The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal

The Mandelbrot set was everywhere, until it wasn’t.

Within a decade, it seemed to disappear. Mathematicians moved on to other subjects, and the public moved on to other symbols. Today, just 40 years after its discovery, the fractal has become a cliché, borderline kitsch.

But a handful of mathematicians have refused to let it go. They’ve devoted their lives to uncovering the secrets of the Mandelbrot set. Now, they think they’re finally on the verge of truly understanding it.

Their story is one of exploration, of experimentation — and of how technology shapes the very way we think, and the questions we ask about the world.

The Bounty Hunters

In October 2023, 20 mathematicians from around the world congregated in a squat brick building on what was once a Danish military research base. The base, built in the late 1800s in the middle of the woods, was tucked away on a fjord on the northwest coast of Denmark’s most populous island. An old torpedo guarded the entrance. Black-and-white photos, depicting navy officers in uniform, boats lined up at a dock, and submarine tests in progress, adorned the walls. For three days, as a fierce wind whipped the water outside the windows into frothing whitecaps, the group sat through a series of talks, most of them by two mathematicians from Stony Brook University in New York: Misha Lyubich and Dima Dudko.

In the workshop’s audience were some of the Mandelbrot set’s most intrepid explorers. Near the front sat Mitsuhiro Shishikura of Kyoto University, who in the 1990s proved that the set’s boundary is as complicated as it can possibly be. A few seats over was Hiroyuki Inou, who alongside Shishikura developed important techniques for studying a particularly high-profile region of the Mandelbrot set. In the last row was Wolf Jung, the creator of Mandel, mathematicians’ go-to software for interactively investigating the Mandelbrot set. Also present were Arnaud Chéritat of the University of Toulouse, Carsten Petersen of Roskilde University (who organized the workshop), and several others who had made major contributions to mathematicians’ understanding of the Mandelbrot set.

And at the whiteboard stood Lyubich, the world’s foremost expert on the topic, and Dudko, one of his closest collaborators. Together with the mathematicians Jeremy Kahn and Alex Kapiamba, they have been working to prove a long-standing conjecture about the geometric structure of the Mandelbrot set. That conjecture, known as MLC, is the final obstacle in the decades-long quest to characterize the fractal, to tame its tangled wilderness.

By building and sharpening a powerful set of tools, mathematicians have wrestled control of the geometry of “almost everything in the Mandelbrot set,” said Caroline Davis of Indiana University — except for a few remaining cases. “Misha and Dima and Jeremy and Alex are like bounty hunters, trying to track down these last ones.”

Lyubich and Dudko were in Denmark to update other mathematicians on recent progress toward proving MLC, and the techniques they’d developed to do so. For the past 20 years, researchers have gathered here for workshops dedicated to unpacking results and methods in the field of complex analysis, the mathematical study of the kinds of numbers and functions used to generate the Mandelbrot set.

It was an unusual setup: The mathematicians ate all their meals together, and talked and laughed over beers into the wee hours. When they finally did decide to go to sleep, they retired to bunk beds or cots in small rooms they shared on the second floor of the facility. (Upon our arrival, we were told to grab sheets and pillowcases from a pile and take them upstairs to make our beds.) In some years, conference-goers brave a swim in the frigid water; more often, they wander through the woods. But for the most part, there’s nothing to do except math.

Typically, one of the attendees told me, the workshop attracts a lot of younger mathematicians. But that wasn’t the case this time around — perhaps because it was the middle of the semester, or, he speculated, because of how difficult the subject matter was. He confessed that at that moment, he felt a bit intimidated about the prospect of giving a talk in front of so many of the field’s greats.

But given that most mathematicians in the broader area of complex analysis are no longer working on the Mandelbrot set directly, why dedicate an entire workshop to MLC?

The Mandelbrot set is more than a fractal, and not just in a metaphorical sense. It serves as a sort of master catalog of dynamical systems — of all the different ways a point might move through space according to a simple rule. To understand this master catalog, one must traverse many different mathematical landscapes. The Mandelbrot set is deeply related not just to dynamics, but also to number theory, topology, algebraic geometry, group theory and even physics. “It interacts with the rest of math in a beautiful way,” said Sabyasachi Mukherjee of the Tata Institute of Fundamental Research in India.

To make progress on MLC, mathematicians have had to develop a sophisticated set of techniques — what Chéritat calls “a powerful philosophy.” These tools have garnered much attention. Today, they constitute a central pillar in the study of dynamical systems more broadly. They’ve turned out to be crucial for solving a host of other problems — problems that have nothing to do with the Mandelbrot set. And they’ve transformed MLC from a niche question into one of the field’s deepest and most important open conjectures.

Lyubich, the mathematician arguably most responsible for molding this “philosophy” into its current form, stands tall and straight, and speaks quietly. When other mathematicians at the workshop approach him to discuss a concept or ask a question, he closes his eyes and listens attentively, his thick eyebrows furrowed. He answers carefully, in a Russian accent.

But he’s also quick to break into loud, warm laughter, and to make wry jokes. He’s generous with his time and advice. He has “really nurtured quite a few generations of mathematicians,” said Mukherjee, one of Lyubich’s former postdocs and a frequent collaborator. As he tells it, anyone interested in the study of complex dynamics spends some time at Stony Brook learning from Lyubich. “Misha has this vision of how we should go about a certain project, or what to look at next,” Mukherjee said. “He has this grand picture in his mind. And he is happy to share that with people.”

For the first time, Lyubich feels he’s able to see that grand picture in its totality.

The Prize Fighters

The Mandelbrot set began with a prize.

In 1915, motivated by recent progress in the study of functions, the French Academy of Sciences announced a competition: In three years’ time, it would offer a 3,000-franc grand prize for work on the process of iteration — the very process that would later generate the Mandelbrot set.

Iteration is the repeated application of a rule. Plug a number into a function, then use the output as your next input. Keep doing that, and observe what happens over time. As you continue to iterate your function, the numbers you get might rapidly rise toward infinity. Or they might be pulled toward one number in particular, like iron filings moving toward a magnet. Or end up bouncing between the same two numbers, or three, or a thousand, in a stable orbit from which they can never escape. Or hop from one number to another without rhyme or reason, following a chaotic, unpredictable path.

