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- Is Particle Physics Dead, Dying, or Just Hard?

QUALIA | VIEW ON WEBSITE

Is Particle Physics Dead, Dying, or Just Hard?

By Natalie Wolchover

January 26, 2026

Columnist Natalie Wolchover checks in with particle physicists more than a decade after the field entered a profound crisis.

The Large Hadron Collider hasn’t found any new physics. Now what?

Kristina Armitage/Quanta Magazine

Introduction

In July 2012, physicists at the Large Hadron Collider (LHC) in Europe triumphantly announced the discovery of the Higgs boson, the long-sought linchpin of the subatomic world. Interacting with Higgs bosons imbues other elementary particles with mass, making them slow down enough to assemble into atoms, which then clump together to make everything else.

A couple of months later, I took a job as the first staff reporter at the nascent science magazine that would become Quanta. Turns out I was starting on the physics beat just as the drama was picking up.

The drama wasn’t about the Higgs particle; by the time it materialized at the LHC there was already little doubt about its existence. The Higgs was the last piece of the Standard Model of particle physics, the 1970s-era set of equations governing the 25 known elementary particles and their interactions.

More striking was what did not emerge from the data.

Physicists had spent billions of euros building the 27-kilometer supercollider not only to confirm the Standard Model but also to supersede it by uncovering components of a more complete theory of nature. The Standard Model doesn’t include particles that could comprise dark matter, for instance. It doesn’t explain why matter dominates over antimatter in the universe, or why the Big Bang happened in the first place. Then there’s the inexplicably enormous disparity between the Higgs boson’s mass (which sets the physical scale of atoms) and the far higher mass-energy scale associated with quantum gravity, known as the Planck scale. The chasm between physical scales — atoms are vastly larger than the Planck scale — seems unstable and unnatural. In 1981, the great theorist Edward Witten thought of a solution for this “hierarchy problem”: Balance would be restored by the existence of additional elementary particles only slightly heavier than the Higgs boson. The LHC’s collisions should have been energetic enough to conjure them.

But when protons raced both ways around the tunnel and crashed head-on, spraying debris into surrounding detectors, only the 25 particles of the Standard Model were observed. Nothing else showed up.

In philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted. Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it. In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.

The absence of any “new physics” — particles or forces beyond the known ones — fomented a crisis. “Of course, it is disappointing,” the particle physicist Mikhail Shifman told me that fall of 2012. “We’re not gods. We’re not prophets. In the absence of some guidance from experimental data, how do you guess something about nature?”

Once the standard reasoning about the hierarchy problem had been shown to be wrong, there was no telling where new physics might be found. It could easily lie beyond the reach of experiments. The particle physicist Adam Falkowski predicted to me at the time that, without a way to search for heavier particles, the field would undergo a slow decay: “The number of jobs in particle physics will steadily decrease, and particle physicists will die out naturally.”

The crisis and its fallout made for years of interesting reporting, but sure enough, the frequency of news stories related to particle physics diminished. I fell out of touch with sources. More than 13 years on, in this first column for Qualia, a new series of essays in Quanta Magazine, I’m taking stock. Is particle physics dying, as Falkowski predicted? Can new physics still be found? What’s the future for particle physicists? Will artificial intelligence help? How much hope is left in the search for answers to the many remaining mysteries of the universe?

Some particle physicists act as if there’s no crisis at all. The LHC is still running and will for at least another decade, and its operators are finding new sources of enthusiasm.

In the last couple of years, data handling at the collider has improved with the use of AI. Pattern recognizers can sort through the outgoing debris of proton collisions and classify collision events more accurately than human-made algorithms can. This helps the physicists to more accurately measure the “scattering amplitude,” essentially the probability that different particle interactions will occur. For instance, AI systems can determine more precisely how many top quarks arise in the aftermath of collisions versus the number of bottom quarks. Any statistical deviations from the predictions of the Standard Model could signify the involvement of unknown elementary particles.

A proton-proton collision documented by the Compact Muon Solenoid at CERN in 2012 shows evidence of the decay of the Higgs boson.

CMS Collaboration; Mc Cauley, Thomas

Novel particles as hefty as Higgs bosons would not be so subtle; they would have shown up already as pronounced bumps on data plots. But as Matt Strassler, a particle physicist affiliated with Harvard University, explained to me, the traces of lighter novel particles could still lie in so-called hidden valleys in the data. “There’s a huge amount of unexplored territory there,” he said. There might exist, for instance, an unstable type of dark matter particle that leaves its mark by occasionally arising and immediately decaying into an excessive number of muon-antimuon pairs. Detecting such an excess would point indirectly to the unstable particle’s existence. “For people who thought all the new physics is at high energies — they’re very disappointed right now,” Strassler said. “I don’t share that view. There are many opportunities for nature to provide clues at low energies.”

So far, though, no such indirect evidence of new physics has been detected. The more accurate the statistics have become at the LHC, the better they match the Standard Model. Michelangelo Mangano, a particle physicist at CERN, the laboratory that houses the LHC, said the collider today is like a tool for exploring the Standard Model’s predictions, and he considers this exploration worthwhile because not all consequences of the equations are easy to calculate. The search for new physics beyond the Standard Model is ongoing, Mangano said, but “the fact that it’s not giving positive results does not mean we are stuck, dead, or wasting our time.”

These questions are so fundamental that of course it’s worth nailing down every amplitude and checking every hidden valley, since we have the tool for the job. But for hunters of new physics, does the game end there?

The community wants to go bigger. CERN physicists want to build a Future Circular Collider, tripling the circumference of the LHC with a 91-kilometer tunnel beneath the Franco-Swiss border, to both probe higher energies and look for subtler signals. This FCC would initially collide electrons, which, unlike protons, are themselves elementary particles, with no substructure. Their clean collisions would allow more precise measurements of scattering amplitudes, making the FCC ultrasensitive to indirect signs of new physics. By the end of the century, the mega-collider would be upgraded to collide protons, as the LHC does now. Proton collisions are messier, but at the FCC they would achieve unprecedented energies — about seven times higher than the LHC can currently muster — so they have a chance, however slim, of revealing heavy particles beyond the LHC’s reach. (In theory, particle masses could range up to a million billion times greater than what the LHC energy scale can produce directly, so there’s no reason to expect them around the next bend.)

We’re not gods. We’re not prophets. In the absence of some guidance from experimental data, how do you guess something about nature?

Mikhail Shifman

As of now, the FCC’s fate is unknown; formal approval and funding commitments by member countries won’t come before 2028.

Meanwhile, U.S. particle physicists are aiming to complement the European strategy by constructing a brand-new type of machine: a muon collider. Muons are elementary like electrons, but they’re 200 times heavier, so their collisions would be both clean and energetic (albeit not reaching the collision energies of the LHC). Both the selling point and the challenge of this newfangled type of machine is that it will require major technical innovations (with all the spin-off potential that can bring), because muons are highly unstable. They must be accelerated and collided mere microseconds after they’re created.

Demonstrating the technology and then constructing the collider would take roughly 30 years, and that’s with federal funding. “We have to figure out how to do it in between 10 and 20 billion [dollars],” said Maria Spiropulu, a physics professor at the California Institute of Technology and co-chair of the committee behind a national report endorsing a muon collider program that came out in June 2025. Over the coming years, the Department of Energy will weigh whether to fund the proposal rather than competing science projects. What hurts its case is the lack of a “discovery guarantee,” which the LHC had with the Higgs boson.

