UGTV (View recorded usergroup presentations by hundreds of your favorite presenters)
References
Reference
Repository
Home > Ajabbi Research > Library >
Home > Handbook >
Last Updated
17/05/2025
CFML Repo
By:Gavin Pickin
CFML Repo: 10/02/2024
'Where CFML engine installers and related files live on for posterity This
repo was created to help folks easily find installers and related files
for ColdFusion, Lucee, and other CFML engines, returning to some of their
earliest releases.
About the CFMLRepo
Created initially in Feb 2014 by Gavin Pickin, the repo has since been
contributed to by many other trusted members of the CFML community,
including Charlie Arehart, Adam Cameron, Dave Epler, Wil Genovese, Frank
Jennings, Travis Peters, Kevin Roche, Carl Von Stetten. The files are
offered on an as-is basis, without warranty.
The repo includes editors/IDEs, hotfixes/updates, docs, and more. (And
while originally called CFRepo, for holding Adobe CF installers, it was
renamed CFMLRepo in 2016 to accommodate the inclusion of other CFML
engines.)
If anyone has a file and would like to contribute, contact me, Gavin
Pickin, on twitter - @gpickin
Using the CFMLRepo
The files in the repo are offered by way of Google Drive, and its web
UI. You will see folders for different engines and related tools within
those often different versions from the past. Note that Drive requires
you to double-click to go into a folder. When viewing files, especially (versus folders), you may find it
helpful to toggle between the grid and list view using the icon in the
top right of the Google Drive UI. You can download a file by either
double-clicking or right-clicking it. You can also select more than one
file at a time, or even download an entire folder. Again, here is the
link:
Design and Build Great Web APIs by Mike Amundsen. 2020, The Pragmatic Programmer.
Enterprise API Management by Luis Weir. 2019, Packt.
Repository
Home > Ajabbi Research > Library >
Home > Handbook >
Last Updated
17/05/2025
Building an API engine
By: Mike Peters
On a Sandy Beach: 06/02/2024
Mike is the inventor and architect of Pipi and the founder of Ajabbi.
"An application programming interface (API) is a way for two or more
computer programs or components to communicate with each other. It is a type
of software interface, offering a service to other pieces of software.[1] A
document or standard that describes how to build or use such a connection or
interface is called an API specification. A computer system that meets this
standard is said to implement or expose an API. The term API may refer
either to the specification or to the implementation. Whereas a system's
user interface dictates how its end-users interact with the system in
question, its API dictates how to write code that takes advantage of that
system's capabilities.
In contrast to a user interface, which connects a computer to a person, an
application programming interface connects computers or pieces of software
to each other. It is not intended to be used directly by a person (the end
user) other than a computer programmer who is incorporating it into the
software. An API is often made up of different parts which act as tools or
services that are available to the programmer. A program or a programmer
that uses one of these parts is said to call that portion of the API. The
calls that make up the API are also known as subroutines, methods, requests,
or endpoints. An API specification defines these calls, meaning that it
explains how to use or implement them." - Wikipedia
Thanks
Martin Catterall got me started with API in 2018 and was a great sounding board.
Priorities
Come up with a robust architecture/data model that can be easily extended using a plugin for each API type.
Any plugin can be a 3rd-party open-source application (GitHub).
Pipi can use this to automatically build internal API.
Do these first
1 REST
2 GraphQL
3 SOAP
4 DDS
5 gRPC
And
Open as possible
Well documented
Comply with standards
Reliable, secure, and fast
Work with multiple types of API
Can be versioned
Can scale
Websites
SOA Tech Magazine
Wikipedia
API
Examples to test
https://github.com/XeroAPI/Xero-OpenAPI
programmableweb.com
RapidAPI.com
apilist.fun
ifttt.com
elastic.io
Postman
Classification
This is my hierarchical list of API types. Is this a good classification?
These are the specification languages used to describe API. This list is incomplete. What is missing? Wikipedia article on interface description languages.