The French Academy, and mathematicians more broadly, had another reason to be interested in iteration. The process played an important role in the study of dynamical systems — systems like the rotation of planets around the sun or the flow of a turbulent stream, systems that change over time according to some specified set of rules.

The prize inspired two mathematicians to develop an entirely new field of study.

First was Pierre Fatou, who in another life might have been a navy man (a family tradition), were it not for his poor health. He instead pursued a career in mathematics and astronomy, and by 1915 he’d already proved several major results in analysis. Then there was Gaston Julia, a promising young mathematician born in French-occupied Algeria whose studies were interrupted by World War I and his conscription into the French army. At the age of 22, after suffering a severe injury shortly after beginning his service — he would wear a leather strap across his face for the rest of his life, after doctors were unable to repair the damage — he returned to mathematics, doing some of the work he would submit for the Academy prize from a hospital bed.

The prize motivated both Fatou and Julia to study what happens when you iterate functions. They worked independently, but ended up making very similar discoveries. There was so much overlap in their results that even now, it’s not always clear how to assign credit. (Julia was more outgoing, and therefore received more attention. He ended up winning the prize; Fatou didn’t even apply.) Due to this work, the two are now considered the founders of the field of complex dynamics.

“Complex,” because Fatou and Julia iterated functions of complex numbers — numbers that combine a familiar real number with a so-called imaginary number (a multiple of i, the symbol mathematicians use to denote the square root of −1). While real numbers can be laid out as points on a line, complex numbers are visualized as points on a plane, like so:

Fatou and Julia found that iterating even simple complex functions (not a paradox in the realm of mathematics!) could lead to rich and complicated behavior, depending on your starting point. They began to document these behaviors, and to represent them geometrically.

But then their work faded into obscurity for half a century. “People didn’t even know what to look for. They were limited on what questions to even ask,” said Artur Avila, a professor at the University of Zurich.

This changed when computer graphics came of age in the 1970s.

By then, the mathematician Benoît Mandelbrot had gained a reputation as an academic dilettante. He’d dabbled in many different fields, from economics to astronomy, all while working at IBM’s research center north of New York City. When he was appointed an IBM fellow in 1974, he had even more freedom to pursue independent projects. He decided to apply the center’s considerable computing power to bringing complex dynamics out of hibernation.

At first, Mandelbrot used the computers to generate the kinds of shapes that Fatou and Julia had studied. The images encoded information about when a starting point, when iterated, would escape to infinity, and when it would become trapped in some other pattern. Fatou and Julia’s drawings from 60 years earlier had looked like clusters of circles and triangles — but the computer-generated images that Mandelbrot made looked like dragons and butterflies, rabbits and cathedrals and heads of cauliflower, sometimes even disconnected clouds of dust. By then, Mandelbrot had already coined the word “fractal” for shapes that looked similar at different scales; the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken.

The images appearing on his computer screen — today known as Julia sets — were some of the most beautiful and complicated examples of fractals that Mandelbrot had ever seen.

Fatou and Julia’s work had focused on the geometry and dynamics of each of these sets (and their corresponding functions) individually. But computers gave Mandelbrot a way to think about an entire family of functions at once. He could encode all of them in the image that would come to bear his name, though it remains a matter of debate whether he was actually the first to discover it.

The Mandelbrot set deals with the simplest equations that still do something interesting when iterated. These are quadratic functions of the form f(z) = z² + c. Fix a value of c — it can be any complex number. If you iterate the equation starting with z = 0 and find that the numbers that you generate remain small (or bounded, as mathematicians say), then c is in the Mandelbrot set. If, on the other hand, you iterate and find that eventually your numbers start growing toward infinity, then c is not in the Mandelbrot set.

It’s straightforward to show that values of c close to zero are in the set. And it’s similarly straightforward to show that big values of c aren’t. But complex numbers live up to their name: The set’s boundary is magnificently intricate. There is no obvious reason that changing c by tiny amounts should cause you to keep crossing the boundary, but as you zoom in on it, endless amounts of detail appear.

What’s more, the Mandelbrot set acts like a map of Julia sets, as can be seen in the interactive figure below. Choose a value of c in the Mandelbrot set. The corresponding Julia set will be connected. But if you leave the Mandelbrot set, then the corresponding Julia set will be disconnected dust.

The first published picture of the set, a rough plot of just a couple hundred asterisks, appeared in 1978 in a paper by the mathematicians Robert Brooks and J. Peter Matelski, who were studying a seemingly unrelated question in group theory and hyperbolic geometry.

It was Mandelbrot who recognized and popularized the set. After using IBM’s computers to graph hundreds of Julia sets, he sought to represent them all simultaneously. In 1980, armed with much more sophisticated computing power than Brooks and Matelski, he ended up generating a far better version of the Mandelbrot set (though still crude by today’s standards). He immediately fell in love and decided to make the fractal as public an image as possible. It’s for this reason that the set was named after him. (Mandelbrot himself was unpopular among mathematicians, because of his habit of jumping from one subject to another without proving deep results, and because he was often strident in his quest to take credit for discoveries like the Mandelbrot set.)

The computer images immediately captured the attention of some of math’s deepest thinkers. “Everybody became very interested, once we could actually see what was going on,” said Kapiamba, who is currently a postdoc at Brown University.

No one had anticipated how rich the world of quadratic equations could be. “It’s like when you open a geode, a simple-looking stone, and inside you find all these crystals — this amazing complex structure,” said Anna Benini of the University of Parma in Italy.

“Mathematicians saw things that they didn’t imagine before,” Avila said. “We all nowadays owe a lot to those explorations.”

Within just a couple of years, Hubbard and the mathematician Adrien Douady had proved a huge number of results about both the Mandelbrot set and the Julia sets it represented. But their proofs were handwritten, “mainly understandable only to Douady and me,” Hubbard wrote. And so in 1983, Douady wrote and delivered a series of lectures to explain those early results. Afterward, he compiled the material from his lectures into a single document, dubbed the Orsay notes. Nearly 200 pages long, it quickly became the field’s bible.

