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Welcome to the Weird Mathematical World of Topology

The quirky rules of topology, a field of math, could help us understand everything from big data to quantum computing.

By Devin Powell
Mar 20, 2019 12:00 AMNov 14, 2019 7:03 PM
William Zuback


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Imagine waking up in a Salvador Dalí painting. You reach over to hit the snooze button on your alarm clock, only to discover that it oozed into a puddle during the night. The sun rising outside your window illuminates an elephant tottering down the street, its legs impossibly skinny stilts a hundred feet tall.

In this fantastic reality, everything is slowly but continuously distorting itself. Your coffee mug morphs into a doughnut, as if made of putty that’s been pinched and pulled. Breakfast is a confusing experience, as everyday items lose their identities.

This may sound like a bad trip. We all know that things in the real world tend to have fixed shapes; the letter L is different from the letter M. But in the funhouse world of topology, a field of mathematics that is slowly reshaping how we think about the world, the usual rules don’t apply. Its practitioners believe that an L is essentially the same as an M, or a C, or a Z. To topologists, objects that can be gently bent, twisted and stretched into each other are, in a sense, fundamentally identical.

For centuries, this Play-Doh way of looking at things held little practical value, but that’s starting to change. Topology is guiding how we make sense of big data today. It has helped physicists discover new materials that conduct electricity unlike anything else on Earth — and new physics hidden inside those materials. It has even inspired Microsoft’s efforts to develop a machine that sounds like science fiction: a quantum computer that promises to solve problems beyond the reach of today’s devices.

“Topology was not applied in a serious way to any serious important problems for a long time,” says Gunnar Carlsson, a Stanford University mathematician and expert in topology. “But it has become a force to be reckoned with in the 21st century.”

Seven Bridges Road

The story of topology begins almost 300 years ago, when one of the smartest mathematicians on the planet heard about a puzzle. The denizens of a faraway European city wanted to know: Could you stroll through their town, Königsberg, and cross each of its seven bridges exactly once?

That mathematician, Leonhard Euler, was an intellectual giant who created much of the mathematical notation we use today and produced arguably the most beautiful equation of all time. He didn’t think much of the Königsberg Bridge Problem at first, calling it “banal” in a letter. Thankfully, the Swiss genius decided to give it his attention, anyway. Euler’s contemporaries couldn’t find such a path — but they could not prove one didn’t exist, either.

William Zuback

Euler took an unusual approach. He ignored the details of the map. Forget about the lengths of the bridges and distances between them. Ignore the size of the city’s central island. Build a clay model of the city, distort it however you like — taking care only to stretch and pinch, never break — and the problem does not change.

All that mattered, Euler said, was the number of bridges and the number of landmasses they connected. Simply by counting these, he showed that the problem was unsolvable. There was no way to walk over every bridge exactly one time, sad news for anyone trying to organize an efficient bridge-themed walking tour of the city, now known as Kaliningrad, Russia.

Euler went on to search for other ways of looking at reality that ignore details you could measure with a ruler. Here’s one: Count and add the number of corners and faces on a closed box. Any box, of any size. Do you have it? Now subtract the number of edges. You should get the number 2. Mold that box into a pyramid or tetrahedron or any other everyday polyhedron. Repeat the same process. The result will always be 2.

Throughout the universe, shapes that look different can still have something in common — something mathematical and abstract.

With his new way of looking at shapes, Euler sowed the seeds of topology. (The name comes from the Greek words for “place” and “study.”) Topologists seek to understand what shapes have in common and what makes one different from another, using simple, numerical characteristics.

“Topology reveals structures that you might start to notice if your eyes weren’t so good,” says mathematician Dan Freed of the University of Texas at Austin. “Structures in which the fine details don’t matter.”

William Zuback


Arguably the greatest achievement of recent topology was proving a wacky-sounding conjecture from French mathematician Henri Poincaré, who took an already strange idea one step further.

Mathematicians knew that, oddly, any everyday object could be transformed into a sphere through gradual deformations, as long as the object has no holes. Poincaré proposed the same was true for objects in more than three dimensions — such as the shape of the universe, perhaps — again, assuming zero holes.

Holes have a special magic in topology. A coffee mug can slowly melt into a doughnut during a topologist’s Dalí-esque breakfast; the hole in the cup’s handle becomes the hole in the pastry. But the doughnut cannot transform into a hole-free sausage, or vice versa. That would require closing (or opening) a hole. Topology allows shapes to gently contort themselves, but not to suddenly tear or merge.

By thinking about holes, topologists have discovered a slew of exotic shapes. One of the strangest, first described in 1882, may be the Klein bottle. It looks like something out of a Dr. Seuss book and only truly exists in four dimensions. Because of the way this container wraps in on itself, it has no volume. Its inside is its outside.

