Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani

[indistinct chatter] - There is a very good feeling behind solving the problems.

You know, when you try hard, hard, and you can't find the solution, and suddenly, you say, "Oh!

That's it."

And I think Maryam Mirzakhani could show this passion to everyone.

[delicate music] - Everywhere you see different types of arts which starts from mathematics.

For example, the buildings, mosques, and the monuments that we have here in Isfahan are full of mathematical ideas.

- When I look at some papers or talk to my collaborators, there's some nice idea.

I feel really fascinated.

It's like listening to music or seeing an amazing drawing.

It's really like art.

And by the: - Maryam Mirzakhani's pioneering work gave mathematicians powerful new tools for understanding the geometry of surfaces.

In 2014, she became the first woman to win the Fields Medal, the most prestigious prize in mathematics.

- The crux of good mathematics is to find problems that are well-motivated and important and somehow centrally connected to lots of other mathematics.

Maryam's work not only solved problems, but it also brought other problems into reach.

- Now I had my half circles here, and I added this half circles which depended on a-1 to a-N.

But that basically, a-1 to a-N, determines a shape like this.

- In mathematics today, we are at a kind of golden era.

Topology, algebraic geometry, ergodic theory, which were sort of developed separately, have been combined to solve long-standing problems.

- Maryam would see influences of all these different fields on one particular problem.

These were all part of the same world, and, you know, they were all connected to one another.

Instead of trying to understand just implications, she was trying to understand how things would interact.

[delicate music] [children chattering indistinctly] [indistinct chatter] - I wasn't very good in math even.

I mean, in middle school one year, I didn't even do well in math 'cause I was basically not very interested.

I was mostly interested in reading novels and books when I was younger.

And I even remember the time that I would every night tell myself some stories.

And night after night, I would continue the story.

I thought one day I would become a writer.

- She would just lock herself up in her room and get, like, a big bowl of apples and just eat those apples one by one and read books, like, the whole afternoon.

- Maryam have two brothers, Orash and Ali.

Orash was actually the one who maybe started her interest in mathematics.

- From her elementary school and from her family, she learned how to study mathematics, how to think mathematics.

- About the time of the Iranian Revolution, these two schools were set up in Iran for exceptionally talented students-- one for boys and one for girls-- originally just in the capital, Tehran.

- Students from all the different school districts of Tehran would take an entrance exam, and then the top maybe 90 or 100 of them would get in.

[indistinct chatter] - When we entered high school, I decided math because I liked it.

And the high school really supported us with resources, teachers, 'cause this school is where Mirzakhani rised from.

- Maryam and I met in 1988.

We had just started middle school, and we became close friends quickly.

And we continued to share a desk for the next seven years, middle school and high school.

- Math teacher was returning our tests to us, and Maryam got her test back, and she had a score of 16 out of 20.

She was unhappy and she said, "You know, that's it for me.

I'm not going to even try to do better."

But things changed.

When we went to the seventh grade, she started to do very well.

- She made a point to solve any problem in hand from several different ways.

It was even a joke that Maryam has solved this problem by four different ways.

- We had a geometry teacher, Mr. Niousha.

And he always talked about this other school he taught at, and these two girls who were brilliant there, and that we should all learn something from them.

Those were Maryam and Roya.

Almost every week in our class, he was talking about what they had been up to and how fast they had solved the problem.

- It was always very motivating to try to compete with the boys.

- I was Maryam's classmate when we were 14, 15 years old.

Maryam was so fast.

It was magnificent that she could solve the problem which was really difficult for all of us.

Sometimes she even couldn't explain.

She could think so fast.

- What made her so successful was her hard work, the fact that she was very focused on her studies and her research, and she practiced a lot.

Practice, practice, practice.

- Maryam, to me, was a child prodigy.

She was not recognized as this, even by her parents, by her teacher.

But since she did very excellent mathematics when she was young, this is the first definition of being a child prodigy.

- One way for attracting young people to mathematics is having the competitions.

So we started mathematics competitions in Isfahan in 1983.

And many students came to mathematics because of these competitions.

- We would hear about the students who had done well in those competitions, so it was always very exciting to try to guess how we would, you know, do compared to them.

- We went to the principal of our school and demanded that we wanted to participate in this competition.

And she was very supportive, and we did participate that year in the national round of the competition.

And there were no girls in the national team before us.

- The main topic of the Mathematical Olympiad is problem-solving.

We have problem-solving classes in four major fields of mathematics.

