Welcome back to another deep dive. Today we're
doing something a bit different for you. Usually
when you picture a mathematical genius, you know,
you think of that classic goodwill hunting movie
trope. Oh, right. The solitary guy just standing
in front of a giant chalkboard. Exactly. Chalk
flying everywhere, furiously writing out equations
at like lightning speed. It's an image that really
suggests true genius is manic. instantaneous
and just totally disconnected from the physical
world. It's a compelling image for Hollywood,
sure. Yeah. But it creates this massive misconception
about how high level mathematics is actually
done. I mean, it's just not realistic. Right.
And that brings us to today's deep dive into
our source material, which is a really comprehensive
Wikipedia biography of Miriam Mirzakhani. One
of the absolute greatest minds of the 21st century.
Without a doubt, because she didn't work like
that movie trope at all. Instead of speed writing
on a whiteboard, she spent her time sitting on
the floor, surrounded by these massive sheets
of parchment paper, just drawing intricate doodles.
Yeah, her young daughter actually thought her
mother's job was simply painting. Which is just
so great. So our mission today for you listening
is to really explore how that visual patient
approach allowed her to map the universe of all
possible geometric shapes. Right. We want to
do that without getting bogged down by all the
academic jargon. Exactly. We want you to walk
away truly understanding her aha moments and
also how her life ultimately shifted the entire
cultural landscape of her home country. So, OK,
let's unpack this. Well. To really grasp the
magnitude of what she achieved mathematically,
we need to understand the crucible in which her
mind was forged. I mean, her story doesn't begin
in some quiet, isolated ivory tower. No, not
at all. It begins in Tehran, Iran, in 1977. Right.
Just a couple of years before the Iranian Revolution,
and then she grew up during the Iran -Iraq War.
So her early childhood was set against a backdrop
of immense national upheaval. Which is a lot
for a kid to process. It is. But as the country
began to stabilize, there was this massive push
to rebuild and foster a completely new generation
of intellectuals. And she ended up testing into
the Tehran Farzanaigan School. Which the sources
say was part of the National Organization for
Development of Exceptional Talents. Yeah, this
wasn't just a slightly advanced high school.
It was a rigorous, highly competitive environment
designed specifically to locate and cultivate
the absolute brightest young minds in the country.
And Meerdakani obviously thrived there. But the
funny thing is the sources note she wasn't actually
interested in math initially. I mean she wanted
to be a writer. She spent all her time just reading
novels. Yeah reading novels. It wasn't until
her older brother came home and told her about
a famous math problem, the one involving adding
numbers from 1 to 100, that she started to see
the puzzle -solving beauty of mathematics. And
from there her trajectory is just staggering.
It really is. In 1994, she becomes the first
Iranian woman to win a gold medal at the International
Mathematical Olympiad in Hong Kong. Scoring 41
out of 42 points. And then she follows that up
the very next year in Toronto by achieving a
flawless score, 42 out of 42, winning two gold
medals. Which is incredible. Yeah. She even co
-published a book on number theory in 1999 with
her close friend and fellow Olympiad medalist,
Roya Beheshti -Zavara. They were the first women
to ever compete in the Iranian National Mathematical
Olympiad. Right. But this is where her story
takes a deeply tragic turn. And it drastically
changes how you view her drive in her later work.
Right. It's easy to look at her perfect Olympiad
scores and assume she just had a computer for
her brain. But how much of her early success
was actually rooted in her specific environment
and, well, her grit? If we connect this to the
bigger picture, her... Genius wasn't just individual.
In March 1998, Mirza Khani, Zavara, and several
other exceptionally gifted students from Sharif
University attended a mathematics conference.
And on the way back? Yeah. On their way back
to Tehran, their bus slipped on a wet road and
fell off a cliff. The sources describe it as
a devastating national tragedy in Iran. Seven
passengers, literally some of the most brilliant
young mathematical minds in the country, were
killed. And Muzikhani and her friend Zavara were
among the few survivors. Wow! Yeah. When you
look at her life from that point forward, you
have to factor in the psychological weight of
that survival. I mean, she lost peers who shared
her incredibly rare gifts. So continuing to pursue
math with that kind of intensity wasn't just
about raw talent anymore. Exactly. It was an
act of profound resilience. She carried the legacy
of an entire generation of Iranian mathematicians
who didn't get the chance to finish their work.