Scientists and technicians inspected and upgraded systems at the Large Hadron Collider during the Long Shutdown 2, which began in 2018.

Maximilien Brice/CERN

Then again, as the mathematical physicist Peter Woit mused on his blog, “Perhaps in our new world order where everything is controlled by trillionaire tech bros, the financing won’t be a problem.”

Deliberations about a Chinese supercollider have come to naught, I’m told. Instead, China has decided to pursue a “super-tau-charm facility”: a lower-energy particle scattering experiment that would cost mere hundreds of millions of dollars instead of tens of billions. The facility will produce a lot of tau particles and charm quarks, partly to study whether taus ever shape-shift into muons or electrons. This kind of switching isn’t predicted by the Standard Model, but it does happen in some theoretical extensions of it.

Okay, we might as well check. We’re desperate for new physics, and the price is good. But by definition it’s very difficult to know which shots in the dark are worth taking.

Adam Falkowski, who sounded the death knell for particle physics back in 2012, used to be known for the sharp commentary he supplied on his blog Résonaances. But the Paris-based particle physicist hasn’t posted anything since 2022. He said that’s partly because he’s been tied up with fatherhood and partly because there hasn’t been much to say.

When we caught up on a video call, Falkowski told me, “I am very skeptical about future colliders. For me it’s very difficult to get excited about it.” He sees momentum behind CERN’s FCC campaign, but personally he worries about the huge costs and timescales, and the fact that “there are absolutely no hints that something is there within the reach of the next collider.”

For his part, Falkowski has turned to the theoretical study of scattering amplitudes, a growing research area focused on the geometric patterns underlying particle interaction statistics, patterns that could point toward a truer perspective on the quantum world. The field seeks to reformulate the equations of particle physics in a different mathematical language in hopes that this language might extend to quantum gravity. “There is a very vibrant program in trying to understand the structure of the physical theories,” Falkowski said. “The hope is that with the help of machine learning, that there can be very fast progress in the coming years. I think that’s where the best things have happened.”

But amplitudeology, as this field is known, is abstract — it’s no atom-smashing experiment. Falkowski said he does think experimental particle physics is dying. He has watched talented postdocs switch to other research areas or take data science jobs. “I’m not sure they are getting the best of the best as they used to,” he said, “because the prospects of returns are so distant. If you want to change the world now, you will do AI; you will do something different from particle physics.”

The ALICE (A Large Ion Collider Experiment) detector at the Large Hadron Collider was designed to study quark-gluon plasma.

CERN, Julien Marius Ordan/Science Source

This brain drain appears to be real. I spoke to Jared Kaplan, co-founder of Anthropic, the company behind the chatbot Claude. He was a physicist the last time we spoke. As a grad student at Harvard in the 2000s, he worked with the renowned theorist Nima Arkani-Hamed to open up the new directions in amplitude research that are being actively pursued today. But Kaplan left the field in 2019. “I started working on AI because it seemed plausible to me that … AI was going to make progress faster than almost any field in science historically,” he said. AI would be “the most important thing to happen while we’re alive, maybe one of the most important things to happen in the history of science. And so it seemed obvious that I should work on it.”

As for the future of particle physics, AI makes worrying about it now rather pointless, in Kaplan’s view. “I think that it’s kind of irrelevant what we plan on a 10-year timescale, because if we’re building a collider in 10 years, AI will be building the collider; humans won’t be building it. I would give like a 50% chance that in two or three years, theoretical physicists will mostly be replaced with AI. Brilliant people like Nima Arkani-Hamed or Ed Witten, AI will be generating papers that are as good as their papers pretty autonomously. … So planning beyond this couple-year timescale isn’t really something I think about very much.”

Cari Cesarotti, a postdoctoral fellow in the theory group at CERN, is skeptical about that future. She notices chatbots’ mistakes, and how they’ve become too much of a crutch for physics students. “AI is making people worse at physics,” she said. “What we need is humans to read textbooks and sit down and think of new solutions to the hierarchy problem.”

Cesarotti was a high school junior when the Higgs boson was discovered. She grew up near Fermilab, the U.S. national lab in Illinois that houses the Tevatron, which was the world’s highest-energy particle collider before the LHC. (The top quark was discovered there in 1995.) This proximity taught her that a particle physicist was a thing you could be. Later, it turned out to be her thing. “What are the fundamental building blocks of the universe — those were the questions that I was most interested in knowing the answer to,” she told me. “But what people said was, ‘Particle physics is dead. Don’t do this.’”

It may have been a fair warning; Cesarotti has yet to land a permanent job as a rising particle physicist. The subfield has continued to shrink, she and others said, as faculty hiring committees and grad students go in other directions. “Definitely all this rhetoric that there was nothing to be found and you should give up on it — people listened,” she said. “And of course that means there are fewer people. It becomes a self-fulfilling prophecy. If you’re pushing all these talented people out of trying to solve these problems into a field that it’s easier to make an impact on, then you’re setting yourself up for failure.”

Cesarotti echoed a sentiment I’d heard from others, which sounds correct to me as well: “Particle physics isn’t dead; it’s just hard.” It’s hard to know what to think about or look for. But the most devoted particle physicists are thinking and looking all the same.

Related:

1. Particle Physicists Detect ‘Magic’ at the Large Hadron Collider

2. How the Higgs Field (Actually) Gives Mass to Elementary Particles

3. Dogged Dark Matter Hunters Find New Hiding Places to Check

4. Inside the Proton, the ‘Most Complicated Thing You Could Possibly Imagine’

5. Contemplating the End of Physics

6. What Is a Particle?

7. A New Map of All the Particles and Forces

8. Axions Would Solve Another Major Problem in Physics

“It was easy for 125 years,” Strassler said. “One thing led to the next. That lucky century has, for now, at least in the medium term, come to an end. That could change tomorrow, or next century, or who knows.”

A hint of a new lightweight particle could, in theory, show up at the LHC, or in some other experiment. Strassler is particularly excited about the study of radioactive thorium-229 decay, which could reveal variations in the fundamental constants. I’m slightly partial to experiments looking for “axions,” dark matter candidates that are so lightweight that they can act a little like light itself.

On the theory side, an obvious solution to the hierarchy problem could drop naturally out of the geometry behind scattering amplitudes. Or, if Kaplan is right, AI systems might someday suggest powerful new ideas for how the 25 particles of the Standard Model fit into a more comprehensive pattern — a possibility I didn’t foresee back when the crisis began.

Clearly, further progress toward the truth remains possible in particle physics. But there’s no discovery guarantee. I’ve had more than 13 years to think about it, and it remains a disturbing prospect: All the empirical clues we can glean about nature’s fundamental laws and building blocks might already be in hand. The universe may plan on keeping the rest of its secrets.

2026 Jan 23

- Monster Neutrino Could Be a Messenger of Ancient Black Holes

ASTROPHYSICS | VIEW ON WEBSITE

Monster Neutrino Could Be a Messenger of Ancient Black Holes

By Jonathan O'Callaghan

January 23, 2026

Primordial black holes could rewrite our understanding of dark matter and the early universe. A record-breaking detection at the bottom of the Mediterranean Sea has some physicists wondering if we just spotted one.