That's number theory, combinatorics, geometry, and algebra.

- I gave them a lot of problems that I found from different sources.

One of these problems that I didn't know how to solve, she kind of wrote down a solution, and she gave it to me.

So I had to go pick up a textbook on complex analysis and read it to just learn the terms that she was using and then see the lemmas she was using are actually correct.

So I had to learn the basics of complex analysis to be able to understand her proof.

- In that program, I used to give two lectures about every year.

And I used to introduce them some open problems.

- Ebad Mahmoodian brought to the attention of the students a problem set.

And it was a good problem in graph theory and combinatorics.

Maryam proved the general case, and they published it.

This was the first paper that Maryam published.

- Roya and Maryam were the first female students who got entered on the list of Olympiad entries for the international team.

- When we participated in the IMO, it was in 1994 in Hong Kong.

Maryam got a gold medal; she got 40 out of 42.

And I got a silver medal.

- I think it showed me somehow the beauty of math, because I got excited about it, maybe just as a challenge.

But then I realized that it's really nice and that I enjoy it.

So it gave me the opportunity to think more about some math problems.

- It was '95.

Olympiad was, I think, in Canada.

She received gold, and this time it was full marks, 42 out of 42.

- She came back downplaying her achievement.

She said that it was the only year that it was so easy to get a full score.

- At that time, there was only three people who had achieved these full marks at the IMO from Iran.

And so, in our eyes, she was some kind of hero and champion.

- She became well known.

Her picture was in the papers.

More women came into mathematics after that.

- Maryam was like a superstar in any home at that time.

Being a girl and having earned a gold medal at Olympiad was very inspiring and very heartwarming for all the students back home and me as well-- knowing that it's not impossible.

- In Iran when we were growing up, there was never any kind of negative perception about women doing math or science.

We never got the impression that math was an unfeminine profession to take.

- The fact that women are on par with men, at least in terms of abilities, is not new in Iran.

I think that aspect has been and is reflected by the composition of the students in universities in Iran today.

- In the higher education, we have always more than 50% are girls.

At Sharif, more than 40% freshman students are the girls.

- Women in Iran are not a privileged group, so they have to try to find a better social situation by entering into art, in science.

To belong to a disadvantageous group has this advantage that you are forced to work hard.

[indistinct chatter] - Maryam, after finishing school, she decided to go to Sharif University.

- As members of the Iranian Olympiad teams, we did not have to take the entrance exam.

We could choose anything that we wanted to do in college.

- Her first choice was to come to the Department of Mathematical Sciences of Sharif University of Technology.

- We attended the university from '95 to '99.

The chair of the department was Professor Tabesh, and we got a lot of support from him.

- We made a center of excellence, let them grow up by their own pace, not control them, not push them.

Early in the morning, they gathered in a circle in front of the math department, talking mathematically with each other.

Roya was there; Maryam was there.

- We had a curriculum that was very open.

The students could move up easily.

They could take graduate courses while they were undergraduates, as Maryam did.

We gave them more freedom than other universities.

- We played a lot of soccer.

Maryam always said she wants to come play, the only girl who actually ever did.

She came up and argued with the person to let her in and play the game.

- She became also a member of the Olympiad committee and she was trainer of the new Olympians.

In sitting with her and looking how she is thinking to design new problems, she has very deep insight about mathematics.

- Her papers were published when she was an undergrad.

Pretty funny story is that she was asked to referee somebody else's grant proposal when she was an undergrad, because she was already building a reputation based on her results.

- I decided to participate Olympiad, and I started by reading some books.

And one of them was Maryam number theory book.

There was something special and different about that book.

It was written by two Iranian women, Maryam and Roya Beheshti.

- It has both the beautiful problems of Mathematical Olympiad and the deep mathematical theorems that you can find in university books.

- Before she writes this, there was no resource for the students to study.

And she wrote something very complete and whole for students who want to learn Math Olympiad.

After many years, we still use it; we still learn from it.

- We went to Ahvaz in the south of Iran to participate in this student conference.

- About 40 students, they went to the city of Ahvaz in the southern part of Iran with a bus.

They had their conference and they had the contest.

Maryam became the first in the contest.

- We left Ahvaz right before midnight.

It was raining, and on the way back, the bus went off the road and it fell down in a river.

And several of our friends were killed.

- In the darkness, the driver did not recognize the road properly, therefore the bus crashed right in a big valley and some of the students died instantly.

- Maryam was in the bus; Roya was in the bus.

Roya got some-- needed some surgery.