“An ant can walk along the entire surface without ever crossing an edge,” writes Cliff Stoll, who may be the world’s biggest Klein bottle enthusiast. Stoll is an astronomer by training, with hair like Dr. Emmett Brown fromBack to the Future, and a glassblower who sells 3D representations of these curvy containers from his house in Oakland, California. He advises customers not to try putting liquid into the bottles; they’re not terribly practical, and they’re a real pain to clean.

A Klein bottle. William Zuback

If this bottle sounds confusing, well, it is. The shapes that topologists play with often have little connection to our daily lives; they defy common sense. Consider the Möbius strip, a loop made by twisting a ribbon once and taping the ends together. It’s an object with only one surface, one edge. While cutting a normal loop of ribbon in half lengthwise produces two smaller loops, if you cut a Möbius strip in half, you’ll get a single, larger loop.

For a long time, these bizarre shapes — and others revealed by topology — were mere curiosities. But then they started showing up in surprising places, like black-and-white digital photographs. About 10 years ago, Carlsson, the Stanford mathematician, was analyzing photographs a colleague had cut up into 9-pixel blocks when a pattern emerged. Plot the blocks as points on a graph with nine dimensions, one for the value of how dark each pixel is, and a shape emerges that looks like a Klein bottle. Carlsson applied this knowledge to invent a new way to digitally compress images to smaller sizes. A company he founded, AYASDI, has used topology to look for patterns in genes involved in different cancers, for instance, and bank transactions that indicate fraudulent activity.

Topology’s practicality doesn’t end there. Biologists hoping to better understand the complexities of genetics have increasingly turned to topology to provide clarity. So have neuroscientists overwhelmed by the sheer amount of data they can collect from the brain — and roboticists trying to teach their creations to perform sophisticated movements.

Topology, once abstract and otherworldly, has begun to rear its head in tangible bits of reality. Just ask Charles Kane.

The Electric Slide Rule

Revolutions often have humble origins, and it was no different with University of Pennsylvania physicist Kane. He has a thing for electrons and spends his days tinkering with computer simulations that model the flow of these negatively charged particles through different materials. It’s hardly glamorous work. But in 2004, Kane noticed something unexpected while futzing with one of his simulations, and it could one day win him a Nobel Prize.

Scientists had long understood why electricity flows through some materials but not others. When electrons move, they travel through a landscape. At the small scales of quantum physics, the branch of physics that describes reality at its most minute, electrons glide across hills and valleys of pure energy, shaped by the atoms that make up a material.

William Zuback

Materials that don’t conduct electricity, called insulators, have deep valleys. Electrons struggle to escape from those pits and stay stuck to their atoms, like billiard balls in the pockets of a pool table.

These electrons form a quantum state that reflects their inability to flow. The quantum state of one insulator, like plastic, can be mathematically molded into that of another insulator, like glass. They’re different only in superficial ways, like Euler’s box and pyramid. In other words, insulators are usually identical, topologically speaking. But not always.

A little over a decade ago, Kane noticed something strange in his simulations: an insulator whose quantum state had the equivalent of a hole. This made it different from other insulators. It was the counterpart of a doughnut, impossible to knead into a bread roll without closing the hole, a deformation this branch of math prohibits. Kane had discovered the first topological insulator.

“It took a long time for people to ask if there is an analogue of the doughnut for electrons in an insulator,” Kane says. “We found one and found that there is a consequence of having this hole: Something that is usually impossible becomes possible.”

Because of the hole in its quantum state, this theoretical material could conduct electricity on its surface (acting like metal there) but not within its interior (where it acted more like plastic). No known material had ever behaved this way.

Once physicists realized they could actually make this simulated stuff in the lab, it set off a race. “The challenge is to find new materials that do not exist in nature,” says M. Zahid Hasan, a physicist at Princeton University who led the team that cooked up the first real topological insulator, in 2007. “To do this, people are trying to rewrite the way we think of physics in topological terms.”

Technically speaking, a team of physicists in California and Germany beat Hasan to the punch with a flat, 2D topological insulator. And decades before, in the ’80s, physicist David Thouless had used topology to explain some odd behavior exhibited by very thin semiconductors in magnetic fields. (Called the quantum Hall effect, this earned Thouless a Nobel Prize in 2016.)

William Zuback

But physicists considered the quantum Hall effect to be a peculiar situation, a one-off, says Allan MacDonald, a physicist at the University of Texas at Austin who studies the effect. Kane’s doughnut discovery extended Thouless’ work dramatically. “He showed that topology could apply not just in the very special circumstance, but very generally,” says MacDonald. That allowed Hasan and his team to make the first 3D topological insulator.

This has tangible benefits as well. Topological insulators have another unusual property. Guided by topology, which downplays details, electrons flowing through these materials don’t mind bumping into imperfections or defects. They don’t tend to lose energy and give off heat, as an electron flowing through a wire would. That means electronics made of these materials could, in theory, consume less power and become much more efficient.

And that’s just one of the technologies that topological materials could energize.

