Years After the Early Death of a Math Genius, Her Ideas Gain New Life

“This is a landmark result,” said Peter Sarnak, a mathematician at Princeton University. “There’ll be a lot more that will come out of this.”

The work, which has not yet been peer reviewed, suggests that hyperbolic surfaces are even stranger and less intuitive than anyone had imagined. It also builds on Mirzakhani’s titanic mathematical legacy, reigniting her dream to illuminate this universe of unimaginable shapes.

A Packed Thesis

As a child growing up in Tehran, Mirzakhani, a voracious reader, hoped to one day write books of her own. But she also excelled in mathematics, and ultimately won two gold medals at the International Mathematical Olympiad, a prestigious competition for high school students. In 1999, after graduating from the Sharif University of Technology, she went to Harvard for graduate school. There she fell in love with hyperbolic geometry. An avid doodler, she enjoyed the challenge of trying to make sense of shapes that by definition could not be drawn.

“A hyperbolic surface is a bit like a puzzle that you can put together locally but you can’t actually ever finish in our universe,” said Alex Wright, a mathematician at the University of Michigan and Mirzakhani’s former postdoctoral fellow. That’s because every piece of the puzzle is curved in the shape of a saddle. You can fit a few pieces together, but never in a way that fully closes the surface — at least not in our flat, three-dimensional space. This makes hyperbolic surfaces particularly difficult to study. Even basic questions about them remain open.

To get a handle on a hyperbolic surface, mathematicians study closed loops that live on it. These loops, called geodesics, come in all sorts of shapes; for a given shape, they carve out the shortest possible path from one point to the next as they return to their start. The more holes a surface has, the more varied and complicated its geodesics can get. By studying how many distinct geodesics of a given length there are on a surface, mathematicians can begin to understand what the surface looks like as a whole.

Mirzakhani became obsessed with these circumnavigating curves. In discussions with colleagues, she brought them up constantly, her usual restraint evaporating. She often spoke breathlessly of geodesics and related objects as if they were characters in a story. “I remember when she would give talks, she would ask these two questions: How many curves are there, and where are they?” said Kasra Rafi of the University of Toronto.

While still in graduate school, she developed a formula that allowed her to estimate, for any hyperbolic surface, how many geodesics there were up to a given length. This formula not only allowed her to describe individual surfaces; it also enabled her to prove a famous conjecture in string theory, and gave her insight into what kinds of hyperbolic surfaces it was possible to construct.

After completing her graduate degree, Mirzakhani went on to make major advances in geometry, topology and dynamical systems. But she never forgot the subject of her Ph.D. thesis.

She hoped to learn more about the creatures that lived in the hyperbolic zoo she had classified. In particular, she wanted to understand what a typical hyperbolic surface looked like. Often, mathematicians first study objects — graphs, knots, sequences of numbers — that they can construct. But their constructions are usually “not at all typical,” said Bram Petri of Sorbonne University. “We tend to draw very special things.” A typical graph, knot or sequence, selected at random, will look very different.

And so Mirzakhani began picking hyperbolic surfaces at random and studying their properties. “She had the perfect tools, so it was very natural,” Wright said.

But she died before she could really pursue this line of inquiry. “She was really just developing the machinery,” Monk said, “and then didn’t have the time to use it.”

Picking Up the Thread

Monk never thought she would be the one to pick up where Mirzakhani had left off. In fact, until she was in her early 20s, she had no intention of pursuing a career in mathematical research. She had planned to become a teacher since she was a child, when she would tutor fellow students to stave off her boredom in math classes. “I was pretty miserable at school,” she said. “I would kind of keep myself busy by being the assistant teacher.”

She enrolled in a master’s program at Paris-Saclay University, one of three women in the 40-person cohort. Near its end, she learned that both of the other women were also planning to leave academia. The exodus made her question whether their plans reflected “our own individual choices and desires,” she said, “or were we more affected than we realized by being in a setting where we were very much the exception.” She felt a duty to the girls she had been planning to teach to become an example of a successful woman in mathematics.

So she decided to pursue a doctorate. “At least one of us has to do it,” she told herself. “Otherwise it’s quite sad.” (Later, one of the other women also got a Ph.D.)

At the suggestion of one of her professors, Monk took a train to Strasbourg to meet Nalini Anantharaman, a potential adviser who, like Mirzakhani, was an expert in multiple fields. In fact, Anantharaman had met Mirzakhani several times over her career — they were about the same age and interested in similar topics. Both also shared a passion for the humanities: Just as Mirzakhani had almost dedicated her studies to literature, Anantharaman had trained as a classical pianist, and hadn’t been sure whether she would go into music or math.