In the Orsay notes, Douady and Hubbard proved several major theorems that were motivated by the computer images they’d seen. They showed that the Mandelbrot set was connected — that you can draw a line from any point in the set to any other without lifting your pencil. Mandelbrot had initially suspected the opposite: His first images of the set looked like one big island with lots of little ones floating in a sea around it. But later, after seeing higher-resolution pictures — including ones that used color to illustrate how quickly equations outside the set flew off to infinity — Mandelbrot changed his guess. It became clear that those little islands were all connected by very thin tendrils. The introduction of color “is a very mundane thing, but it’s important,” said Søren Eilers of the University of Copenhagen.

Douady’s interest in the Mandelbrot set was contagious. He would host elaborate meals, parties and concerts at his apartment, and was known to walk barefoot through the corridors of the universities he taught at in France — and to sing, loudly, in public. (He was often mistaken for a busker.) In his later years, he never read math papers; he instead invited their authors to visit and explain the work to him directly.

“I would compare him with painters of the Renaissance who had a school of disciples around them,” said Xavier Buff, a mathematician at the University of Toulouse and one of Douady’s former doctoral students. “It was very exciting.”

A key part of the Orsay notes was a humble statement that would soon become the most important question about the Mandelbrot set: the MLC conjecture.

MLC posits that the Mandelbrot set isn’t just connected; it’s locally connected — no matter how much you zoom in on the Mandelbrot set, it will always look like one connected piece. For instance, a circle is locally connected. An extremely fine-toothed comb, on the other hand, is not. Though the entire shape is connected, if you skip over the shaft and instead zoom in on the tips of some of its teeth, you’ll just see a bunch of separate line segments.

Despite being a straightforward statement about the Mandelbrot set’s geometry, MLC quickly gained a reputation for being incredibly hard. Many mathematicians were hesitant to work on it. It seemed so technical and time-consuming — a risky problem to set one’s sights on. More than one mathematician ended up leaving mathematics because of it. Avila actively steers his students away from MLC and related areas of research until they have time to learn all the mathematics required to make headway. “I quote The Lion King and say, ‘Look, there is the whole of dynamics. All you can see is your domain. But there’s that dark corner that you should not explore … because if you do explore this part, you get trapped and never get out,’” he said. “There’s so much you need to learn to get into this.”

But some mathematicians couldn’t resist.

Only Connect

Misha Lyubich grew up in the 1960s in Kharkiv, the second-largest city in Ukraine. Stalin was dead; Nikita Khrushchev briefly held power, but was soon replaced by Leonid Brezhnev. The Soviet economy flourished, only to stagnate as the decade wore on. Tensions with the West were at an all-time high.

Lyubich’s father was a professor of mathematics at Kharkiv University, his mother a programmer; he remembers other mathematicians coming to his home when he was young, where math was always in the air, a frequent topic of conversation. “Life all around me was mathematics,” he said.

As a Jew in the Soviet Union — where “there were state policies which tried to eliminate Jews from being actively involved in various fields,” Lyubich said — he had trouble getting into top universities. He applied to Moscow State University but was rejected. Despite being a top student and one of the highest-ranking participants in the Soviet Union’s prestigious Math Olympiad competitions, he was told he hadn’t passed his oral exam. The examiners refused to tell him where he’d gone wrong.

He ended up attending Kharkiv University, one of the top undergraduate institutions that accepted Jewish students on merit. His father taught subjects that students would typically only be able to find at Moscow universities. (Moscow was the center of mathematical progress in the Soviet Union.) “It was a unique opportunity that my father was providing at that time … to get a broader vision of mathematics,” Lyubich said. In particular, his father encouraged him to start thinking about problems in complex dynamics — a field that wasn’t getting attention in the Soviet Union at all. “At that time, we didn’t see anybody working in this area,” Lyubich said. He quickly got hooked: It was in those university years that he started to think about math “essentially nonstop.”

Though he graduated second in his class, he struggled to get into graduate programs. He ended up more than 2,000 miles away at Tashkent State University in Uzbekistan, where his father had colleagues. He continued to study complex dynamics, isolated from and unaware of the work Douady and Hubbard were doing in France. “I was kind of alone,” he said. “It was quite lonely.”

University students were required to do agricultural labor during the autumn months. “The universities essentially emptied in October and November,” Lyubich said. And so he found himself picking cotton — Uzbekistan was the Soviet Union’s main cotton supplier at the time — in the fields outside Tashkent. From sunrise to sunset, in 90-degree heat, he bent over the plants, which stood only a couple feet high. He considered himself fortunate, though. Undergraduates had to meet a quota — high enough that “it required skill,” he said, and turned into back-breaking work that “would not have been possible for me to do.” Graduate students didn’t have to.

And so, “I was just walking around the cotton fields thinking about mathematics,” Lyubich said. In particular, he started to think about the parameter space of complex quadratic equations.

Even though the first computer images had already emerged in the West, Lyubich had no access to them. Instead, the basic features of the Mandelbrot set took shape in his mind — the fractal’s central heart-shaped region, called the main cardioid, and aspects of the set’s backbone, which bisects the shape horizontally along the x-axis. “I just built up a picture in my mind and tried to understand it,” he said. “I had no idea how deep the questions hidden inside of this picture were.”

In March 1982 — while Lyubich was still a graduate student — John Milnor, one of the most distinguished American mathematicians of his generation (then a professor at the Institute for Advanced Study), visited Moscow to give a talk. Because the university was flexible about where Lyubich spent his time, so long as he completed his exams and dissertation (as well as his cotton-picking duties), he often went to Moscow to attend seminars and meet with mathematicians who worked there. It just so happened that he was there when Milnor visited. After Milnor finished his talk, he and Lyubich spoke for a bit.

Due to the language barrier, they either wrote things down or had one of Lyubich’s colleagues help translate. It became clear to Lyubich that related work was happening on the other side of the Iron Curtain. “It was my first contact with Western mathematics in this direction,” he said.