GraphQL
SDL
REST
API Blueprint
OData
OpenAPI
RAML
SOAP
WSDL
CFML
Taffy (The REST Web Service Framework for ColdFusion and Lucee)
Home > Ajabbi Research > Library > Subscriptions > Quanta Magazine
Home > Handbook >
Last Updated
17/05/2025
The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal
By: Jordana Cepelewicz
Quanta Magazine: 26/01/2024
Jordana Cepelewicz is the math editor at Quanta Magazine. She previously covered mathematics and biology as a staff writer. Her writing has also appeared in Nautilus and Scientific American. Before entering the world of science reporting, Jordana did editorial work at Harper’s Magazine, Politico and Tea Leaf Nation. She graduated from Yale University in 2015 with bachelor’s degrees in mathematics and comparative literature..
In the mid-1980s, like Walkman cassette players and tie-dyed shirts, the
buglike silhouette of the Mandelbrot set was everywhere.
Students plastered it to dorm room walls around the world. Mathematicians
received hundreds of letters, eager requests for printouts of the set. (In
response, some of them produced catalogs, complete with price lists; others
compiled its most striking features into books.) More tech-savvy fans could
turn to the August 1985 issue of Scientific American. On its cover, the
Mandelbrot set unfolded in fiery tendrils, its border aflame; inside were
careful programming instructions, detailing how readers might generate the
iconic image for themselves.
By then, those tendrils had also extended their reach far beyond
mathematics, into seemingly unrelated corners of everyday life. Within the
next few years, the Mandelbrot set would inspire David Hockney’s newest
paintings and several musicians’ newest compositions - fuguelike pieces in
the style of Bach. It would appear in the pages of John Updike’s fiction,
and guide how the literary critic Hugh Kenner analyzed the poetry of Ezra
Pound. It would become the subject of psychedelic hallucinations, and of a
popular documentary narrated by the sci-fi great Arthur C. Clarke.
The Mandelbrot set is a special shape, with a fractal outline. Use a
computer to zoom in on the set’s jagged boundary, and you’ll encounter
valleys of seahorses and parades of elephants, spiral galaxies and
neuron-like filaments. No matter how deep you explore, you’ll always see
near-copies of the original set — an infinite, dizzying cascade of
self-similarity.
That self-similarity was a core element of James Gleick’s bestselling book
Chaos, which cemented the Mandelbrot set’s place in popular culture. “It
held a universe of ideas,” Gleick wrote. “A modern philosophy of art, a
justification of the new role of experimentation in mathematics, a way of
bringing complex systems before a large public.”
The Mandelbrot set had become a symbol. It represented the need for a new
mathematical language, a better way to describe the fractal nature of the
world around us. It illustrated how profound intricacy can emerge from the
simplest of rules — much like life itself. (“It is therefore a real message
of hope,” John Hubbard, one of the first mathematicians to study the set,
said in a 1989 video, “that possibly biology can really be understood in the
same way that these pictures can be understood.”) In the Mandelbrot set,
order and chaos lived in harmony; determinism and free will could be
reconciled. One mathematician recalled stumbling across the set as a
teenager and seeing it as a metaphor for the complicated boundary between
truth and falsehood.
Mathematicians working in the field of
complex dynamical systems are patiently unraveling the Mandelbrot set's
mysteries and may be on the verge of solving a fundamental conjecture that
would allow them to describe the set completely.
Video: Mathematicians working in the field of complex dynamical systems
are patiently unraveling the Mandelbrot set’s mysteries and may be on the
verge of solving a fundamental conjecture that would allow them to
describe the set completely.
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.
The mathematicians Misha Lyubich (right) and Dima Dudko have spent
decades exploring the Mandelbrot set.
Karen Dias for Quanta Magazine
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.
Alex Kapiamba (left) and Jeremy Kahn often work together on a
blackboard in Kahn’s backyard, near Brown University’s campus in
Providence, Rhode Island, as they try to gain control of the geometry of
the Mandelbrot set.