In 2015, both mathematicians ended up visiting the University of California, Berkeley, for a semester. Mirzakhani’s daughter and Anantharaman’s son were close in age, and the two mathematicians occasionally met at a local playground, where they talked about motherhood while their children played.

Anantharaman knew that Mirzakhani had begun experimenting with random hyperbolic surfaces toward the end of her life. She was now hoping to build on that work.

One way to characterize a hyperbolic surface is to measure how connected it is. Imagine you’re an ant walking on a surface in a random direction. If you walk for a while, are you equally likely to end up anywhere on the surface? If it’s well connected, with plenty of possible paths between its various regions, then the answer is yes. But if it’s poorly connected — like a dumbbell, which consists of two large regions attached by a single narrow bridge — you might instead spend a long time wandering on just one side before you find a way to cross to the other.

Mathematicians measure how connected a surface is using a number called the spectral gap. The bigger its value, the more connected the surface. Even though it’s still impossible to imagine the surface, the spectral gap offers a way to think about its overall shape. “This is like a way of quantifying the sentence, ‘What does the surface look like?’” Rafi said.

While the spectral gap can theoretically be any value between 0 and 1/4, most of the hyperbolic surfaces that mathematicians have been able to construct have a relatively low spectral gap. It wasn’t until 2021 that they figured out how to build surfaces with any number of holes that had the highest possible spectral gap — that is, surfaces that were maximally connected.

But even though there are relatively few known hyperbolic surfaces with a high spectral gap, mathematicians suspect that they’re common. There is a vast and largely unexplored universe of hyperbolic surfaces. While mathematicians usually can’t construct individual surfaces in this universe, they hope to understand the general properties of a typical surface. And when they look at the population of hyperbolic surfaces as a whole, they expect that most have a spectral gap of 1/4.

That’s the problem Anantharaman hoped to assign her new graduate student. Monk, eager to work closely with a female mentor and to set ambitious goals for herself — “if I’m going to do a Ph.D., I’ll really do it,” she remembers thinking — signed on.

Writing the Sequel

In 2018, just one year after Mirzakhani’s death, Monk began her graduate studies with Anantharaman. Her first step was to learn everything she possibly could about Mirzakhani’s work on hyperbolic surfaces.

It was known that if you could get an accurate enough estimate of the number of closed geodesics on a surface — those looped paths that Mirzakhani had studied so intensively — you would be able to compute the surface’s spectral gap. Monk and Anantharaman needed to show that almost all hyperbolic surfaces have a spectral gap of 1/4. That is, the likelihood of picking a surface with an optimal spectral gap would approach 100% as the number of holes in the surface increased.

The pair started with the formula for counting geodesics that Mirzakhani had come up with during her Ph.D. The problem was that this formula underestimates the number of geodesics. It counts most, but not all, of them — it misses more complicated geodesics that cross themselves before returning to their start, like a figure eight encircling two holes.

But using Mirzakhani’s limited formula, Monk and Anantharaman saw a way to prove a relatively large spectral gap. “It looked almost like a miracle,” Anantharaman said. “It’s still quite mysterious to me that it works so well.”

What if she and Monk could sharpen Mirzakhani’s formula to count the more complicated geodesics, too? Perhaps they could get their count to be accurate enough to translate into a spectral gap of 1/4, something that mathematicians before them had hoped to do, too.

Anantharaman suddenly remembered an email she had received from Mirzakhani just a couple of years before she died, posing a series of questions about the relationship between the spectral gap and counting geodesics. “At the time, I didn’t really know why she was asking all these questions,” Anantharaman said. But now she wondered whether Mirzakhani might have been planning to take a similar approach.

Monk spent part of her time in graduate school figuring out a way to extend Mirzakhani’s formula to more complicated geodesics. While doing so, she also wrote long, detailed descriptions of key concepts that Mirzakhani had not fully explained in her original papers. “I feel like some of her ideas were just put on the table for someone to kind of explain them to the community because she didn’t have a chance to do it,” she said.

By 2021, Monk had figured out how to count up all sorts of geodesics that had previously been inaccessible. She and Anantharaman knew that, with some additional work, they could probably use their new formula to get a better estimate of the spectral gap. But rather than publishing a partial result, they were determined to achieve the full 1/4 goal.