After returning home, Milnor spread the word about some of Lyubich’s research. “The communication was very poor, but it was my good luck that I met Milnor,” Lyubich said. And so later, Douady sent Lyubich a copy of the Orsay notes, where Lyubich first learned about the MLC problem.

Lyubich wouldn’t truly start thinking about MLC for a few more years, though. He was working on other problems, and after completing his doctorate in 1984, he and his wife, also a mathematician, moved to Leningrad (now St. Petersburg), where he was once again barred from academic jobs because he was Jewish. Over the next five years, he worked instead as a high school teacher, as a programmer at what he called a “quasi-research institute” (focused on medical technologies), and finally as a modeler at a scientific institute that did comprehensive studies of the Arctic and Antarctic. With each new job, he inched closer and closer to being able to focus on his mathematical interests in dynamical systems.

Throughout those years, he kept working on his math problems. He attended seminars, met with other mathematicians, and continued to produce results. “I never stopped,” Lyubich said. “You see, if you stop, it is very difficult to recover. You should not stop.”

It was draining. Lyubich recalls feeling particularly exhausted after teaching high schoolers all day, only to then force himself to spend the rest of the evening working on math. “I was frustrated that I could not dedicate myself fully to mathematics, which is what I wanted to do,” he said. But “I sort of decided for myself that I would do mathematics, no matter what.”

“I was lucky that perestroika came and I was allowed to leave,” he added. “I don’t know for how long I would be able to keep this going.” In 1989, he and his wife obtained a visa that allowed them to leave the Soviet Union as refugees. With just a few hundred dollars in their pockets, they made their way first to Vienna, then to Italy, where they applied to move to the United States. After spending a few months at a refugee camp in Italy, waiting for their paperwork to be processed — during this time, Lyubich made extra income by giving guest lectures at local universities — he and his wife finally arrived in New York. There, Lyubich had a job waiting for him: Milnor (with whom Lyubich had kept in touch) had invited him to work at the new Institute for Mathematical Sciences he was starting at Stony Brook University.

While in Italy, Lyubich gained access to email for the first time — and it was there that he received an email from Douady. (Douady was an early advocate of using email for mathematical discussions and collaborations. “He worked a lot exchanging ideas with faraway collaborators, which was something new in the ’80s,” said Pierre Lavaurs, one of his former graduate students.)

The email informed Lyubich and other mathematicians in the field that Jean-Christophe Yoccoz had proved local connectivity at almost all points in the Mandelbrot set: MLC was true for values of c that did not reside inside an infinite nest of smaller self-similar copies of the full set. (Yoccoz would later be awarded the Fields Medal, considered math’s highest honor, in part for this work.)

In the email, Douady went on to say that the full solution to MLC was just around the corner. He wasn’t the only one who felt optimistic. “There were people who thought they could deal with the local connectivity of the Mandelbrot set in just a few years,” said Davoud Cheraghi of Imperial College London.

Instead, decades of work remained. MLC turned out to be a very subtle, almost impossibly difficult problem that only a handful of mathematicians were able to keep working on. It would require tools from all over math, and the development of a new theory that would forever change the field of complex dynamics.

Leading the way, armed with the persistence that had been a part of his mathematical journey all along, was Lyubich.

A City Within a City

We tend to think of math as the purest of the sciences — when we think of it as a science at all. The subject has a reputation for being abstract, detached, driven by beauty and logic. It doesn’t get its hands dirty or concern itself with anything as concrete as “applications.” (It’s even in the name: We distinguish “pure math” from “applied math.”) The way math papers are written doesn’t help: Only the final proofs and theorems are usually published, not the meandering process that led to them.

But this is a modern conception of mathematics, one that only started to solidify in the late 19th century. It’s a conception that grew as mathematicians sought to make their definitions more rigorous, and as writing formal proofs became the only way for them to get jobs and build careers. It was further bolstered in the 1930s, when a powerful, secretive group of mathematicians began to publish joint work under the pseudonym Nicolas Bourbaki. Their ethos came to dominate mathematical thinking, intent on stripping the discipline to its foundations and making it as formal as possible.

Yet long before this, mathematicians — just like physicists or biologists or chemists — relied on experimentation to discover and prove new phenomena. They made guesses, discarded hypotheses, looked for patterns by trial and error. They performed computations, made observations, gathered data. They took note of similarities, of certain numbers or sequences arising in unexpected places.

The giants of 18th- and 19th-century math — Euler, Gauss, Riemann — were all experimentalists who relied on massive amounts of computation, done laboriously by hand. Gauss conjectured the prime number theorem (a crucial formula that describes how the primes are distributed among the integers) a century before it was actually proved. That’s because, as a teenager, he pored over tables of primes and decided to count how many of them there were in blocks of a thousand numbers, all the way up to a million. (No doubt Gauss would have been thankful for today’s computers.) Similarly, Riemann posed his eponymous hypothesis, the biggest open problem in mathematics, only after doing pages of calculations. Those pages weren’t discovered for decades; until then, many mathematicians heralded the Riemann hypothesis as an example of what could be achieved by “pure thought alone.”

There’s no such thing. All thinking, mathematical or otherwise, is influenced by the world around us, by the technologies and philosophical movements and aesthetics of our time.

In this regard, Bourbaki’s philosophy — its requirement for total rigor, and its emphasis on general statements over concrete examples — represented a detour of sorts. Mathematicians’ perspective on Bourbaki is divided. Some claim it gave certain fields a much-needed push toward rigor. Others say it was confining, closed-minded, cutting math off from other sources of inspiration.

Since the 1970s, the pendulum has begun to swing back, pushed by modern computers, which have offered mathematicians entirely new ways to experiment and play. “I think people generally agree that the Bourbaki thing was sort of a mistake,” Eilers said. “This very abstract view, this is not so human-friendly … this is just not how the field should evolve.”