- So I spent a couple of nights in the hospital there.

My liver was damaged, so there was a surgery.

And there were two other people, Kia and Iman.

- People would come and visit and see how I am doing, and Maryam was one of them.

She became very emotional, and I had no concept of why.

Only later I realized that she had--she had assumed that I had passed away.

- This was a great loss to our science community.

We never forget that.

- [indistinct speech] - It was kind of assumed that if you were at the top of the class and you wanted to continue and go for a PhD, you would leave the country.

- Professor Shahshahani suggested that in order to increase the chances of all of us getting admitted to good universities, we should each choose a school to which the others would not apply.

Maryam chose Harvard and I chose MIT because we wanted to be close to each other.

- In the fall of 1999, I got a phone call from Professor Joe Harris.

He was part of the admissions committee for the math department for the grad school.

And he was calling me to ask me if I know anything about Maryam Mirzakhani who had applied for admission to Harvard grad school.

- Cumrun Vafa, he called me and he said, "Were you serious about this letter of recommendation?

"Because they're considering giving her, you know, admission."

- He told me that we should expect great things from her.

And so I was kind of surprised by his certainty.

- It turned out that five of them got this first choice.

[sweeping orchestral music] - I think Maryam's story was something similar to me.

It was about a girl from my country in the real world.

- Living independently on your own as a girl coming from another country with a whole different cultures and whole different traditions, it was very challenging.

Just renting a place to live, going over the bills like credit cards-- things like that that you would have never had to deal with.

- You're used to speaking your native tongue.

You think you speak a foreign language, and then you arrive and you realize that that's not quite the case.

- It was the first time we were coming to the U.S.

I didn't know anything about the city, the universities, really nothing.

I was a little bit worried, but this went away as soon as I arrived.

- Maryam went to Harvard and Roya went to MIT, so they were both in Boston.

And Maryam's and Roya's friendship continued in Boston.

- I would sometimes go to Boston and meet those two friends.

And they were actually inseparable.

Every time I see Maryam, Roya was with her.

Every time I see Roya, Maryam was with her.

They were like sisters.

- I never felt even, you know, homesick the first year I was there because there were so many good friends there.

- There was the Iranian Studies group that we had started, and we set up seminars and talks and then some movie nights.

- I went to her house to help her cook.

She was making biriyani, some Iranian rice dish where, like, you know, you burn the bottom of the rice a little bit.

So she was very busy into that.

She put on some music, started dancing around while she was stirring the rice.

She was a little bit reserved.

Even though, like, you know, we were some of the closest friends of hers in graduate school, I'm not sure that she ever opened up to us about her political beliefs or religious beliefs.

- This point here is not prime, because if I try to draw...

I first met Maryam in my research seminar.

A lot of times, graduate students gave presentations.

So it was really like an open classroom.

- She started working with Curt McMullen in her second year.

She was a little bit frustrated that the problem that she had started working on, she couldn't solve it.

She was unhappy.

- She wasn't quite sure that McMullen, her advisor, was paying enough attention to her.

- So I suggested she give a talk on this curious formula of McShane.

And she gave a very beautiful presentation.

She would always arrive at my office with a long list of questions and describe these speculative mathematical scenarios.

We would just spend our time kind of musing about whether or not these things made sense and how you might approach them.

- A lot of Mirzakhani's work had to do with the geometry of surfaces, like the surface of a ball or the surface of a doughnut, what mathematicians call a torus.

And in her doctoral dissertation, she focused on a very basic question about surface geometry, which is about what happens if you start somewhere on the surface and walk in a straight line.

Depending on your surface, your path might close up on itself.

For example, if you're walking straight on the surface of the Earth, you're going to go around a great circle like the equator and end up back where you started.

And the same thing happens on a torus-- a doughnut surface-- if you walk straight toward the central hole, you're gonna dip down into the hole and underneath and back around and end up back where you started.

But if you were to walk straight in a different direction on the torus, your path might wander around forever and never get back where it started.

Mirzakhani's doctoral dissertation focused on the straight paths that eventually close up and never cut across themselves-- what mathematicians call "simple closed curves."

These curves are easy to categorize on a sphere or a torus but not easy to categorize on a torus with more than one hole.

- Simple loops on this object are much more complicated, possibly, because you can--you can go through one of these holes, go around, come back the back side...

The first thing that she did in her thesis was, she started with a very natural problem.

You have a surface that's more complicated than a sphere or a torus.

It has a couple of handles, as we say in mathematics; its genus is bigger than one.