Adam Wasilewski for Quanta Magazine
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.
Lyubich has nurtured generations of mathematicians at the institute he
now runs at Stony Brook University.
Karen Dias for Quanta Magazine
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.
In the 1910s, the French mathematicians Pierre Fatou (left) and Gaston
Julia pioneered the study of iterated functions that would later give rise
to the Mandelbrot set.
Left (Fatou): Collection familiale. Right (Julia): Deutsches Museum,
Munich, Archive, PR 01671/01
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:
Merrill Sherman/Quanta Magazine
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.
Benoît Mandelbrot, known today for coining the term “fractal,” also
studied the behavior of financial markets and geological
phenomena.
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.
Merrill Sherman/Quanta Magazine
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.
Pick a point in the Mandelbrot set to the left, and the corresponding Julia
set will appear in the right panel. (Some Julia sets far from the Mandelbrot
set are too faint to be seen.)
Paul Chaikin for Quanta Magazine
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.
Merrill Sherman/Quanta Magazine
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.
The first published plot of the Mandelbrot set, produced on a dot-matrix
printer, appeared in a paper by Robert Brooks and J. Peter Matelski in
1978.
“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.
Merrill Sherman/Quanta Magazine
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.
Adrien Douady (left) and John Hubbard were the first mathematicians to
show that the Mandelbrot set is connected.
George M. Bergman (Douady)/Archives of the Mathematisches
Forschungsinstitut Oberwolfach
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.
Lyubich in Leningrad in 1986. For years, he worked on math in his spare
time, unable to get an academic job because of Soviet
antisemitism.
Courtesy of Misha Lyubich
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.) Seminal work by the French mathematician
Jean-Christophe Yoccoz linked the MLC conjecture, one of the biggest open
problems in complex dynamics, to the mathematical theory of
renormalization.
Gerd Fischer/Archive of the Mathematisches Forschunginstitut
Oberwolfach
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.
The secret math society known as Bourbaki influenced much of
mid-20th-century mathematics. The effects of its emphasis on abstraction
and rigor continue to be felt today.
Nicolas Bourbaki
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.
Today, Lyubich is the director of the Institute for Mathematical Sciences
at Stony Brook.
Karen Dias for Quanta Magazine
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.”
Merrill Sherman/Quanta Magazine
“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 π .
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.
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.”
In the late 1980s, Benoît Mandelbrot was arguably the most famous
mathematician alive.
IBM
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.
Merrill Sherman/Quanta Magazine
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.
At age 13, Jeremy Kahn was already exhibiting prodigious mathematical
talent.
Courtesy of Carol Kahn
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.
This deep, deep dive into the Mandelbrot set reveals the fractal nature
of its boundary, as patterns endlessly appear.
This deep, deep dive into the Mandelbrot set reveals the fractal nature
of its boundary, as patterns endlessly appear.
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.
With Lyubich, Kahn applied a technique called renormalization to
understand some of the most devious regions of the Mandelbrot set.
Adam Wasilewski for Quanta Magazine
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.
These sketches show calculations involving renormalization, a technique
that originated in physics that has since been developed into a rigorous
mathematical theory. Renormalization plays a central role in complex
dynamics and has far-reaching applications.
Adam Wasilewski for Quanta Magazine
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.
During their frequent collaborations, Lyubich, from Ukraine, and Dudko,
from Belarus, often converse in Russian.
Karen Dias for Quanta Magazine
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.
Zoom into a spot near the cusp of the Mandelbrot set’s main cardioid,
and you’ll see a pattern that looks like a parade of elephants.
Maths.Town
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.
Alex Kapiamba, currently a postdoctoral fellow at Brown University,
showed that the marching elephants get arbitrarily close together as we
approach a valley on the right side of the Mandelbrot set.
Adam Wasilewski for Quanta Magazine
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 story of the Mandelbrot set shows how computers can open up
entirely new mathematical vistas, ripe for exploration.
Karen Dias for Quanta Magazine
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.”