You know, you would think that surviving something
like that might make a person want to rush through
life, like to publish as fast as possible just
to outrun the clock. But when she moved to the
United States to do her graduate work at Harvard,
She does the exact opposite. It slows down. She
really does. She began her PhD under the supervision
of Curtis T. McMullen, who is a mathematician
who had won the Fields Medal, which is the highest
honor in math. Right. And at Harvard, she became
known for her relentless questioning. She would
sit in McMullen's office taking notes in Persian,
just relentlessly drilling down into the foundations
of an idea. She actually described herself as
a slow mathematician. She explicitly said, you
have to spend some energy and effort to see the
beauty of math. She wasn't looking for shortcuts.
No, not at all. And she had this incredibly vivid
quote about her process. She said, discovering
a mathematical proof is like being lost in a
jungle and trying to use all the knowledge that
you can gather to come up with some new tricks.
And with some luck, you might find a way out.
That visual of being lost in a jungle perfectly
captures the nature of the specific mathematics
she was studying. I mean, she worked in topology
and geometry. Wait, wait, let's break that down
for a second. When most people think of geometry,
they think of rigid triangles or, you know, calculating
the area of a circle. What does topology actually
mean in this context? Right, so topology is often
called rubber sheet geometry. Rubber sheet geometry.
Yeah, it's the study of geometric properties
that don't change even when a shape is stretched,
twisted, or bent out of proportion. Okay. The
only rules are you can't tear the surface and
you can't glue bits together. So to a topologist,
a coffee mug and a donut are the exact same shape.
Because they both have exactly one hole. Exactly.
You can continuously deform a mug into a donut
in your mind without ripping it. Okay, so she's
working with shapes that are just constantly
morphing and bending. It makes sense why she
needed those massive sheets of parchment paper
on the floor. You can't just crunch a quick algorithm
to understand a surface that is continuously
stretching. No, you can't. You have to visualize
it. You have to draw it out. It sounds like she
wasn't trying to win a sprint. She was like a
botanist wandering through a dense jungle mapping
every single leaf. Which is honestly so comforting
for you listening to hear that one of the world's
smartest people embraced feeling lost. Precisely.
And the specific objects she was studying were
Riemann surfaces and their moduli spaces. Okay,
here's where it gets really interesting. Let's
talk about moduli spaces. Well, to understand
her major breakthrough... we need to introduce
a classic physics problem that mathematicians
love, the billiards problem. Right, the sources
actually mention mathematician Jordan Ellenberg
using this exact analogy to explain her work.
Yes, imagine a perfectly frictionless billiard
table. You hit a ball and it bounces off the
walls forever. In a standard rectangular table,
the path of the ball is fairly easy to predict.
But what if the table is shaped like a weird
polygon with strange angles? Right. Tracking
that ball as it bounces chaotically off the walls
becomes a massive headache. The math just gets
incredibly messy. Because every time it hits
an edge, you have to calculate the angle of reflection
and map the new trajectory. Exactly. So mathematicians
use a brilliant trick. Instead of having the
ball bounce off the wall, imagine you take the
entire billiard table and reflect it across that
edge like a mirror. Oh, wow. So the ball doesn't
bounce at all. Right. It just rolls straight
through the wall into a newly attached mirrored
copy of the table. So every time it would have
hit a wall, you just build another imaginary
table next to it and let the ball keep rolling
in a perfectly straight line. Yes. And if you
keep doing this, reflecting the table over and
over and gluing the corresponding edges together,
that flat jagged polygon eventually folds up
into a closed continuous surface. Which usually
ends up looking like a multi -hole donut. Exactly.
A Ryman surface. That is wild. So tracking a
chaotic bouncing ball on a flat table is mathematically
the exact same thing as tracking a perfectly
straight line wrapping around a multi -hole donut.
You got it. On a curved surface, the equivalent
of a straight line is called a geodesic. If you
want to walk from New York to London on the curved
surface of the earth, the shortest path, the
geodesic, is an arc. Great. Now, mathematicians
were obsessed with finding a specific type of
path on these multi -hole donuts, a simple closed
geodesic. Simple closed meaning what, exactly?
Let's make sure we're clear. Sure. Closed means
the straight path eventually loops perfectly
back onto its starting point, like a rubber band
wrapped around the donut. And simple means the
path never crosses over itself. Mm -hmm. So no
figure eights. Just a clean, continuous loop.
OK, I'm with you. And mathematicians wanted to
know how many of these simple closed loops could
exist on a given surface. Right. It was an incredibly
difficult open problem. University of Chicago
topologist Benson Farb actually called counting
these loops the key object to unlocking the structure
and geometry of the whole surface. Because as
the loops get longer and longer, they wrap around
the donut in more and more complex ways. How
do you count something that is stretching into
infinity? And this is where Mirzakhani's genius
fully reveals itself in her 2004 PhD thesis.