And she counted the number of simple curves on this surface, the number of loops that don't cross themselves and whose length is bounded by constant-L. - To understand the simple closed curves on a surface, you have to know the surface's precise geometry.

And when we talk about a torus or a torus with two holes, we've said something about the shape of the surface but not everything.

We could mean a really plump torus or maybe a long, skinny torus.

And the same thing is true for surfaces with more than one doughnut hole.

These surfaces have a more complicated geometry called hyperbolic geometry.

But once again, we get to choose certain details about the geometry, like how thick or thin the handles are.

And the geometry we choose will determine how many simple closed curves there are of different lengths and where they live.

There's this concept mathematicians have called the moduli space that lets you consider all the different possible geometries on a surface in one fell swoop.

The moduli space is like a map, and every point on the map is the address for a different possible geometry that could live on your surface.

Your favorite plump doughnut is gonna live at some address on the map, and a long, skinny doughnut is gonna live at some other address.

And two doughnuts that are roughly the same thickness will live in the same neighborhood on the map, while a doughnut with a different thickness is gonna live further away.

And when Mirzakhani wanted to count the simple closed curves on some specific surface, she ended up looking at all the different surfaces at the same time.

She looked at the moduli space.

- And to do that, she had to, in turn, compute the volume of these so-called moduli spaces.

When she computed these volumes, she got some simple-looking polynomials with pis and rational numbers in them.

And there was an interpretation she could give to these numbers and to the recursive formulas she found for them, which led to a new proof of a famous result in string theory or mathematical physics that was originally called Witten's Conjecture.

- Maxim Kontsevich, who gave the other proof of Witten, at some point told me that he believed that Maryam's proof of this conjecture was perhaps more elegant than his.

So this shows that even though it was not the first proof of this conjecture, it was a very elegant proof.

- Those three papers that made up her thesis were published in the three top journals in mathematics, which is an incredibly rare achievement for a grad student to make it into any one of the those journals.

It's the "Annals of Mathematics," "Inventiones Mathematicae," and the "Journal of the American Math Society."

People are still reading them and extracting ideas from them.

- People are different, and there are different styles of doing math.

But if you want to get a reasonable good idea, you have to really spend a lot of time just thinking patiently and not getting disappointed somehow, coming back to the same problem.

And staying somehow confident that maybe one day you will have a good idea.

- She was extremely focused.

And I think she had this attitude that, I--you know, when she was, you know, trying to solve a problem that, you know, "I can do it.

I'm going to, you know, solve it."

- Whenever she saw an obstacle, that was just a puzzle that you had to go around or over.

Like, she didn't really take obstacles too seriously, I think.

- Rather than asking questions, she would start describing sort of elaborate stories or mathematical narratives.

And these narratives were very ambitious and speculative.

They were almost like science fiction.

She could make these very vague speculations about the shape of the unknown mathematical frontier quite elaborate and detailed in imagining how things might fit together.

And she was very adept at finding the right question.

And I think this came from this sort of speculative intuition that she had.

She would sort of think ahead as to what the shape of the theory might be that was yet to be discovered.

- Maybe a couple of months after she had met Jan, she told me that, "You know, I have been going out with this guy and"... - She has met this tall Czech man who's a student as well.

Not a lot of detail.

Maryam didn't share a lot of details.

- She was graduating in 2004, and we met in the fall of 2003.

She was very modest; you know, a normal girl.

She didn't talk about her work at all.

At first I had no idea that she was so accomplished.

I knew that she was a PhD student at Harvard.

- They kind of held hands at the corner of the room.

And I was like, "Oh, this is new.

This is not"--this wasn't happening as far as I knew.

The fact that Maryam started dating a non-Iranian was kind of breaking several boundaries in our community.

It was beautiful that Maryam didn't have those barriers as much internalized.

- They had both a civil marriage and also a Muslim marriage at Princeton.

- We went into the mountains.

They got married there.

We hiked back, and we cooked.

That was--the wedding meal was salmon with peach and mango.

- My impression was that Maryam and Jan were very happy together.

[regal music] - We got wind of her thesis and her achievements at Princeton and quickly realized we--this is somebody we have to recruit.

- She got a very prestigious postdoc position, the Clay Fellowship, and she decided to go to Princeton with it.

- So I first met her at Princeton when she was a postdoc and I was a visitor at the Institute of Advanced Study.

We kind of just kind of hit it off right away.

I really enjoyed talking to her.

We started collaborating.