Strings of optical modules like this make up the underwater KM3NET neutrino detector.

Courtesy of KM3NET

Introduction

Nearly three years ago, a particle from space slammed into the Mediterranean Sea and lit up the partially complete Cubic Kilometer Neutrino Telescope (KM3NET) detector off the coast of Sicily. The particle was a neutrino, a fundamental component of matter commonly known for its ability to slip through other matter unnoticed.

The IceCube observatory in Antarctica, a comparable detector that has been running for more than a decade, has found hundreds of cosmic neutrinos — but none quite like this one. Some 35 times more energetic than any neutrino seen before, the particle might have shot out from a highly active galaxy — a blazar — or a background source of cosmogenic high-energy particles that scientists suspect pervade the cosmos.

But those aren’t the only possibilities. The day after the KM3NET collaboration announced the detection, the physicist David Kaiser walked into a room full of his colleagues at the Massachusetts Institute of Technology with a bold proposition: What if the monster neutrino came from an exploding primordial black hole?

Such black holes “could form before there were even atoms, let alone stars,” said Kaiser, who has been heavily involved in the hunt for these hypothetical objects.

The idea that the neutrino came from a primordial black hole is a long shot; Kaiser said he was “half-joking” when he suggested it. But in the absence of a definitive explanation, it remains intriguing, not least because the existence of primordial black holes could mean they play a role in dark matter.

So the question is, did we just spot one?

In a Split Second

The idea of primordial black holes was first proposed in 1966 by the Soviet physicists Yakov Zel’dovich and Igor Novikov, and it was cemented by the British astrophysicist Stephen Hawking in 1971. Hawking and his student Bernard Carr, of Queen Mary University of London in the U.K., then worked out the concept of primordial black holes in detail in 1974.

A primordial black hole, or PBH, is loosely defined as a black hole that formed in the first split second of the universe. The hypothesis goes that during the rapid expansion of space, there might have been spikes in the density of space-time that were so high that they would have collapsed into black holes. These black holes would have spanned a range of masses, depending on the size of the spikes. Some could have been as small as an atomic nucleus.

David Kaiser, a physicist at the Massachusetts Institute of Technology, is searching for evidence of primordial black holes.

Allegra Boverman

The idea has its problems. Incorporating PBHs into the early universe requires physicists to adjust the parameters of their models very precisely to fit with observations, said Wenzer Qin, a theoretical physicist at New York University. “You have to tune the knob just right,” she said. “You need the density spikes to be 10,000 times larger than you would have predicted in standard cosmological theory.”

The same year Hawking and Carr explained the concept of primordial black holes, Hawking also put forward his idea of Hawking radiation, a process in which black holes might lose mass and energy by emitting photons and other fundamental particles through the interaction of quantum physics and gravity at the black hole’s edge.

Giving off Hawking radiation, a primordial black hole that was originally the size of an atomic nucleus would meet its doom in the modern universe, slowly dwindling before ending in a sudden, extreme burst of particles. “Very little mass gets radiated over the majority of the black hole’s lifetime,” said Alexandra Klipfel, a doctoral student working with Kaiser on PBHs at MIT. “But then, right at the end, it emits a majority of its mass in a very rapid explosion. It heats up really, really quickly, a runaway process that ends in a big explosion of ultra-high-energy particles.”

That process occurs “because the temperature of the black hole is inversely proportional to the mass,” Klipfel said. “The lighter the mass, the hotter the temperature.”

Because the energy released in Hawking radiation doesn’t favor one type of particle over another, the final burst would include all 17 fundamental particles in the Standard Model, our benchmark model explaining the cosmos. In that moment, as the black hole was extinguished, trillions upon trillions of particles would explode into space, “including neutrinos and quarks and all kinds of exotic things,” Kaiser said.

Alexandra Klipfel co-wrote a paper about the possible origins of the powerful neutrino.

Josu Aurrekoetxea

The tiniest primordial black holes would have lasted only moments, but more massive ones could still be around. “A black hole with a mass of about 10^14 [100 trillion] grams has a lifetime equal to the age of the universe,” Klipfel said.

Scientists have ruled out (or constrained) the possible masses of PBHs that could be hiding in the universe today. Too small and the PBHs would have evaporated already. Too big and their gravitational effects would have been spotted warping the light coming to Earth from distant stars and galaxies. The best window left for most PBHs, it seems, ranges from about 100 quadrillion (10^17) grams — the mass of an average asteroid — to 100 sextillion (10^23) grams, the mass of a moon. A smaller subpopulation of PBHs of 100 trillion (10^14) grams or lower, the mass of a small asteroid, would be in their final stages of evaporation.

If primordial black holes do exist in the asteroid-mass window, not only can they tell us about the conditions at the dawn of the cosmos, but they might also answer another open question in astrophysics. If they still exist in the present universe, they might constitute some or all of the missing mass we can infer in the rotation of galaxies and the structure of the cosmos that scientists more commonly link to undetected particles of dark matter.

“They’re one of the few good theories for what dark matter could be,” Qin said. “So it’s important to keep looking for them.”

Monster Moment

To the surprise of Kaiser’s team, the idea that the neutrino that crashed into the KM3NET detector originated in the explosion of a primordial black hole worked mathematically, and on September 18, 2025, Kaiser and Klipfel published a paper in Physical Review Letters explaining the mechanism. They found that, if a primordial black hole with the initial mass of a small asteroid exploded about 2,000 astronomical units away — 2,000 times the distance between the Earth and the sun — it could have produced the powerful neutrino.

“We found about an 8% chance of this happening,” Klipfel said. “It’s a low-probability event. But it’s not a completely impossible event.”

This visualization shows a particle passing through the KM3NET detector and activating a series of optical modules.

Courtesy of KM3NET

The PBH, at the lower 100-trillion-gram end of the asteroid-mass window, would have steadily emitted Hawking radiation over the 13.8-billion-year lifespan of the universe until its explosive finale.

Had the PBH been much closer, we likely would have seen its flash in gamma rays or other radiation that would have signaled its final explosion. If it had been too far away, the neutrinos would have been too spread out for one to hit Earth.

Models in which PBHs account for most or all of the dark matter also predict that enough of the smaller PBHs would have the correct mass for one to fly past our solar system right as it exploded, producing a burst of energetic particles, including the neutrino that hit our planet in just the right spot to trigger the KM3NET detector.

Carr, who solidified the concept of PBHs with Hawking, finds the idea enticing and is hopeful that KM3NET’s spectacular neutrino is a sign that we might discover PBHs after all. “I’ve been working on primordial black holes for 50 years, and there was no purported evidence for them, just constraints, which is really sad,” Carr said. “This could be evidence for black hole explosions.”

Cosmic Detection

Not everyone is convinced. “I don’t know where this KM3NET neutrino comes from, but I would bet an awful lot of money that it has nothing to do with primordial black holes,” said Dan Hooper, a cosmologist at the University of Wisconsin, Madison. He argues that if such exploding black holes existed, they would be easy to see, “and we don’t see those,” he said. “I think you can completely rule out that hypothesis.”

Hooper is not completely opposed to the idea of PBHs themselves, however, and he has worked to understand how they might have formed during the period of cosmic inflation. “There are lots of ways they could have been made in the early universe,” he said.