She realized that trying to count the loops on
just one single surface was completely the wrong
approach. We had to zoom out. Yes. You had to
look at the moduli space. This is a crucial concept.
So a moduli space isn't a physical space, right?
No. It is an abstract mathematical landscape
where every single point in this space represents
a different possible shape that your multi -hole
donut could take. So if you stretch the donut
to be longer, you move to a new point in the
moduli space. If you pinch it so a hole gets
smaller, you move to another point. Exactly.
So she wasn't just studying one billiard table.
She was studying the meta universe of all possible.
billiard tables. And her big aha moment was realizing
that counting the individual simple loops on
a single surface is directly connected to calculating
the total volume of that entire moduli space.
Wow. By shifting her perspective from the micro,
the single loop, to the macro, the universe of
all possible shapes, she unlocked the whole problem.
And the sources outline the actual formula she
discovered. She proved that the number of simple
closed loops of a certain length grows proportionally
to a polynomial formula, the length L to the
power of 6G minus 6. Let's translate those variables
so we can see why it's so elegant. Yeah, please.
So L is the maximum length of the loop you're
trying to count. And g is the genus of the surface,
which is just the mathematical term for the number
of holes it has. So if our surface is a donut
with two holes, g equals two, which means six
times two minus six equals six. So for a two
-hole shape, the number of simple loops grows
proportionally to the length to the sixth power.
Exactly. She didn't just guess this. She proved
the exact underlying architecture governing how
these loops behave. That's just brilliant. And
she didn't stop there. She also provided a new
proof for a long -standing conjecture by physicist
Edward Witten regarding the mathematics of string
theory. So she tied her geometric discoveries
directly to theoretical physics. It's just astonishing,
but she kept going deeper into that jungle, which
brings us to another major concept she tackled,
William Thurston's earthquake flow. Yeah, this
sounds dramatic, but it's actually a very precise
mechanical tool for mathematicians. Okay. Remember
how we said moduli space is the map of all possible
shapes a surface can take? Right, the meta universe.
Mathematicians need a systematic way to travel
through that space. They need a rule -bound way
to deform one shape into another. The earthquake
flow is one of those methods. How does it work?
You take your multi -hole surface and you cut
it along one of those closed geodesic loops we
just talked about. Okay, so you physically slice
the doughnut along a loop. Then you slide the
two edges past each other by a specific distance,
much like tectonic plates sliding past each other
along a strike -slip fault line during an earthquake.
Ah, hence the name. Exactly. Then you glue the
edges back together. You have now created a mathematically
distinct surface. You have moved to a new point
in modulized space. But why do mathematicians
actually want to do that? Like, what's the benefit
of cutting and twisting these shapes? Because
it allows them to map the dynamics of the space
itself. Mirzakhani proved long -standing conjectures
about how these theoretical earthquakes behave
over time. OK. She showed that as you apply these
earthquake flows, the transformations are not
entirely random. They are measure preserving
and ergodic, meaning they explore the space in
a highly structured, predictable way. Which perfectly
sets up her crowning achievement a decade later
in 2014, the magic wand theorem. She developed
this alongside mathematicians Alex Eskin and
Amir Mohammadi. The magic wand theorem is arguably
her most profound legacy. Imagine you are inside
that moduli space, that universe of all possible
shapes, and you decide to zoom off in a random
direction, stretching and warping the shape continuously.
It feels like that would eventually devolve into
complete chaos, like you would just end up in
a jagged fractal mess of unpredictable shapes.
That's exactly what everyone assumed. But Mirza
Khani, Eskin, and Mohammadi proved the opposite.
They proved that the closure of these complex
paths in moduli space are surprisingly regular.
Wait, really? They don't just splatter randomly
into chaos. No, they always land in neat, rigidly
algebraic subspaces. The International Mathematical
Union put in beautifully. They said it was astounding
to find rigidity in an inhomogeneous world. That
is the core of her genius. She proved that even
in a system that allows for infinite mind -bending
transformations, there are strict, beautiful
underlying rules. She mapped the hidden order
inside the chaos. And the global mathematical
community recognized that immediately. In August
2014, Mariam Mirzakhani was awarded the Fields
Medal in Seoul. The Fields Medal is widely considered
the Nobel Prize of Mathematics. It is the absolute
pinnacle of achievement in the field. Her win
was historic on multiple fronts. I mean, she
became the first woman ever to win the prize
since it was established in 1936. Wow. and she
also became the first Iranian to win. But the
sources reveal a pretty heartbreaking reality
playing out behind the scenes of this triumph.