She was absolutely amazingly brilliant, obviously.

She was very young.

She had already had quite a bit of success, but she was--somehow it didn't quite go to her head.

She was still the nicest person I've ever met.

[wind whistling] She came to Chicago once when we tried to hire her.

And that coincided with the biggest storm in the history of the city.

Temperature was somehow something like 20 below or something like this.

She never came back.

[laughs] - Actually, before she left Princeton we had already promoted her to tenure.

For various reasons, the usual two-body problem, she'd moved to Stanford where they made her a permanent offer.

[lilting music] - Her teaching load was reduced for some time.

She had quite a lot of freedom.

So we had a very good time at the beginning in California.

- The most satisfactory part of the teaching, for me, comes from discussing things with students outside of the classroom, more than covering a certain material during the lecture.

- I remember coming to her once.

And I'd completely destroyed my proof.

I'd found a hole, and then the whole thing just, like, unraveled.

So I was like, "Maryam, this thing has unraveled."

And so--and then she was like, "No, it's okay."

And we just started talking about it.

And by the end of that conversation, we'd worked something out, where it was like, "Oh, okay, this is the thing to focus on; this will probably work."

- To talk to younger mathematicians, it's very exciting, because they are more optimistic about things.

They don't know all the technical difficulties necessarily, so they are more open, and they are more willing to try new directions.

- It's unquestionably the best part of the job: to find somebody that you can connect with and have them explain their ideas to you and be surprised and happy at their ideas and try to explain your ideas to them.

[clears throat] Yeah, I mean, she was the best part of Stanford for me.

- Basically these angles are the same.

And then it hits here, and the same.

And it goes on.

- In the middle of, you know, explaining something, she suddenly would get too excited and then lose track of what she had been talking about.

She would jump to something without defining things.

- When she was trying to describe some mathematical idea that she was grappling with, she would make these large hand gestures.

She was very animated.

She had these beautiful blue eyes, and, like, you know, you would see a light in them when she was excited about some mathematical thought that she was having.

- She had this catchphrase, like, "How can it be?"

If something had to be true, she'd be like, "How can it be that this isn't true?

"How can it be?

"How can it be that this particular thing doesn't work?"

- She bought these large format notepads and she started using them mostly on the floor.

I think this is all about distributions of closed geodesics.

You can see some surfaces here with holes on the hyperbolic surface.

But certainly what you can see here is not the full complexity of the problem that she was thinking about.

The full thing was only in her head.

[inquisitive music] - One of her main achievements, which was recognized in her Fields Medal, was, she solved a problem called Ratner's theorem for moduli space.

And she did this working with two other mathematicians, Eskin and Mohammadi.

- This problem that she solved stems from a very simple question: if you hit a ball on a billiard table, what are the possible ways that it could travel?

Maybe it's gonna roll into a corner or maybe it's gonna bounce around in a cycle or maybe it's gonna follow a more complicated trajectory.

And here we might be talking about our usual rectangular billiard table or we might take a more complicated shape like a triangle or a hexagon.

And for most of these different shaped billiard tables, it's really hard to understand what the different possible trajectories are.

But there's a way to transform this problem into a problem about the geometry of surfaces, which was one of Mirzakhani's specialties.

So imagine that we put a mirror along every side of our billiard table.

When a ball hits one of the walls, it's going to bounce off, but in the mirror, it's going to look as if it's traveling along a straight line.

When it hits a wall in that looking-glass world, it just rolls straight into another looking-glass world, and so on.

So from this perspective, it looks as if the ball is just rolling forever along a straight line.

We've simplified the path of the ball, but we've made the world that it's traveling through more complicated.

- And instead of looking at the reflection, just continue the path.

And so knowing things about this shape is the same as knowing things about the original billiard.

- Now, for the kind of billiard tables Mirzakhani studied, it turns out that if you unfold your billiard table into this series of looking-glass worlds, you're eventually going to get to one that's a perfect copy of your original billiard table, just shifted over.

And any further unfolding is just going to give you repeats of things you've already seen.

We can imagine that we take the last table before the repeats started and glue it to our original table, instead of unfolding it onto a new table.

And if you do all these different gluings, you end up with a surface.

And this surface gets a very precise geometry that comes from the billiard table.

This geometry is not the only billiard table geometry we could imagine our surface having.

Maybe there's some other billiard table that glues up into the same surface but with a different geometry that maybe makes the handles fatter or skinnier.

Just as with Mirzakhani's earlier work, we can talk about a moduli space.