Workers prepare to deploy a string of optical sensors, which will be attached to the seafloor and unspooled from the spherical frame that contains them.

Courtesy of KM3NET

Others, like Evan McDonough, a theoretical physicist from the University of Winnipeg, are more confident that PBHs exist today. “My personal hunch is that there is probably at least one PBH out there,” he said. “Whether or not they exist in some sizable amount is the big question.”

Priyamvada Natarajan, a theoretical astrophysicist from Yale University, says that although she thinks PBHs cannot account for more than a tiny fraction of dark matter, she is interested in them as a tool to explore ideas for dark matter outside its two most popular candidates, weakly interacting massive particles (WIMPs) and the more wavelike axions. “PBHs are really forcing us to rethink the dark matter problem and move away from our fixation with WIMPs and axions,” she said. “In that sense they’re scientifically really important. They’ve allowed us to break out of a mindset and be more open.”

If primordial black holes exist, there are several ways we might find them. For example, we might spot a signal in gravitational waves from the merger of PBHs smaller than the mass of a star but still huge enough to produce detectable gravitational waves in our instruments. Black holes of such mass would have needed to form through primordial means.

Andrea Thamm of the University of Massachusetts, Amherst and her colleagues, meanwhile, proposed in September that exploding black holes might be more common than theorists thought and could be spotted in the near future, if we invoke some exotic physics.

In their model, they suggest that particles known as dark photons and dark electrons — dark matter variants of normal matter — could have reduced the rate at which lower-mass PBHs in the universe emitted Hawking radiation. That could mean many more PBHs are in the final stages of evaporation today. If that’s so, Thamm and colleagues suggest that within certain constraints, there is a more than 90% chance of spotting the evaporation of a primordial black hole in the next 10 years.

A “backbone cable” connects multiple optical modules in the KM3NET detector.

Courtesy of KM3NET

“Within a fairly short time window [of the final explosion], around 1,000 seconds, there would be very highly energetic photons [emitted],” said Thamm, who said existing experiments like the High-Altitude Water Cherenkov observatory in Mexico could look for such events. “And then it would very suddenly stop once the primordial black hole has exploded.”

Another novel idea, proposed by Kaiser, is that if PBHs do indeed fall into the asteroid-mass range and constitute most or all of dark matter, they should occasionally fly through our solar system at hundreds of kilometers a second. In this scenario, they might produce noticeable gravitational effects.

Using spacecraft orbiting Mars, Kaiser said, we could precisely calculate the distance from Earth to the red planet and look for any wobbles that might indicate a PBH flying past. At any given time, he said, there would be at least one PBH in the solar system, possibly producing detectable Hawking radiation. Every three to 10 years, one would get close enough to Mars to produce a tiny but measurable change in the planet’s motion.

In the tens of centimeters, “Mars will begin rocking away from its otherwise well-tracked orbit,” Kaiser said. He is planning to work on the idea with a team of astronomers over the next couple of years.

Related:

1. Dogged Dark Matter Hunters Find New Hiding Places to Check

2. A New Map of the Universe, Painted With Cosmic Neutrinos

3. Cosmic Map of Ultrahigh-Energy Particles Points to Long-Hidden Treasures

Alongside that, Kaiser wants to keep an eye out for other high-energy neutrinos to test the idea that some might come from exploding black holes. Proving that this is true is difficult because PBH neutrinos would look just like any other neutrinos produced from another source, unless they could be associated with a flash of gamma rays in the sky from an exploding PBH.

There is still a long way to go before we can say whether primordial black holes exist, let alone whether they make up dark matter. But for now, scientists cannot rule out the possibility that one slipped past Earth nearly three years ago, sending a lone emissary into an underwater neutrino detector.

“The numbers are outrageously congruent, considering I walked in half-joking,” Kaiser said.

Editor’s note: Priyamvada Natarajan is a member of Quanta Magazine’s advisory board.

2026 Jan 21

- How Animals Build a Sense of Direction

NEUROSCIENCE | VIEW ON WEBSITE

How Animals Build a Sense of Direction

By Yasemin Saplakoglu

January 21, 2026

Researchers documented the activity of neurons that shape directional navigation as bats explored a remote island off the coast of Tanzania.

Researchers brought Egyptian fruit bats to a remote island to study how a network of cells in the mammalian brain constructs a directional sense in the wild.

Nachum Ulanovsky

Introduction

On a remote island in the Indian Ocean, six closely watched bats took to the star-draped skies. As they flew across the seven-acre speck of land, devices implanted in their brains pinged data back to a group of sleepy-eyed neuroscientists monitoring them from below. The researchers were working to understand how these flying mammals, who have brains not unlike our own, develop a sense of direction while navigating a new environment.

The research, published in Science, reported that the bats used a network of brain cells that informed their sense of direction around the island. Their “internal compass” was tuned by neither the Earth’s magnetic field nor the stars in the sky, but rather by landmarks that informed a mental map of the animal’s environment.

These first-ever wild experiments in mammalian mapmaking confirm decades of lab results and support one of two competing theories about how an internal neural compass anchors itself to the environment.

“Now we’re understanding a basic principle about how the mammalian brain works” under natural, real-world conditions, said the behavioral neuroscientist Paul Dudchenko, who studies spatial navigation at the University of Stirling in the United Kingdom and was not involved in the study. “It will be a paper people will be talking about for 50 years.”

Follow-up experiments that haven’t yet been published show that other cells critical to navigation encode much more information in the wild than they do in the lab, emphasizing the need to test neurobiological theories in the real world.

Neuroscientists believe that a similar internal compass, composed of neurons known as “head direction cells,” might also exist in the human brain — though they haven’t yet been located. If they are someday found, the mechanism could shed light on common sensations such as getting “turned around” and quickly reorienting oneself. It might even explain why some of us are so bad at finding our way.

A Sense of Direction

How the mammalian brain navigates the environment has been a source of fascination for scientists for at least half a century. Its study has led to the discovery of “extremely interesting phenomena, several of which have won Nobel Prizes,” said Nanthia Suthana, a neuroscientist at Duke University.

In the early 1970s, John O’Keefe, a neuroscientist at University College London, discovered cells in the rat hippocampus, the brain’s memory hub, that responded to specific locations in the rodents’ enclosures. He called them “place cells.” A few decades later, May-Britt Moser and Edvard Moser of the Norwegian University of Science and Technology discovered, in a nearby brain area, cells that create a coordinate system for the brain, which they called “grid cells.” The three researchers were awarded a Nobel Prize for their discoveries.

Neuroscientists camped on a seven-acre island in the Indian Ocean to study how bats orient themselves in a large, wild environment. Though the island is small, the bats can’t see from one end to the other.

Nachum Ulanovsky

Together, these two cell types can create a map of an animal’s surroundings. But knowing where you are in space isn’t enough to get you somewhere else. “You also need to know what direction you’re facing,” said Jeffrey Taube, a neuroscientist at Dartmouth College. “You need those two key pieces of information. One without the other doesn’t do you much good.”

In 1984, Jim Ranck, a neuroscientist at the State University of New York Downstate in Brooklyn, New York, was investigating what happens when information from place cells leaves the hippocampus when he accidentally discovered what became known as head direction cells. These cells didn’t seem to care where the animal was located; instead, they responded to the direction the animal was facing. “It was a very serendipitous but obviously wonderful finding,” said Taube, who did his postdoctoral work under Ranck.