In the experimental spirit of Gauss and Riemann, mathematicians posed one of today’s most famous open problems — the Birch and Swinnerton-Dyer conjecture, a question about elliptic curves that, if solved, comes with a $1 million reward — only after using a computer to generate mountains of data. Many other problems have arisen in similar ways. “This is how the sausage is made,” said Roland Roeder of Indiana University–Purdue University Indianapolis. “It’s not as advertised as it should be.”

Mathematicians have used computers to look for counterexamples to both established conjectures and nascent hypotheses. They’ve used them to find, and fix, mistakes in old proofs. They’ve turned to them to forge new connections between disparate fields. And in many areas, mathematicians have come to rely on computers to make key calculations and perform other steps in the mathematical argument itself.

In the case of the Mandelbrot set, computers helped to jump-start an entire field.

To hear mathematicians tell it, computers have allowed them to treat the Mandelbrot set like a city — a physical space to explore. They’ve spent hours, days, years strolling its neighborhoods and streets, getting lost, familiarizing themselves with the terrain. “You start to understand more and more and more, and every time you come back, it’s like coming back home,” said Luna Lomonaco of the National Institute for Pure and Applied Mathematics in Brazil. “It really becomes part of you.”

This familiarity is clear whenever you speak to mathematicians in the field. They navigate different computer programs with ease, zooming in to specific spots to show different properties. Dudko describes these images as “like a language in complex dynamics.” Buff can predict exactly where he thinks a small copy of the set will pop up before it becomes visible, just based on how certain branches and tendrils look. Chéritat was once asked to reproduce a decades-old poster of a region deep within the Mandelbrot set, without any additional information — and he did it. Douady could apparently look at a Julia set and know which value of c in the Mandelbrot set it came from. Hubbard still refers to Julia sets as “old friends.”

“Studying the Mandelbrot set really feels like an experimental field of math. It almost feels like an applied field of math, as opposed to a pure field of math,” Kapiamba said. “You’re just taking something that is out there, and then trying to dissect and analyze it in a way that to me feels like you have some natural phenomenon that you’re trying to uncover.”

“It’s not something you create. It’s something which is there, and that you explore,” Buff added. “It’s clearly there on my computer. I visit the Mandelbrot set. And maybe there are some places in the Mandelbrot set that I have not discovered yet.”

This area of study is riddled with such discoveries. There was the discovery of smaller copies of the set within itself, and of specific patterns in the way its antennae, hairs and other decorations appear. There was the discovery of the Fibonacci sequence, encoded in the set — as well as an approximation of $latex \pi$. And there was the discovery of Mandelbrot sets in other contexts entirely, as in the search for numerical solutions to cubic equations.

“Computers show us stuff that’s tantalizing, that’s crying out for someone to come and explain it,” said Kevin Pilgrim of Indiana University Bloomington. Which in turn motivates the right questions, if not the answers.

When computers revealed all those smaller copies of the Mandelbrot set within itself, Douady and Hubbard wanted to explain their presence. They ended up turning to what’s known as renormalization theory, a technique that physicists use to tame infinities in the study of quantum field theories, and to connect different scales in the study of phase transitions. It had previously held little interest for mathematicians; by their standards, it wasn’t even rigorous.

But in the 1970s, the physicist Mitchell Feigenbaum brought renormalization theory into the world of dynamics, using it as a way to explain a particular self-similar pattern that emerges when you iterate quadratic equations using real numbers.

Douady and Hubbard realized that renormalization was precisely what they needed to explain the more complicated self-similar patterns they were seeing on their computer screens. And so they figured out how to apply renormalization theory to complex dynamics.

Since then, work on MLC by Lyubich and his colleagues has pushed that theory further than anyone thought possible.

A Name for Every Dot

Once Lyubich arrived in New York in February 1990, months after he’d left Moscow, he had the chance to learn more about the work that Douady had written so excitedly about in his email.

At first, it wasn’t the MLC result that fascinated Lyubich, but rather the techniques Yoccoz had developed to prove it. “Somehow, it clicked very well with me,” he said. He had been interested in real dynamics, and in answering questions that had arisen based on Feigenbaum’s work on renormalization. For most of the 1990s, Lyubich focused on developing Yoccoz’s methods further, to address those open problems. By the end of the decade, he felt that he’d “essentially gotten the full description of dynamics on the real line, using this machinery,” he said.

As a natural consequence of this work, Lyubich ended up proving MLC for many, though not all, of the cases that Yoccoz’s result had not covered.

That wouldn’t have come as a surprise. Yoccoz’s proof showed MLC for all points on the Mandelbrot set except those known as “infinitely renormalizable” parameters — points that lived inside infinitely nested baby Mandelbrot copies. His result instantly turned MLC into a problem that was intimately connected to renormalization theory.

That link was exciting. On the surface, MLC seemed to belong to an entirely different corner of the field. “Renormalization theory had developed completely independently,” Lyubich said. “And then everything became part of the same story.”

And so Lyubich also grew interested in addressing the MLC problem.

Even before renormalization entered the fray, there were already signs that MLC was a question with deeper resonances.

In the Orsay notes, Douady and Hubbard showed that if MLC is true, then it also has implications for properties of the interior of the Mandelbrot set. Not every point inside the set behaves the same way. Points in the main cardioid correspond to functions that, when iterated from a starting value of zero, converge to a single number. Points in other lobes correspond to functions that end up oscillating between a particular number of different values. The largest lobe on top of the main cardioid, for example, represents functions that oscillate between three values. For carefully chosen points, however, a function might produce sequences that remain bounded but never oscillate — they keep jumping between new, distinct values.

But if MLC is true, Douady and Hubbard showed that such non-oscillating sequences must be rare — a property called “density of hyperbolicity” that mathematicians want to prove or disprove for any dynamical system they happen to be studying. “It’s basically the most important question in dynamics, not just complex dynamics,” Lomonaco said.

Density of hyperbolicity deals with the Mandelbrot set’s interior. But MLC would also enable mathematicians to assign an address to every point on the set’s boundary. “It gives a name to every dot. And then, once you have been able to name every dot of the boundary of the Mandelbrot set, you can hope to really understand it completely,” Hubbard said.