But this time, our moduli space is a map of billiard geometries and related structures called flat geometries.

So the torus we made out of a rectangular billiard table lives at one address on the map, and the torus we built from a triangular table lives at a different address.

For understanding a particular billiard table, Mirzakhani used an approach that looks at all the different billiard geometries at the same time in the moduli space.

Our particular billiard table lives at some point in the moduli space map, but there are going to be other billiard table geometries that have a special affinity for our original billiard table.

If we pick some trajectory on the table, we could imagine squashing the table in the direction of the trajectory and stretching it out in the perpendicular direction.

That's going to produce a new billiard geometry, so it's going to move us to a new address in the moduli space map.

Or we could shear the billiard table in the direction of the trajectory, and that's gonna produce yet another geometry.

These deformations move us to new points in the moduli space but not to every possible point-- only the ones that have this kind of special affinity for our original billiard table.

And these points that we get to are called the orbit of our original billiard geometry.

So we've translated our original problem about a ball rolling around on a billiard table into a problem about a point rolling to different locations in the moduli space.

And if you could understand what this orbit looks like in the moduli space, that would tell you something about what the trajectories can look like in your original billiard table.

- You can understand how many closed trajectories there are, how many ways a billiard ball can bounce back to itself.

You can understand how a billiard ball spreads around on the table.

All of these things come from understanding the orbit closure.

- This is a problem that concerns surfaces with any number of handles.

And it's kind of a big guiding conjecture in the field.

And nobody knew if we would ever make progress on it.

- And one of the goals is to understand the behavior of these orbits.

Now, Ratner's theorem is a certain dynamical system which comes from homogeneous dynamics in which you can actually understand the orbit of every single point.

- I was lucky enough to make the first breakthrough on this problem by showing that it was true for surfaces with two handles, surfaces of genus two.

And I remember Maryam coming to my office one day and saying, "Well, why did you just do genus two?"

[laughs] And then--and I sort of-- you know, it was sort of like saying, "Why did you just climb Mount Everest?

Why didn't you climb all the mountains in the Himalayas?"

- Trying to prove a theorem is like trying to climb a mountain which nobody has ever climbed.

So we're climbing up this mountain.

And at some point we thought that we were kind of slowly going up.

And at some point we kind of felt like we could see the top, but then there is this ravine in front.

And then there was a difficulty which we really couldn't overcome.

It was a little bit discouraging.

We basically sank two years of immense effort.

And we were stuck.

- They thought that the problem is basically solved, and they realized, "No, there are complications."

And it took another several years of very hard work.

- It was a bit depressing.

But she stayed very positive.

But somehow, yeah, I mean, basically it was like, you know, "I have no idea.

Do you?"

"No, I have no idea."

[laughs] - At some point, they made an announcement that they solved the problem.

And they used some result which contained a typo.

And this typo was dramatic, because part of the proof just evaporated.

- There was about a year and a half where we had made absolutely no progress.

Until the very end, we were not sure we had it.

We really didn't know if the whole project would even be done.

So it was incredibly intense experience.

And I mean, Maryam was just absolutely amazing, both in terms of mathematics, but also I think there was just a lot of kind of mental stability aspect, which is necessary for this kind of work.

And she was amazing at that as well.

- Sometimes you'd see she was happy.

Like, she said, "Oh, I have this idea, "and I hope it's correct.

"Actually, let me not think about it for a few hours "before I discover that it was wrong, "because I want to enjoy the feeling that it might be correct."

- Good mathematicians have the courage to imagine that they can solve a problem and to imagine an answer.

Having the insight or faith that the solution can be figured out is really very important.

Certainly, I think her personality was ideally suited to thinking about problems for a really long time and persevering on them.

- We climbed all the way down to the bottom of the mountain, and then we found another approach, going up the other side.

And it took us maybe another two or three years to actually get the thing done.

- So what Mirzakhani and her collaborators did was, they essentially climbed all the mountains.

They proved this theorem not just for a torus with two handles but a torus with any number of handles.

And the theorem they proved was so powerful that mathematicians started calling it the magic wand theorem.

- Consider the points in this sophisticated moduli space associated to this particular point, and you touch it with this magic wand.

It grows; it makes grow the orbit inside this moduli space, and the geometric properties of this manifold are responsible for all geometric and dynamical properties of this particular initial surface.

So really, for me, it's an image like in Cinderella story.

- It's a huge breakthrough.

A bunch of problems that you could not even approach, you couldn't solve at all, you couldn't even think about them, you can now solve just by applying this theorem.