In the years since, neuroscientists have characterized how head direction cells work in rodents. The neurons receive inputs from the external world, through the things we see, hear, and touch, and also from the internal world, especially from the vestibular system, a network in the inner ear that tracks head movements. It’s thought that as an animal moves around, it keeps track of its movement relative to the landmarks around it, learns to associate certain landmarks with certain directions, and uses this information to constantly update its mental map. Neuroscientists have come to call this system the head direction circuit, or internal compass.

“It’s not a compass in a magnetic sense, but it is a compass in an absolute sense,” Dudchenko said. “What does a compass do? It keeps orientation relative to where you are, or where you’re standing, or what environment you are in.”

Mark Belan/Quanta Magazine

These head direction cells are connected in a ringlike system called a ring attractor network. In mammals, this network is not a physical ring (though it is, strangely, in fruit flies), but it can be schematically represented as such. The ring is always active. When an animal faces a particular direction, certain cells in the ring fire. When the animal turns, those cells turn off and others activate in a continuous fashion.

“As the animal keeps turning its head 360 degrees, a sequence of different cells will fire, each of them tuned to a specific direction,” said James Knierim, a neuroscientist at Johns Hopkins University who was not involved in the new research. (He co-authored an accompanying perspective on the paper for Science.)

The big question, Knierim said, was whether these cells would remain faithful to their assigned directions, as a magnetic compass does, in the real world, where animals live in large territories. Previous work had generated two competing theories. The “global compass” hypothesis claims that each head direction cell commits to a direction during continuous navigation through a large environment: A cell that fires when an animal faces northeast will always fire for northeast. The “mosaic” hypothesis suggests that head direction cells reset and change their compass direction as an animal moves through different regions of a large environment, so that north-indicating cells in one region may represent east in another part.

All the research on this question had been done in small, enclosed spaces. To understand how the compass really works, the scientists needed to go outside.

A Natural Laboratory

Everything we know about what’s going on in the brains of mammals as they navigate their environments comes from lab experiments. But they give an incomplete view. In a small box on a lab bench, an animal sees “immediately everything there is to see,” said Nachum Ulanovsky, a behavioral systems neuroscientist at the Weizmann Institute of Science in Israel. “It’s not real navigation in the challenging sense, like you would navigate in a city.”

Nachum Ulanovsky calibrates GPS devices (here, mounted to a mop) to specific locations on Latham Island. The system tracks the brain activity of bats in flight as they build a neural map of the novel environment.

Courtesy of Nachum Ulanovsky

When walking around a city, on the other hand, we constantly integrate information about space and time, and from our own memories. We need a mental map, sure, but we also must deal with environmental interference: We need to avoid a cyclist, run across a street before the light turns red, and step over trash without slamming into other people. We need to know how to get from point A to point B, even if we’ve never been there before. And we need to know how vastly different environments — meandering sidewalks, a park with many trails, a fifth-floor apartment — connect to one another.

This kind of complex environment is hard to simulate in the lab. But studying the sense of direction outside the lab, in an uncontrolled setting, can be even harder. So, despite the excitement around the neural basis of navigation, “none of these neurons — neither place cells, nor grid cells, nor head direction cells — had been studied in the real world, outdoors,” Ulanovsky said. “So I had, for many years, this dream that we would like to do that. But for years, it stayed as a dream because how do you even approach this?”

In 2016, his team built a 200-meter-long tunnel at the Weizmann Institute and developed wireless systems to record the brain activity of Egyptian fruit bats as they flew through it. The team reported in Science that place cells behaved differently in the tunnels than they had in the lab — a hint that a more complex experimental environment would be key to really understanding mammalian navigation.

In 2016, frustrated by a small, confining lab, researchers at the Weizmann Institute in Israel built a 200-meter-long tunnel to study bat navigation in a semi-realistic setting.

Nachum Ulanovsky

But a tunnel was still too confined for Ulanovsky. He wanted to create conditions closer to the real world. The answer came to him in 2018 as he was scuba diving on the Great Barrier Reef in Australia. “Being on an island there, it hit me that that’s a solution,” he recalled. “Suppose I find an island somewhere in the world” to use as a wild laboratory.

He searched for an island far away from land (so his bats couldn’t escape and create ecological problems) that was not too big and not too small. It had to be uninhabited by people and mostly barren (so bats wouldn’t hide in tall trees), and it couldn’t be a nature reserve (to avoid permitting issues). “The conjunction of these things is pretty rare,” Ulanovsky said. His team homed in on 30 or 40 islands across the world that might work. Only one was in the home range of the Egyptian fruit bats they study: Latham Island, in the Indian Ocean 25 miles east of Tanzania.

Latham Island, a plot of land the size of about four soccer fields, was small enough for the researchers to contain and track the bats — and big enough to ensure that the bats couldn’t see from one end to the other.

Ulanovsky’s team was ready to watch as bats learned to navigate a complex habitat more like the one they evolved in. They implanted microwires, each a few micrometers thick, in the brains of six Egyptian fruit bats to record neural activity; the wires connected to a data logger, which stored the data. They brought the bats to the island on a boat, along with everything the scientists needed to sustain themselves for a few weeks, including tents, chairs, tables, generators, and refrigerators. They released the bats, usually at night, and tracked their positions as they flew across the island. At the end of every night, the researchers re-captured the bats to download data on the activity of head direction cells and other cells involved in navigation. By the end of the experiments, performed over two seasons in 2023 and 2024, the researchers had data from 301 flights.

Latham Island is a small, treeless plot of land that’s uninhabited aside from seabirds and, sometimes, a few fishermen. In the absence of light pollution, at night the researchers’ camp sat below “the most amazing stars that you can imagine,” Ulanovsky said.

Shaked Palgi and Orian Las

On the first couple of nights, as the bats began to explore Latham Island, their head direction cells fired crudely. Some fired when the bats faced generally south, others while they faced generally east, west or north. But by night five or six, as the place grew more familiar to them, the cells had stabilized to fire in coordination with precise directions and did not change depending on where the animal was on the island.

Because they could not see the entire island at once, their brains seemed to be stitching together small parts of the island into a global whole. The findings suggest that the global compass hypothesis is indeed correct, as some experiments have predicted. This makes sense, as “a compass should be a compass,” Dudchenko said. “If you move to the next room, it should still be pointing in the right direction.”

How did these cells anchor themselves to particular directions? They weren’t adjusting to celestial cues; the bats’ brain activity remained stable as the moon moved across the sky and when the moon and stars were covered by clouds. Nor were the head direction cells anchoring themselves to the Earth’s magnetic field, as some preliminary experiments by Ulanovsky’s team had suggested. The team hypothesizes that the bats anchored themselves to landmarks in their environment, such as the coastline, the experimenters’ tents, and their perches. As they got to know the new space, the landmarks became part of their internal maps and cued the head direction cells to fire.

The findings confirmed decades of lab work suggesting how this head direction cell system worked in smaller environments. “It was an open question, one way or the other, whether the cells behaved the same way in large, natural environments,” Knierim said. He and others applauded the study for recording the activity of these cells out in the wild, in a much bigger and more complex space than experiments could simulate. “In this area of neuroscience, there’s just nothing like that,” he said.