In this way, MLC tells mathematicians that the picture they have of the set isn’t missing anything. But without a proof, there could still be some regions, tucked away in the deepest corners of this infinitely complex landscape, that have not yet appeared on computer screens — that behave in some fundamentally different way. It would mean that mathematicians are still missing part of the story.

Think Deeply About Simple Things

Jeremy Kahn grew up in New York City in the 1970s, the son of a social worker and a science writer. As a child, he quickly proved to be something of a math prodigy. He skipped years ahead in the subject. In sixth grade he scored a 790 on the math section of the SAT. And he wrote his own computer programs to explore various mathematical concepts in greater depth. When he was 13 years old, he became the youngest person (at the time) to win a spot on the U.S.’s International Mathematical Olympiad team. He participated in the competition throughout high school, winning two silver medals and two gold. During this time, he also started taking math courses at Columbia University, and he re-proved several theorems (without knowing they’d been proved) on a blackboard he kept in his bedroom.

After he graduated from high school, he went to Harvard University to major in math. There he became captivated by the Mandelbrot set. By his senior year, he was devoting all his energy to understanding it. Since no one at Harvard was working on it at the time, he would bike over to Boston University to learn from a mathematician there about fractals and dynamical systems. After he graduated and enrolled in a doctoral program at the University of California, Berkeley, he focused on hyperbolic geometry — a field that mathematicians had previously connected to complex dynamics, back when the Mandelbrot set was first becoming popular.

Kahn wanted to strengthen that connection. As a graduate student, he re-proved Yoccoz’s famous MLC result, building on seminal work done by the mathematicians Dennis Sullivan and Curt McMullen. He also began to think about how to apply ideas from hyperbolic geometry to renormalization.

Kahn’s classmate Kevin Pilgrim remembers seeing him fill massive sheets of paper with drawings of curves and annuli, of geometric objects that degenerated and grew distorted. “He started to think very, very deeply about these things,” Pilgrim said. “And when I say ‘deeply,’ I mean for 15 years.”

“Jeremy’s tenacity for thinking really hard about something is pretty amazing,” he added.

Kahn thought particularly hard about renormalization. He studied Lyubich’s work, and Douady and Hubbard’s.

In all these contexts, renormalization is a way to relate different scales of a dynamical system to one another. Consider the dynamics of one quadratic equation. Points will bounce around the complex plane in certain ways. Renormalization allows you to describe the dynamics of all those points by focusing on just a small subset of them.

“Renormalization acts like a super powerful microscope that allows you to understand structures which lie at the deepest level,” said Romain Dujardin of Sorbonne University in France.

The extent to which you can do this depends on the equation you’re iterating. Sometimes you simply cannot describe its dynamics in terms of a smaller part of the system. Or you might be able to use the microscope of renormalization to magnify things once, or twice, or 10 times, before reaching a point where you can no longer say anything meaningful about the smaller scales.

But for the functions associated with infinitely renormalizable parameters, it’s possible to keep applying renormalization forever.

It’s a delicate procedure. “It cannot be done in a random way,” Lyubich said. You have to rigorously show that you can move from one scale to another without losing too much precision.

The first step toward doing that involves gaining a rough sort of control over the geometry of the different scales. It’s this step that can then be used to show MLC for a given value of c in the Mandelbrot set.

As a graduate student, Kahn was already thinking about how to apply his knowledge of hyperbolic geometry to the problem. His research garnered attention, and in his third year of graduate school, he accepted a tenure-track job at the California Institute of Technology.

Everything seemed to be lining up perfectly.

And then he froze.

At Caltech, he couldn’t write. He had results from his time in graduate school — but every time he sat down at a computer, he would lose any willpower he had. “I wasn’t good at writing,” he said. “I wasn’t good even at sitting down to write. So I wasn’t getting the stuff written up.”

He couldn’t focus his mathematical attention either. “I would sometimes lose myself in the extremes of wanting to prove truly great theorems, like MLC, or P versus NP. And then I’d come back to reality,” he said. “I was lost, and unhappy.”

In four years at Caltech, Kahn didn’t write a single paper. He lost his job.

And so, in the fall of 1998, at just under 30 years old, his once-promising career in tatters, “I kind of wandered back home” to New York, Kahn said.

He called Milnor, asking for advice. Milnor put him back in touch with Lyubich, whom Kahn had met a few times in graduate school. And so, “I just showed up at Stony Brook,” Kahn said. “Misha was incredibly welcoming.” The two would discuss math for hours. Kahn recalls going to Lyubich’s house all the time, eating dinner with his family — by then, Lyubich and his wife had a daughter; they would later have a second — and soon becoming friends. “He really took me in,” Kahn said. “He was this world-famous mathematician, and he treated me as an equal, not some lost child.”

“He became practically a second father to me,” he added.

Lyubich found a temporary position for Kahn at Stony Brook, without teaching duties. From the late 1990s into the mid-2000s, Lyubich helped the younger mathematician out. When Lyubich spent a year working at the University of Toronto, he found a place for Kahn; when he returned to Stony Brook, he did the same. When Kahn left academia to work at a hedge fund for a year, only to decide that it wasn’t for him, Lyubich helped him out once again. When Kahn’s father was diagnosed with cancer and later died, Kahn wasn’t able to work. But he eventually made his way back to Lyubich, and Lyubich welcomed him.

To hear Lyubich tell it, he recognized that Kahn had very interesting, sometimes brilliant ideas. “He just had this psychological block he needed to overcome,” Lyubich said. “So I kept supporting him as much as possible.”

Although Kahn still often felt lost during these years, he and Lyubich developed what Kahn called “quite an intense collaboration.” It kept him grounded. The two mathematicians unified their approaches to renormalization, which also allowed them to prove MLC for many more parameters.

“The sort of collapse of my career gave the opportunity for me to just follow Misha around” and get this work done, Kahn said. “It was putting off a lot of elements of living, not deliberately, but in effect for the sake of proving these theorems.”

