28.880 --> 00:19:32.400 work and her life. How do you mean? Well, Mirza 00:19:32.400 --> 00:19:35.059 Khani proved that in the hyper -complex, shape 00:19:35.059 --> 00:19:37.660 -shifting universe of moduli spaces, there is 00:19:37.660 --> 00:19:41.160 a surprising hidden rigidity and order. Chaos 00:19:41.160 --> 00:19:43.880 resolves into elegant algebra. Right. But looking 00:19:43.880 --> 00:19:46.400 at her life cut tragically short yet permanently 00:19:46.400 --> 00:19:49.200 altering mathematics and Iranian societal norms, 00:19:49.680 --> 00:19:51.539 perhaps she proved the exact opposite about the 00:19:51.539 --> 00:19:53.660 human condition. Oh, wow. Yeah, in the rigidly 00:19:53.660 --> 00:19:56.059 structured, seemingly inflexible systems of our 00:19:56.059 --> 00:19:59.119 society, a single brilliant mind can introduce 00:19:59.119 --> 00:20:01.880 a beautiful, unpredictable change that shifts 00:20:01.880 --> 00:20:02.980 the shape of the world forever.