Then they got stuck.

Revisiting the Tome

There was one particularly gnarly type of geodesic that kept getting in their way. These geodesics would wind around the same region of a surface for a long time, forming convoluted tangles. The tangles appeared only on a small number of ornery surfaces, but when they did, they appeared in droves. If Monk and Anantharaman included them in their total count, it would throw off the computation they needed to perform to translate the count into the spectral gap — giving them an output smaller than 1/4.

The situation seemed hopeless, Monk said.

Her dejection only deepened when two independent teams published papers a couple of months apart in which they proved a spectral gap of 3/16. The news didn’t bother Anantharaman; she only cared about getting to 1/4. “When I start working on something, I kind of fall in love with a distant goal,” she said — apparently a trait she shared with Mirzakhani.

But Monk, still in the last year of her Ph.D. and needing a result that would let her finish her thesis, wondered if they should have settled for less. “I was a bit disheartened that we hadn’t thought of doing that,” she said.

Alex Wright, who was on one of the teams that achieved the 3/16 result, understood her perspective. “It’s pretty unusual for a graduate student to be working on a problem that ambitious,” he said. And it didn’t seem as if anyone was going to figure out a way to achieve 1/4.

But Anantharaman had an idea: to turn to a different area of math, called graph theory, for inspiration. Remember that Anantharaman and Monk were trying to show that most hyperbolic surfaces are as connected as possible. Two decades earlier, the mathematician Joel Friedman proved that most graphs — collections of vertices and edges that appear all over mathematics — have this property.

But Friedman’s result was not easy to translate. “Its an infamously hard result with a super long proof that resisted simplification,” Wright said.

Anantharaman had tried to read Friedman’s proof when she and Monk began their project. But like so many other mathematicians, she found it impenetrable. “At the time, I really didn’t understand it at all,” she said. Now she returned to it in search of new clues.

She found them. Certain steps of the proof looked familiar, like a graph-theoretic analog of what she and Monk had been trying to do with their hyperbolic surfaces. In fact, she realized, Friedman had encountered complicated paths between vertices in his graphs that, like her tangled geodesics, prevented him from getting the best estimate of the spectral gap. But somehow he had found a way to deal with these paths, and Anantharaman couldn’t quite understand how.

In May 2022, she and Monk organized a workshop and invited Friedman to speak about his work. “They really needed a technique that was deep in the bowels of my proof,” he said.

He had essentially found a way to prove that he could remove the graphs with problematic paths from his calculations entirely. After speaking with Friedman, Monk and Anantharaman realized they could do the exact same thing. There was a lot of work left to do: It would be difficult to convert Friedman’s method into something that would work for hyperbolic surfaces. But their doubts were assuaged. “It was very exciting,” Monk said. “At this point, it was quite clear that we could finish.”

A Growing Legacy

In early 2023, the two mathematicians wrote a paper that sketched out what they had done so far. In it, they proved a record 2/9 spectral gap. “That felt like a very nice intermediate step,” Monk said.

The following year, they adapted Friedman’s methods, and wrote up a plan for how they would use it to get to 1/4. Last month, they finally completed the proof, showing that a randomly selected hyperbolic surface is likely to have the maximal spectral gap. The result tells mathematicians more about hyperbolic surfaces than they have ever known. Other researchers now hope to use the pair’s techniques to answer other major questions, including one about important surfaces in number theory and dynamics.

This kind of work “instantly creates an avalanche of results that go together,” said Anton Zorich, a mathematician at the Institute of Mathematics of Jussieu in Paris.

It also allowed Monk and Anantharaman to gain a deep familiarity with Mirzakhani’s research. Although Monk has still never watched any of Mirzakhani’s recorded lectures or heard her voice — preferring her to remain “a bit of a mystery in my mind,” she said — she feels as if she knows Mirzakhani through her proofs. “When you read the works of someone in detail, you end up understanding things beyond the sheer content of the work, about how they were thinking,” Monk said.

She’s honored to have been able to extend Mirzakhani’s legacy, and mathematicians are excited to see what that legacy will bring next.

“I’m sad she can’t see it,” Wright said of his former mentor.

Zorich agreed. “She was supposed to be there to appreciate this,” he said. “I have no doubt she would be extremely happy.”

Clarification: March 3, 2025

This article has been updated to clarify that one of the other women in Monk’s master’s program did ultimately get a Ph.D.