Just one year prior to receiving the medal, in
2013, she had been diagnosed with breast cancer.
So while she is stepping onto the world stage,
being recognized for mapping the universe of
shapes, she's quietly undergoing severe medical
treatments. And despite her incredible resilience,
the cancer eventually spread to her bones and
liver. Mary Mirzakhani passed away on July 14,
2017 at Stanford Hospital. She was only 40 years
old. It is a profound and premature loss for
science. But the impact she had in those 40 years
didn't just stay in the realm of abstract geometry.
No, not at all. Her life and her death triggered
immediate and massive ripples in the physical
world, particularly in Iran. So what does this
all mean? She studied these abstract earthquakes
in geometry, but it sounds like she caused a
literal cultural earthquake in her home country.
What's fascinating here is the cultural reaction
detailed in the sources. The Iranian president
at the time, Hassan Rouhani, publicly praised
her, noting how her scientific achievements made
Iran's name resonate globally. And demonstrated
the immense capability of Iranian women. Yes.
But the reaction in the press is what really
highlighted her cultural impact. Iran has strict
societal rules regarding the hijab, and media
protocol generally dictates that women must be
shown with their hair covered. Right. However,
upon her passing, several major Iranian newspapers,
as well as President Rouhani and his public tributes,
broke those taboos. They published photographs
of Mirza Khani with her hair uncovered. Now,
to maintain impartiality, we should note the
sources mentioned that some newspapers adhered
to the strict rules by digitally manipulating
images or using paintings to obscure her hair.
But the fact that so many mainstream outlets
chose to publish her true image was noted internationally
as a really significant break from protocol.
And her impact extended beyond media representation
all the way to national legislation. Her death
actually sparked a massive political debate in
Iran regarding citizenship laws. Because under
Iranian law at the time, citizenship was passed
down patrilineally, meaning through the father.
Exactly. But Mirza Khani's husband Jan Vondrak
is a Czech mathematician whom she met in the
U .S. And they had a daughter together named
Anahita. Because Anahita's father wasn't Iranian,
she could not easily get Iranian nationality
to visit her mother's home country. So following
Mirza Khani's death, The sources state that 60
Iranian members of parliament immediately urged
the speeding up of an amendment to the law. Wow.
They pushed to allow the children of Iranian
mothers married to foreigners to be granted Iranian
nationality, specifically citing the desire to
make it easier for Mirza Khani's daughter to
visit Iran. When you step back and look at that
trajectory, it is staggering. I mean, she dedicated
her life to sitting on the floor, deep in the
mathematical jungle, studying abstract earthquakes
and the rules of imaginary morphing donuts. Yet
her actual life caused a literal cultural earthquake.
She inadvertently forced a reevaluation of strict,
real -world societal rules simply through the
undeniable magnitude of her existence. It's incredible.
And her memory continues to be honored around
the world today. International Council for Science
declared her birthday, May 12th, as International
Women in Mathematics Day. High schools and university
libraries in Iran have been named after her.
There is an Iranian microsatellite, named in
her honor, and an asteroid. Asteroid 321357 bears
her name. That's so cool. In 2024, the International
Astronomical Union even named a lunar crater
after her. And the Breakthrough Prize Foundation
created the Maryam Mizikani New Frontiers Prize,
which is a $50 ,000 award to support early career
female mathematicians. She fundamentally altered
what the world picked when they think of a mathematical
genius. She really did. For you listening, if
you are tackling a complex problem right now,
whether you're trying to learn a new coding language,
analyzing a massive data set, or just figuring
out a complicated life decision, Mirza Khani's
process is a master class. True mastery doesn't
always look like furious speed. Sometimes it
requires you to sit down. lay out the pieces,
and just be willing to wander slowly through
the jungle until the underlying structure reveals
itself. That is the enduring lesson of her slow
math. And if there is a final thought to mull
over, building on the mathematics we've explored
today, it is the profound contrast between her
work and her life. How do you mean? Well, Mirza
Khani proved that in the hyper -complex, shape
-shifting universe of moduli spaces, there is
a surprising hidden rigidity and order. Chaos
resolves into elegant algebra. Right. But looking
at her life cut tragically short yet permanently
altering mathematics and Iranian societal norms,
perhaps she proved the exact opposite about the
human condition. Oh, wow. Yeah, in the rigidly
structured, seemingly inflexible systems of our
society, a single brilliant mind can introduce
a beautiful, unpredictable change that shifts
the shape of the world forever.