Mathematica ex Machina

If Chetan Nayak ever builds the computer of his dreams — one inspired by topology — he could become the world’s most dangerous hacker, stealing credit card numbers with ease. Or, if he used his powers for good, he could create a search engine light-years ahead of Google, help chemists design new drugs and aid physicists in understanding the building blocks of the universe.

These are some of the long-standing promises of quantum computing, which seeks to store bits of information not as 1s and 0s, as is the case in conventional computers, but in weird quantum states that can be partly 1 and partly 0 at the same time.

But you don’t need to worry about Nayak; he’s a researcher, not a hacker. And despite decades of work, quantum computers have yet to live up to their potential. Google has created a chip that has 72 quantum bits; IBM’s best effort sports 50; and Intel has a 49-qubit device. None of these machines can do anything more than the $200 chip in your laptop, packed with billions of transistors. Quantum devices are still puny, and significant barriers remain to supersizing them.

The problem is that these futuristic devices store information in fragile quantum states of individual subatomic particles. And these states are notoriously fickle; the slightest disturbance or defect can easily corrupt their information. This limits the computing power of quantum devices, which must spend most of their resources on correcting errors caused by contact with the outside world.

Nayak is betting topology can solve this problem by protecting quantum information from the outside world. “On paper, topology can be exploited,” he says. His employer, Microsoft, agrees and has set up a facility called Station Q at the University of California, Santa Barbara, dedicated to building a new kind of quantum computer. Step inside, and you’ll find chalkboards covered in the equations of topology, written by physicists and mathematicians just steps away from the Pacific Ocean.

Despite ramping up its efforts in the past year, Station Q has yet to make a chip that draws on the special benefits of topology. “The experiments aren’t quite there,” Nayak says. But he’s confident in the approach.

Miniature Universes

Thanks to the unusual mathematics that governs their behavior, topological materials effectively house miniature universes that play by rules different from the outside world. Particles that do not exist in nature can appear in strange forms inside these materials.

Consider the Weyl fermion. Nearly 90 years ago, while playing around with the equations of quantum physics, as one does, German physicist Hermann Weyl showed that this massless and charged particle could, in theory, exist. But it has never shown itself among the elementary particles that make up the universe or appeared in experiments searching for new particles by smashing other particles together.

William Zuback

But something that behaves like the Weyl fermion did show up in 2015, in a topological material made of exotic elements known as a Weyl semimetal. The researchers studying this stuff (including Hasan) found a sort of mathematical hole in the quantum state of its electrons, something akin to Kane’s topological insulators. This hole causes electrons to come together in groups and behave like a single particle, like a flock of birds collectively forming a shape in the sky. As a group, the electrons behave like a massless Weyl fermion.

Physicists hope that exploring the behavior of Weyl fermions could lead to the discovery of new quantum phenomena — maybe even new kinds of matter. And because it has no mass, this so-called quasiparticle can move through a material faster than everyday electrical currents, a potentially useful trick for new kinds of quantum electronics and lasers.

Another quasiparticle could help Microsoft get closer to its quantum computer, along with an even newer topological material: the topological superconductor. Regular superconductors conduct electricity without any resistance; powerful magnets, like those in MRI machines, depend on superconductors. A topological superconductor is even weirder. This material should house something resembling a particle called a Majorana fermion, predicted in 1937 by an Italian physicist playing with the same quantum math that had intrigued Weyl. Like a Weyl fermion, a Majorana fermion has no mass. It also has no charge, despite being made of a bunch of negatively charged electrons. And whereas other particles have an antimatter twin — identical in mass but opposite in charge — Majorana fermions are, technically, their own antiparticle. They’re weird.

As a quasiparticle in a topological superconductor, the Majorana would have superpowers useful for quantum computing. The electrons responsible for its existence could essentially split in two, like birds splitting their flock. Because of the symmetries of topology governing the materials, the split particle is protected; even if you tamper with one of the fermion’s halves, it won’t affect the information that’s shared between them, which is preserved. So a quantum computer that exploited this behavior would not have to deal with the errors that plague today’s devices. In theory.

There’s only one problem. No one has ever made a topological superconductor, at least not for sure. In 2012, researchers in the Netherlands found what could be a sign of Majorana’s ghost, in tiny wires. The evidence has continued to accumulate, but the jury’s still out.

“There has been a great debate as to whether this can be explained by other phenomena,” says Hao Zhang, a postdoc with the team at Delft University of Technology, which is working with Microsoft. “I think our work has convinced most of the community.”

We’ll have to wait and see.

The Shape of Things to Come

From high-tech topological superconductors to handmade Klein bottles, Euler never could have guessed at the repercussions of the Königsberg Bridge Problem. Despite decades of finding useful applications, researchers today still have only a hazy understanding of where topology might lead them.

“History shows us that mathematics often gets applied in ways you wouldn’t think about,” says Freed, the mathematician at UT. “We’re just beginning to apply powerful tools and techniques of topology that were developed decades ago.”

The world is growing ever more complex. And topology — with its emphasis on simplicity — may be the secret to understanding it.

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