Kahn and Lyubich’s work marked a massive breakthrough in renormalization theory, and in MLC. But “the Mandelbrot set is tremendously devious,” Lyubich said, because it is not exactly self-similar, and it exhibits different kinds of self-similarity. As Avila put it, “it has different personalities as you move inside it.” These different kinds of self-similarity correspond to very different dynamics and therefore require different types of renormalization to relate one scale to another.

Kahn and Lyubich had developed one type, but they’d pushed their techniques as far as they could. “They hit a wall, and they knew that they’d hit a wall,” Mukherjee said.

To prove MLC for other parts of the Mandelbrot set, they would have to get a similar kind of geometric control, but using some other type — or types — of renormalization.

And Kahn and Lyubich disagreed on how best to proceed.

Progress stalled.

They each started to work on other problems. Kahn turned back to hyperbolic geometry. Lyubich thought about ways he could apply the MLC work to other parts of complex dynamics (and even to questions in physics).

“This is why, in a way, you’re never really stuck,” said Lyubich, who in 2004 became the director of Stony Brook’s Institute for Mathematical Sciences. “If tomorrow someone will find a one-line proof of MLC in all cases, would it annihilate everything we have done before? No. There are so many problems that rely on this technique.”

That’s part of the reason he never felt frustrated when things didn’t seem to be progressing quite so smoothly on the MLC front. “Every step in MLC is an opening to many other problems,” he said.

Meanwhile, Kahn made significant advances in hyperbolic geometry. Tenure offers began to come in. Hoping to make a fresh start, he moved to Providence, Rhode Island, in 2011 to take up a professorship at Brown University.

Neither Lyubich nor Kahn stopped thinking about MLC, but they drifted apart, busy with their own responsibilities.

Other mathematicians working in complex dynamics started to move in different directions — focusing on parameter spaces even more complicated than the Mandelbrot set, and on the connection between complex dynamics and number theory.

But in recent years, Lyubich and Kahn have each taken on apprentices and renewed their efforts to prove MLC.

Squaring Up

About a decade ago, Lyubich began working with Dima Dudko.

Dudko grew up in the 1980s in Belarus, where his mathematical prowess quickly became obvious to those around him. (He represented Belarus in the International Math Olympiad 15 years after Kahn aged out. Like Kahn, he won a gold medal.) Later, when he was a graduate student in Germany, his adviser consulted Lyubich about what problem Dudko should work on for his dissertation. They decided on a question about the Mandelbrot set that they didn’t expect Dudko to be able to answer. The statement would follow automatically from MLC; they figured that, without MLC to help him, he’d be able to make partial progress on it at best.

Dudko found a way around MLC and solved the problem completely.

After finishing his graduate program in 2012, he continued to work in Germany as a postdoc — but also started collaborating with Lyubich. With a third mathematician, Nikita Selinger of the University of Alabama, Birmingham, they developed a new renormalization theory. Lyubich and Dudko then used it to show that MLC holds for some of the most difficult infinitely renormalizable parameters in the Mandelbrot set — precisely the ones that Lyubich and Kahn’s methods couldn’t be applied to. (Lyubich’s former student Davoud Cheraghi and Mitsuhiro Shishikura of Kyoto University have also been developing techniques to address some of these outstanding cases.)

“This case is so different that it took another couple decades,” Lyubich said. It also took some original thought. Dudko, who led the recent MLC seminar with Lyubich in Denmark, is seen as a star in the area, and he has an intriguing way of looking at things. This is perhaps best exemplified by how he sometimes sketches the Mandelbrot set as a bunch of squares, rather than the circles that most mathematicians tend to draw.

“It’s taken me by surprise that it’s possible to solve these problems,” Lyubich said. “What we have been doing recently, it goes beyond anything I had done before.”

In an effort to assemble all of these results in one place, Lyubich has been writing a series of textbooks about the Mandelbrot set, MLC and related work in complex dynamics. So far, he’s produced over 700 pages, split into two volumes out of a planned four. “Hopefully, when I finish with volume 4, MLC will be there,” he said.

Like Lyubich, Kahn has found a younger protégé. The idea of recruiting Alex Kapiamba first came to Kahn in a dream. He was at a conference in 2022. For several months, he, Lyubich and Dudko had been meeting regularly to discuss progress on MLC — something that was immediately reflected in the dream, where the three of them were on a bus. “And then I see this fourth person get onto the bus, and that’s the whole dream, essentially,” Kahn said. “And then I wake up, and I’m like, Alex Kapiamba is this fourth person.”

The next day, he arranged to meet with Kapiamba to discuss his research. Kapiamba now works with Kahn as a postdoc at Brown, and will move to Harvard in the fall.

When I met Kapiamba last year, his arm was in a sling; he’d dislocated his shoulder a few days earlier playing ultimate Frisbee. (He played semiprofessionally for the Detroit Mechanix while in graduate school, and continues to play in a club league.) He was modest about how much he thought he’d be able to contribute to the MLC effort. “It’s sort of a little scary,” he said. “I definitely feel some imposter syndrome.”

“I just want to get in and do a little bit before it’s too late,” he added.

Kapiamba hadn’t set out to study mathematics. As an undergraduate at Oberlin College in Ohio, he started as a biochemistry major; it was only at the end of his junior year, after he took a topology course, that he grew interested in math. “In biochemistry, what I really liked was understanding the structure of things,” Kapiamba said. “And math is really just trying to study structure in its barest form. It really felt like it was the parts of biology or chemistry that I really enjoyed, distilled down into a pure form. I could just do that part.”

After graduating in 2014, he was unsure about what he wanted to do. He moved to Washington, D.C., to be near his family, and found jobs working at a bakery and as a tutor. During this time, he began to contemplate pursuing a career in math. He soon quit his baking job, and for the next two years, he continued to tutor while studying higher-level mathematics on his own time — reviewing the material he’d learned during his undergraduate years (“to get a different vantage point,” he said) and taking online courses. “I wanted to feel very prepared,” he said. In 2016, he enrolled in a master’s program at the University of Michigan.