- This work of Mirzakhani and Eskin is the central achievement in Teichmueller dynamics.

And everything that's happening in the field, you see her work is cited almost instantly in a lecture.

[light music] - We had a happy life.

And it was not all about math.

We liked music.

We liked sports.

We traveled together.

It was, in many ways, a very normal life.

- She was so friendly.

And she would just instantly was, like, peppering me with questions about how you manage life as a parent and as a mathematician.

- She was very much devoted to Anahita.

And family in general meant very much to her.

- Maryam had a lot of things happen in her life.

She--she got pregnant, she had a baby, and then in the final stages, like, once we were getting out-- gotten over the main hump, she was actually diagnosed with cancer.

- Sadly Maryam got sick just before Anahita turned two.

She was diagnosed with breast cancer, and then the sequence of treatments, you know, surgeries.

If she had to choose between doing her job on a given day and, you know, doing something important for Anahita, she would actually pick Anahita.

- She stayed very positive.

After her treatment, we thought that she was completely recovered.

She started to again focus on work.

- I think right after her thesis, everybody was saying that this person should win a Fields Medal.

[applause] - We have our fourth Fields medalist, Maryam Mirzakhani, for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

- It was fantastic that Maryam was awarded the Fields Medal and became the first woman to be so recognized.

So it was a great moment when she actually was able to attend the ceremony, receive the Fields Medal from the president of Korea.

And she brought her family with her, Jan and Anahita, and they were sitting in the front row when I was giving the laudation.

- I think mathematics is, in my opinion, necessary for all the advances in sciences and technology, so... - She had told me very early on she didn't want to give a whole series of interviews.

And we both knew-- I mean, the first woman to get a Fields Medal, that this would be something that everybody would want to talk about.

We are here this year... - Ingrid Daubechies, who was the president of the International Mathematical Union at the time, organized a group of women who would surround Maryam kind of wherever she went and protect her.

- We formed what we called the MM shield, the Maryam Mirzakhani shield.

Whenever Maryam was coming out publicly at ICM, two people of our little group would be with Maryam.

One of them could intercept a journalist, and said, "Oh, "you're very interested in women in mathematics?

Let me tell you about my career," and so on, while the other one could escort Maryam discreetly away.

- A lot of people were contacting me so that I could, you know, ask her to, you know, go give a speech or sign a letter.

She really didn't want to.

- She did talk about some pressure coming from Iran, people that thought that she should now play a very large political role.

And she didn't want to play a political role in Iran.

- With the amount of attention that she gets, every word has outsize weight.

You can't just, in passing, make a comment about something you like or you don't like.

Different groups will take this, and they would, you know, create a cause out of this.

She was very aware of these dangers of just putting herself out there.

And so she decided not to do that.

- When she received the Fields Medal, the parents heard the news from the media rather than from her.

And when they asked her, "Why didn't you tell us about this Fields Medal?"

She said, "I didn't think it was a big deal."

- Yeah, she did not like publicity.

She did not even like to let people know when she comes back to Iran.

When she used to come to see her family, she didn't like to let other people know.

- A few months after she received the Fields Medal, I heard from her that she's visiting Harvard, and I knew that the Iranian students would have loved to have a big celebration to honor her achievement.

- When she came to the gathering for the lunch, everybody was super excited.

You had somebody who has always been in your mind since middle school, and finally you are gonna meet her.

- She became a symbol of hope for the young generation in Iran in some sense, because she was showing that even coming from times of war and lots of restrictions and limitations, you can do important things.

For the Iranian community, I think that was a really proud moment.

To this date, I think Maryam is one of the few figures that really unites the whole country.

[playful flute music] - The fact that she was the first woman to earn the Fields Medal, I'm sure that will be an inspiration for more women to come into mathematics and sciences.

- There are so few women in mathematics, so we always celebrate when we hear of a fantastic new young woman.

The mathematics we do is exactly the same mathematics.

It's not that there is a female way of doing mathematics.

We also feel it's sad that so many young women think mathematics is not for them when they could be mathematicians.

- One thing that does stop people from going into math, and maybe particularly girls, is this idea that math will take over your life and you won't be able to do anything else.

Yeah, math will take over your life, but if you really keep a focus on what's essential in life and what's really important, you can be a complete human being and still be a mathematician.

- I think it will change after some time.

And--and-- but it's not something that will change by-- in two or three or five years.

It can take decades, but things like this change.

And there are examples of these kind of changes.