Beyond the Island

Already, this real-world approach is bearing fruit. In November 2025, at the Society for Neuroscience meeting in San Diego, Ulanovsky presented early data showing that the brain cells of bats navigating Latham Island encoded more information than they do inside the lab — for example, place cells not only recorded the bat’s location but also activated based on how fast the bat was going.

All bat species can see, Ulanovsky said, but the Egyptian fruit bat, his study species, has “excellent vision, much better than rats or mice.” The bats probably use a combination of vision and echolocation to learn the layout of the land and tune their inner compass.

Yuval Barkai

These preliminary findings make an “even better argument for doing natural experiments,” Dudchenko said. “They suggest a new approach to how we do neuroscience.” Instead of crafting experiments that control for complexity, neuroscientists should embrace it, he said.

As neuroscientists look beyond the lab, they’re also hoping to look beyond rats and bats. If you’ve spent any time navigating a city, you’ve surely employed your own head direction system. Knierim recalls walking in Manhattan; he thought he was heading east. “When I hit the corner, and I’m expecting to see Second Avenue, and I see Lexington Avenue [instead] — my whole head, you know, my own perception of the world just spun around,” he said. “I can literally feel it inside.” When he realized his internal map was misaligned, he could feel it twist around him as his mental space caught up with his physical one.

Not much is known about the neural basis of our own sense of direction. Head direction cells have not yet been located in humans, though there is some evidence that they exist. “We do have the same brain structure [as rodents and bats], so it’s not too crazy to think that those brain structures then have similar function,” Dudchenko said. Certainly, our experiences navigating our environments suggest that we have a sense of direction (some more than others).

Related:

1. Bats Use the Same Brain Cells to Map Physical and Social Worlds

2. How Animals Map 3D Spaces Surprises Brain Researchers

3. Shrinking Bat DNA and Elastic Genomes

The lack of human studies is a “major gap that we’re trying to fill,” said Suthana, the Duke neuroscientist. With consent from epilepsy patients, Suthana and her team connected a new device to electrodes already implanted into their brains for presurgical monitoring. Then she recorded navigation cells in humans exploring a seminatural environment — a hospital room and hallway — to collect data on how navigational cells track the body and head as a person moves. This was the first time such a study had been performed in human subjects.

“Moving into these wilder, naturalistic environments really has the ability for us to test things or find things we would never see in the lab,” she said. While 15 minutes wandering a hospital hallway isn’t exactly the wild, her team is working toward the goal of recording high-resolution brain activity in even more complex environments. “Maybe not on a remote island, but who knows?”

2026 Jan 20

- Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle

TOPOLOGY | VIEW ON WEBSITE

Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle

By Elise Cutts

January 20, 2026

The Bonnet problem asks when just a bit of information is enough to uniquely identify a whole surface.

For the first time, mathematicians have found an example of a compact doughnutlike surface (as seen above) that shares its local geometric information with another surface, despite having a completely different global structure.

Mark Belan/Quanta Magazine; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)

Introduction

Imagine if our skies were always filled with a thick layer of opaque clouds. With no way to see the stars, or to view our planet from above, would we have ever discovered that the Earth is round?

The answer is yes. By measuring particular distances and angles on the ground, we can determine that the Earth is a sphere and not, say, flat or doughnut-shaped — even without a satellite picture.

Mathematicians have found that this is often true of two-dimensional surfaces more generally: A relatively small amount of local information about the surface is all you need to figure out its overall form. The part uniquely defines the whole.

But in some exceptional cases, this limited local information might describe more than one surface. Mathematicians have spent the past 150 years cataloging these exceptions: instances in which local measurements that usually define just one surface in fact describe more than one. But the only exceptions they managed to find weren’t nice, closed-up surfaces like orbs or doughnuts — instead, they stretched on forever in some direction, or had edges you could fall off of.

Nobody could find a closed-up surface that broke the rule. It began to seem as though there simply weren’t any. Perhaps such surfaces could always be uniquely defined by the usual local information.

Now, mathematicians have finally uncovered one of those long-sought exceptions. In a paper published in October, three researchers — Alexander Bobenko of the Technical University of Berlin, Tim Hoffmann of the Technical University of Munich, and Andrew Sageman-Furnas of North Carolina State University — describe a pair of very twisty, closed-up surfaces that, despite having the same local information, have completely different global structures.

Finding them took years of toil, a few very overheated laptops, and an unexpected clue from a seemingly unrelated corner of geometry.

Geometric Misfits

Mathematicians have all sorts of ways to describe a surface locally, but two are especially useful.

One captures information about the surface’s “extrinsic” curvature. Choose a point on your surface. At that point, there are infinitely many directions in which you can calculate how quickly the surface bends in space — what’s known as its curvature. Focus only on the directions where you get the biggest and smallest curvature values, then take the average of the two. The number you get is called the mean curvature. You can compute the mean curvature for any given point on the surface to gain a better understanding of how it’s situated in the space surrounding it.

Another kind of measurement captures information about the surface’s “intrinsic” curvature — a geometric property that doesn’t depend on the space that the surface lives in. Consider a flat sheet of paper. You can wrap it into a cylindrical tube without stretching or tearing it. If two points are connected by a curve on the sheet of paper, that curve will have the same length on the cylinder. This means that the sheet of paper and the cylinder have the same “metric,” or notion of distance. But try to wrap the sheet of paper around a sphere, and that’s no longer the case. You’ll have to stretch, cut, or crinkle the paper, and the lengths of curves between points will change. The two surfaces therefore have different metrics.

Mark Belan/Quanta Magazine

In 1867, the French mathematician Pierre Ossian Bonnet showed that if you know both the metric and the mean curvature at every point on a surface, that’s enough to tell you what the surface is. Most of the time.

But most of the time is not all the time, and that’s the kind of caveat that makes mathematicians itch.

In the 150 years since Bonnet’s proof, mathematicians have discovered various kinds of surfaces that defy his rule of thumb. These surfaces have the same metric and mean curvature, yet they don’t have the same global structure.

Pierre Ossian Bonnet asked when it was possible for two different shapes to be defined by the same local data.

Public Domain

But all these surfaces are what mathematicians call non-compact. They don’t wrap up nicely the way spheres, doughnuts, and other “compact” surfaces do. Rather, a non-compact surface might stretch out infinitely in some direction (like a plane or cylinder), or have edges where it suddenly ends (like a piece cut out from a larger shape).

Compact surfaces are more restricted. They have to satisfy various constraints to twist back on themselves and close up perfectly. So it seemed reasonable to think they might be uniquely defined by their metric and mean curvature. In 1981, the mathematicians Blaine Lawson and Renato de Azevedo Tribuzy proved that this is true for the sphere and any surfaces topologically equivalent to it — that is, any compact surfaces that have no holes.

When it came to compact surfaces with a hole (topological doughnuts called tori), there was a bit more wiggle room. The mathematicians showed that a given metric and mean curvature could correspond to at most two different tori.

No one could find examples of such “compact Bonnet pairs,” however, and so for decades, the prevailing view was that tori were like spheres, and that a given metric and mean curvature would define a single torus. “People believed that for a long time,” said Robert Bryant of Duke University, “because they couldn’t construct any examples.”

But they were wrong.

A Pixelated World

Alexander Bobenko has spent the past 20 years chewing on mathematical doughnuts. In the 2000s, he tried to prove that compact Bonnet pairs do indeed exist. But after realizing that the problem would take him more than a few months to solve, he set it aside to focus on questions he thought he could make faster progress on.

He turned to an area of mathematics that seemed unrelated to the Bonnet problem. But that area would end up being the key to solving it.

Bobenko started to think about “discrete” surfaces, which are a bit like pixelated low-resolution versions of smooth surfaces. Mathematicians study discrete surfaces because they have important geometric properties in their own right, as well as practical applications in computer science, physics, engineering, and more.

To get a discrete surface, take a finite collection of points and connect them by lines to form a shape with flat faces. By choosing different points, you can represent a given smooth surface in different ways. Here are some examples of how you might represent a sphere, for instance:

Mark Belan/Quanta Magazine

Some discrete surfaces are better representations than others. Bobenko and his frequent collaborator Tim Hoffmann have dedicated nearly two decades to developing a theory for how to preserve the most salient geometric features of smooth surfaces using discrete ones.

In the 2010s, Andrew Sageman-Furnas, then a doctoral student at the University of Göttingen, joined the effort — and brought the Bonnet problem back into the mix.

Sageman-Furnas had been drawn into discrete mathematics through his interest in the mechanics of woven fabrics like fishing nets, which are essentially discrete surfaces. Along the way, he’d asked a discrete version of the Bonnet question: When will local information uniquely define a discrete surface, and when won’t it? By adapting a known method for generating exceptions to Bonnet’s rule, Sageman-Furnas, along with his adviser Max Wardetzky and Hoffmann, found a recipe for concocting exceptions in the discrete case.

As in the smooth case, these exceptions were always non-compact. But because discrete surfaces don’t contain infinitely many points, it’s possible to study them using computers. Might it be possible, Sageman-Furnas wondered, to use computational brute-force methods to find a compact Bonnet pair in the world of discrete geometry? If so, then perhaps the discrete case could lead the way to a smooth solution as well.

And so he joined Bobenko and Hoffmann in Berlin as a postdoctoral researcher in Bobenko’s group and got to work.

Surface Safari

In the spring of 2018, Sageman-Furnas began a computer search for a special type of surface — one that could be transformed into a Bonnet pair, akin to how a sourdough starter acts as a base for whipping up different kinds of bread. This “starter” surface would be like the ones he had used to make discrete Bonnet pairs as a graduate student. Except this time, he required it to be a torus. That is, it had to be compact with one or more holes.

He disappeared for weeks, if not months, Hoffmann recalled. When the younger mathematician finally reemerged, he had found what he’d been looking for: a very spiky shape that looked more like an origami rhino than a torus.

The “rhino.”

Mark Belan/Quanta Magazine; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)

But a torus it was. And according to Sageman-Furnas’ computer program, it had all the other properties required of a starter surface that would generate Bonnet pairs. Even more important, when Sageman-Furnas generated those pairs on his computer, they were also tori. The transformations from the rhino to the Bonnet pair didn’t seem to twist the rhino open into non-compact surfaces. The surfaces stayed compact.

“When you start to do computational exploration and design,” Sageman-Furnas said, “you can get new examples that are far outside of what you thought was possible.”

But was it too good to be true? Computer programs make rounding errors: Sageman-Furnas’ rhino might appear to meet the desired criteria, and the Bonnet pair it generated might appear to be tori, but that could all be a mirage, an artifact of small computational errors. Without a rigorous proof, the mathematicians couldn’t be sure.

“He showed up, and he showed us some weird-looking geometric object that really looked like it could have been numerical crap,” Hoffmann said. “Tongue in cheek, probably my most precious contribution to the whole project was that at the time I said, ‘I’ve seen worse.’”

Andrew Sageman-Furnas (left), Tim Hoffmann (center), and Alexander Bobenko constructed a pair of novel shapes that settled a long-standing conjecture.

From left: Courtesy of Andrew Sageman-Furnas; N. Kutz; Courtesy of Alexander Bobenko

It took some time, but Hoffmann and Sageman-Furnas were eventually able to convince themselves that the rhino was worth taking seriously. And if it was possible to find such a likely example of a discrete Bonnet pair, maybe the smooth case wasn’t so hopeless after all. Hoffmann and Sageman-Furnas spent that sweltering summer scouring the rhino for clues, sometimes sitting in video chats for eight to 12 hours at a time, searching for unusual properties and geometric constraints that might help them narrow down where to look for smooth Bonnet tori.

As September rolled around, they finally found a new lead that felt so promising that it drew Bobenko back into the problem he’d abandoned decades earlier.

Closed Loops

The clue had to do with particular lines that loop around the rhino along its edges.

These lines were already known to provide important information about the rhino’s curvature — tracing out the directions in which it bent and folded the most and least. Since the rhino is a two-dimensional surface that lives in three-dimensional space, the mathematicians had expected these lines to carve out paths throughout 3D space as well. But instead, they always lay either in a plane or on a sphere. It was exceedingly unlikely that these alignments had happened by chance.

“That suggested to us that there was really something special happening,” Sageman-Furnas said. It was “spectacular.”

Unlike discrete surfaces, smooth surfaces don’t have edges. But you can still draw “curvature lines” that trace out the paths of maximum and minimum bending. Sageman-Furnas, Bobenko, and Hoffmann decided to look for a smooth analogue of the rhino whose curvature lines were similarly restricted to living in planes or on spheres. Perhaps a starter surface with those properties could give rise to smooth Bonnet tori.

But it wasn’t clear if such a surface even existed.

Jean Gaston Darboux came up with formulas that, more than a century later, turned out to be the missing link in work on the Bonnet problem.

Public Domain

Then Bobenko realized that more than a century ago, the French mathematician Jean Gaston Darboux had laid out almost exactly what the mathematicians now needed.

Darboux had come up with formulas for generating surfaces that had the right kinds of curvature lines. The problem was that his formulas wouldn’t produce curvature lines that looped back on themselves. Instead, they “look like spirals and go to infinity,” Bobenko said. “No chance to get them closed.” Which meant that while the curvature lines might live on planes and spheres, the overall surface wouldn’t be a torus.

After years of toil, the mathematicians — using a combination of pen-and-paper techniques and computational experiments — figured out how to adjust Darboux’s formulas so that the curvature lines would close up. They’d finally found their smooth analogue of the rhino (although the two didn’t look much alike).

Moreover, as they’d hoped, this smooth rhino could generate a pair of new tori that had the same mean curvature and metric data but different overall structures. The team finally had their answer to the original Bonnet problem: Some tori can’t be uniquely defined by their local features after all.

But when they worked out what this Bonnet pair actually looked like, they found that the two tori were mirror images of each other. “Technically, this wasn’t an issue,” Sageman-Furnas said. “Formally, it solved the problem.” But, he added, it was still unsatisfying.

And so over the next year, they tried to tweak their smooth rhino in various ways. Ultimately, they realized that if they dropped the requirement that one set of curvature lines had to sit on spheres, they could construct a new smooth rhino that did what they wanted. They then used this surface to generate a new Bonnet pair — this time, two very twisty tori that were much more obviously different surfaces but still had the same metric and mean curvature.

The team’s final compact Bonnet pair.

Mark Belan/Quanta Magazine; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)

The result came as a surprise to Rob Kusner, a mathematician at the University of Massachusetts, Amherst. According to him, it demonstrates that even tori — some of the nicest, best-studied surfaces — can’t always be perfectly described by their local characteristics.

“It’s an example of something where our intuition wasn’t good enough,” said Bryant, the Duke mathematician.

Still, the two tori that the mathematicians found are a bit strange: They pass through themselves like figure eights. Bobenko now hopes to prove that there are Bonnet tori that don’t intersect themselves.

Related:

1. Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture

2. A New Proof Smooths Out the Math of Melting

3. An Old Conjecture Falls, Making Spheres a Lot More Complicated

The Bonnet tori are a welcome validation of Bobenko and Hoffmann’s decades of work on discrete surfaces. Traditionally, the geometry of smooth shapes has advanced much faster, dragging the less developed theory of discrete geometry along behind it. But in this work, the discrete theory charged ahead and was ultimately what made progress on the smooth side possible.

According to Hoffmann, this highlights the fact that while discrete surfaces might seem like less sophisticated models of their smooth counterparts, they have a mathematical life of their own. The discrete world can be just as rich as the smooth one, if not richer, revealing extra symmetries and connections that might otherwise get lost.

“People sort of forgot about this discrete aspect,” Hoffmann said. But “there are still things to gain from it.”

2026 Jan 16

- Why There’s No Single Best Way To Store Information

EXPLAINERS | VIEW ON WEBSITE

Why There’s No Single Best Way To Store Information

By Ben Brubaker

January 16, 2026

The math of data structures helps us understand how different storage systems come with different trade-offs between time, memory, and other resources.

In data storage, sometimes it’s best to embrace a bit of disorder.

Kristina Armitage/Quanta Magazine

Introduction

Just as there’s no single best way to organize your bookshelf, there’s no one-size-fits-all solution to storing information.

Consider the simple situation where you create a new digital file. Your computer needs to rapidly find a place to put it. If you later want to delete it, the machine must quickly find the right bits to erase. Researchers aim to design storage systems, called data structures, that balance the amount of time it takes to add data, the time it takes to later remove it, and the total amount of memory the system needs.

To get a feel for these challenges, imagine you keep all your books in a row on one long shelf. If they’re organized alphabetically, you can quickly pick out any book. But whenever you acquire a new book, it’ll take time to find its proper spot. Conversely, if you place books wherever there’s space, you’ll save time now, but they’ll be hard to find later. This trade-off between insertion time and retrieval time might not be a problem for a single-shelf library, but you can see how it could get cumbersome with thousands of books.

Instead of a shelf, you could set up 26 alphabetically labeled bins and assign books to bins based on the first letter of the author’s last name. Whenever you get a new book, you can instantly tell which bin it goes in, and whenever you want to retrieve a book, you will immediately know where to look. In certain situations, both insertion and removal can be a lot faster than they would be if you stored items on one long shelf.

Of course, this bin system comes with its own problems. Retrieving books is only instantaneous if you have one book per bin; otherwise, you’ll have to root around to find the right one. In an extreme scenario where all your books are by Asimov, Atwood, and Austen, you’re back to the problem of one long shelf, plus you’ll have a bunch of empty bins cluttering up your living room.

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Computer scientists often study data structures called hash tables that resemble more sophisticated versions of this simple bin system. Hash tables calculate a storage address for each item from a known property of that item, called the key. In our example, the key for each book is the first letter of the author’s last name. But that simple key makes it likely that some bins will be much fuller than others. (Few authors writing in English have a last name that starts with X, for example.) A better approach is to start with the author’s full name, replace each letter in the name with the number corresponding to its position in the alphabet, add up all these numbers, and divide the sum by 26. The remainder is some number between zero and 25. Use that number to assign the book to a bin.

This kind of mathematical rule for transforming a key into a storage address is called a hash function. A cleverly designed hash function ensures that items will usually end up distributed relatively evenly across bins, so you won’t need to spend as much time searching in each bin.

If you want to reduce retrieval time further, you can use more bins. But that leads to another trade-off: Those bins will take up space even if they end up empty.

This trade-off between space and time is an inherent feature of hash tables — it’s the price you pay for avoiding the tension between insertion and retrieval time that plagues simpler data structures. More than 70 years after hash tables were invented, computer scientists are still discovering new things about their fundamental properties. Recently, they finally devised a version that strikes an ideal balance between space and time. And last year, an undergraduate student disproved a long-standing conjecture about the minimum amount of time needed to find a specific item in a hash table that’s almost full.

A Heap of Priorities

Hash tables work well when you can’t anticipate which piece of data you’ll need to retrieve next. But that’s not always the case. Imagine you’re trying to complete tasks on a to-do list, but you’re constantly being assigned new tasks with different deadlines. You want to be able to quickly add new items to the to-do list, but you don’t care about retrieving items until they become your top priority.

In this case, your best bet is a type of data structure called a heap. As the name suggests, a heap is a somewhat haphazard approach to data storage. It’s basically a mathematical version of a pile of stuff: Some items are stored above others, and these higher items are easier to access. The highest-priority item is always at the top of the heap, where you can instantly pluck it off. Lower layers will be more disorganized, but you don’t need to worry about the relative positions of these low-priority items.

The simplest implementation of this basic idea uses a mathematical object called a binary tree, which is a network of nodes with a special shape: There’s a single node at the top, and each node is connected to two nodes directly below it.

Let’s imagine a binary tree that contains the items in a to-do list. Each node can store a single item, and each item is labeled with a number that represents its due date. High-priority items get smaller numbers.

Mark Belan/Quanta Magazine

Each new item is put into an empty slot in the current lowest layer.

Once the new item goes in, compare its due date to that of the item in the node directly above it. If the new task is due sooner, swap the items. Keep swapping until the new item ends up directly below an item that’s more urgent.

This procedure ensures that the highest-priority item will always rise to the top. What’s more, the procedure is extremely fast. Even in a nightmare scenario where you have 1,000 tasks on your to-do list and keep getting new assignments, storing them in a heap ensures that it takes no more than nine swaps to move each new item up to the appropriate position. Whenever you complete the most urgent task and remove it from the heap, you can quickly pull up your new top priority from the layer below.

Related:

1. Scientists Find Optimal Balance of Data Storage and Time

2. Computer Scientists Establish the Best Way to Traverse a Graph

3. For Algorithms, a Little Memory Outweighs a Lot of Time

Within computer science, heaps are widely used in algorithms for finding the shortest path from a given starting point in a network to every other point. In 2024, a team of researchers used an ingenious new heap design to transform a classic shortest-paths algorithm into one that is theoretically optimal for any network layout.

There’s no shortage of self-help books filled with contradictory advice about the best way to organize your belongings. If computer science offers any lesson, it’s that there is no perfect solution — every approach comes with trade-offs. But if some items are more important to you than others, don’t be afraid to leave a bit of a mess.

Table of Contents

Quanta Magazine: 2026 Jan 26

2026 Jan 26

Is Particle Physics Dead, Dying, or Just Hard?

2026 Jan 23

Monster Neutrino Could Be a Messenger of Ancient Black Holes

2026 Jan 21

How Animals Build a Sense of Direction

2026 Jan 20

Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle

2026 Jan 16

Why There’s No Single Best Way To Store Information