As a master’s student, he started to work on a question about the geometry of the Mandelbrot set near the cusp of its main cardioid, where a parade of elephants marches out of a shallow valley. As you approach the valley, the elephants seem to get closer and closer together. And so it’s been conjectured that as you approach the valley’s deepest point, the distance between the elephants will shrink to zero. “I was like, obviously,” Kapiamba said, motioning at his computer screen, where he’d zoomed in on the elephants for me to see. They really did look as if they were touching.

A key part of his argument rested on an offhand remark made in an old doctoral thesis paper. The 73-page dissertation, written entirely in French, was completed in 1989 but never published. Its author had left mathematics just one year later, after growing disillusioned and frustrated with the problem he’d hoped to solve: MLC.

Kapiamba combed through the text, often getting lost in its pages without realizing the clock had long since ticked past midnight, relying on the French he knew from high school and Google Translate. He lamented that he hadn’t been raised to speak French. Both his father, who’s from the Democratic Republic of Congo, and his mother, who met him there while serving in the Peace Corps, spoke the language fluently. But the couple had moved to Maryland shortly before Kapiamba was born, and in an effort to help his father learn English as quickly as possible, they only spoke English at home.

Eventually, Kapiamba realized that he wasn’t failing to grasp some step in the thesis paper’s logic. Its author had made a mistake. His claim was likely correct, but the reasoning behind it didn’t hold up. And so Kapiamba set his sights on fixing the error.

He let things simmer, the way he waits for bread to rise. (He still bakes to focus his mind. He enjoys the opportunity it gives him to make something with his hands.) Over the next few years, he finally figured out the proof. To do so, he had to strengthen a theorem that Yoccoz had used in his original MLC proof, about the size of the elephants.

The work took the complex dynamics community by complete surprise. Computer images had already indicated that certain regions of the Mandelbrot set seemed to shrink much, much faster than Yoccoz’s theorem suggested, meaning that his statement could be strengthened. “If you just plot some pictures and look at them, you can see, oh, it seems like the bound Yoccoz gives us is very, very bad,” Kapiamba said. But no one had been able to improve it.

Until Kapiamba. His work only applied to certain regions in the Mandelbrot set; mathematicians hope that the stronger version of Yoccoz’s statement can be shown for the entire set. Even so, “people got really excited,” Benini said. “Everyone working on this knows this must be true; they just didn’t know how to prove it.”

Lomonaco and other mathematicians have already used Kapiamba’s result to prove theorems of their own. But it’s also seen as a potential linchpin in a future proof of MLC.

A Laboratory and a Guide

Last year’s conference marked the last time mathematicians will gather at the old military base in Denmark. Roskilde University, which sponsors the workshop series, gave up its lease on the location this year.

If Lyubich, Kahn, Dudko and Kapiamba can combine their different approaches to finally prove MLC, it will mark the end of another era — an era that began when Mandelbrot and Hubbard and Douady first saw the fractal appear on their computer screens.

The last half-century of exploration of the Mandelbrot set was made possible by the development of computer graphics. The math that generates the fractal is simple: You really only need to know how to add and multiply. But the drawings that made the set famous could not have been done by hand. They relied on carrying out those easy computations millions of times, something that wasn’t feasible without computers.

In principle, a visionary mathematician might have held a snapshot of the set in their mind’s eye hundreds of years ago. But in the unfolding of history, though genius can sometimes glimpse over the horizon, technology has modulated what can be imagined. Fatou, for instance, “was able to formulate conjectures without having been able to see the Mandelbrot set,” Buff said. But Fatou could only go so far. However powerful his imagination might have been, there is a world of richness swirling beneath the Mandelbrot set that was inaccessible to him, but readily visible to an average person today.

Lyubich does not tend to use computers in his work. “My way of thinking is very visual,” he said. “It’s very geometric. I think in terms of pictures — but I just draw more or less primitive pictures, by hand or in my mind. I never use computers in any substantial way.” (He jokes that perhaps the programming job he briefly held in Leningrad before emigrating is to blame. “It repelled me,” he said.) Nevertheless, he lives in a world steeped in computation. Back in Uzbekistan’s cotton fields, he too only got so far by letting his imagination run wild. “It was Douady and Hubbard who viewed the next level of depth,” he said — using the computers available in the 1980s. In the decades since, Lyubich has seen his collaborators use computers as a laboratory and as a guide. In his one joint paper with Milnor, he recalls, Milnor ran several computer experiments to help steer their proof in the right direction. And Dudko returns again and again to the computer while working with Lyubich. “He’s very good at interpreting what he sees,” Lyubich said, “to translate these pictures into mathematical language and formulate very deep conjectures.”

Galileo discovered the moons of Jupiter not only because he had developed the right theory to make sense of what he saw, but because he had a telescope. Similarly, there are entire swaths of the mathematical universe that remain hidden until technological change makes them visible. They can no more be discovered with pure thought than Jupiter’s moons can be discerned by squinting.

If the computational revolution of the 1970s and ’80s opened up the continent of the Mandelbrot set for exploration, mathematicians might today be on the cusp of another such tipping point. Artificial intelligence is only beginning to be used to formulate substantive conjectures and prove significant mathematical results. It is hard — perhaps impossible — to gauge its potential with confidence. (“We’ve got to try to train a neural network to zoom around the Mandelbrot set,” Kapiamba joked.) But if the story of the Mandelbrot set is one of how mathematicians can use pure thought to survey a vista opened up by technology, the next chapter remains to be written.

“I never had the feeling that my imagination was rich enough to invent all those extraordinary things,” Mandelbrot once said. “They were there, even though nobody had seen them before.”

Correction: January 30, 2024
This article originally stated that Jeremy Kahn recently submitted work from his time at Caltech for publication. The recently submitted work was done in 2001, three years after he left Caltech. The article also originally misstated the date of a dream by Jeremy Kahn. The dream took place in 2022, not 2019.

Clarification: January 30, 2024
An image depicting what it means to be globally connected but not locally connected did not clearly show that locally disconnected tines of a comb are those that are arbitrarily close to the comb’s left edge. That image has been updated.