- Maryam's example shows that the pursuit of knowledge is a borderless, timeless, and, yes, genderless adventure.

Her example will convince many others who may not have otherwise done so pursue science.

- People look at her and see what she accomplished.

And girls will look at that and say, "That can be me."

- I think she's raised the bar for girls, but beside that, she's opened up a new way.

She has proved to us that it doesn't matter being the first one.

You can do anything that you want to when you put your desire and your effort in.

- She was one of us.

She was just like us in this school.

That helps us to think that why can't we do what she does?

And makes us feel way better about what we are.

- It is a great honor for me to be here to have the opportunity today to talk to you about some of the mathematical work of Maryam Mirzakhani.

Given a hyperbolic Riemann surface X and a positive number L, how many closed geodesics does X have, of length, say, less than or equal to L?

[indistinct chatter] - This is part of an international initiative to celebrate May 12th, the birthday of Maryam Mirzakhani.

And there are more than 100 different events taking place in over 30 countries in the world where women are gathered to honor the memory of Maryam Mirzakhani.

- Maryam was on the scientific advisory board.

MSRI was always very close to her scientific life.

She participated in those meetings until she really wasn't able to do it.

- In April of 2016 she was diagnosed with a stage IV cancer.

She knew that she didn't have much time left.

- These are pictures from the last time I visited her.

We discussed math.

We discussed life.

We discussed the cancer.

We discussed everything.

She was still funny.

She was silly.

She was goofy.

She had a sense of humor about everything.

If you look at the picture, you see a person full of life, full of energy.

I do miss her, all the time.

Yes.

- Most of what I did for maybe a very long time was essentially collaborate with her.

And so now she's gone, and it's very difficult.

[waves crashing] - It was just a tremendous loss.

I miss that powerful voice that she had.

She was just pure good.

[laughs] [waves crashing] - I knew her from the time when she was not that famous.

And she didn't change.

She was just a very good person.

[solemn music] [bells jingling] - Iranians are very proud of their rich ancient tradition, especially in science, mathematics, astronomy, and so on.

And we haven't had very many people to cheer about in the last century.

So when somebody like that appears, it's going to be a national phenomenon.

It's going to be something people are really proud of.

- Media, all the newspapers, first page.

They covered everything about her.

They published a stamp in her memory.

The post company in Iran published a stamp.

- She might be the first figure who is shown without a scarf in public memorials in Iran.

It shows that even the government has agreed, we don't want to distort her image.

If she didn't have a scarf, we shouldn't put a scarf on her.

- We changed the name of the hall of Mathematics House to Maryam Mirzakhani for the young generation.

They encourage to do mathematics and somehow they may be able to become a person like Maryam Mirzakhani.

- Lot of good high schools are trying to initiate Mirzakhani Fellowship, Mirzakhani Scholarship to remind students of her and to just keep her legacy alive.

- People are thinking about her work.

And they're discovering that her thoughts can be applied to other problems.

And they're rediscovering the beauty in her work.

- In 2019, Alex Eskin's collaboration with Mirzakhani on the magic wand theorem earned him the Breakthrough Prize, one of the highest honors in mathematics and science.

- The Breakthrough Prize in mathematics is awarded to... Alex Eskin.

- I am extremely honored and humbled to receive the Breakthrough Prize.

And I am deeply in debt to my late coauthor, Maryam Mirzakhani.

Without her brilliance, strength, and persistence the work recognized by the prize committee could not have been done.

- It's our great honor to announce that beginning next year the Breakthrough Prize Foundation will establish a new award, the Maryam Mirzakhani New Frontiers Prize.

[cheers and applause] - This is certainly the collaboration which I enjoyed the most.

Part of it was just her enthusiasm.

She just loved mathematics, she loved talking about mathematics, she loved thinking about mathematics.

It was just incredibly infectious.

[uplifting music] - So I fold it a little bit.

I've got to move it over a little bit more.

And as I continue to fold, you can see these green edges coming together.

She tackled big questions.

Her impact will be lasting.

- I expect to see her work popping up in unexpected places.

It's that kind of work.

It's going to show up in places that I can't foresee right now.

- I wish there would be more mathematicians like this.

Life would be different.

More people like this.

Life would be completely different.

- The amazing thing about Maryam is that she was just getting started.

This was just the beginning.

Ten years of work is nothing for a mathematician.

There would have been so much more to come.

[bright music] And by the: To learn more about "Secrets of the Surface" and Maryam Mirzakhani's mathematical quest, go to: