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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.

2026 Jan 14

- String Theory Can Now Describe a Universe That Has Dark Energy

THEORETICAL PHYSICS | VIEW ON WEBSITE

String Theory Can Now Describe a Universe That Has Dark Energy

By Steve Nadis

January 14, 2026

In an unprecedented step, researchers crafted a detailed model compatible with the universe’s accelerated expansion.

Scientists have struggled to make string theory compatible with the expanding universe.

Nash Weerasekera for Quanta Magazine

Introduction

In 1998, astronomers discovered dark energy. The finding, which transformed our conception of the cosmos, came with a little-known consequence: It threw a wrench into the already daunting task of finding a version of string theory that describes the universe we live in.

Dark energy is a “positive” energy that causes our universe to expand at an accelerating rate. But the best-understood models of string theory describe universes with energy that is either negative or zero.

Of the various criticisms made of string theory through the years — that it only works in a 10-dimensional universe, that its fundamental constituents, tiny strings, are too small to ever be observed — this was perhaps the most troubling. String theory appeared to be useful only for describing a universe with a negative “anti-de Sitter” geometry, whereas we live in a universe with a positive “de Sitter” geometry.

Then last year, two physicists offered a stripped-down but precise formula for how string theory could give rise to a universe similar to ours — a de Sitter universe undergoing accelerated expansion.

“It is the very first example [from string theory] of an explicit de Sitter space,” said Thomas Van Riet of KU Leuven in Belgium.

The new work, by Bruno Bento and Miguel Montero of the Institute for Theoretical Physics in Madrid, describes a universe with a dark energy that should weaken over time — a result that matches preliminary cosmic observations from the past few years.

But the universe they describe is not exactly like ours. While their original hope was to reduce the high-dimensional world of string theory to our own four-dimensional world, they ended up with an extra dimension. “What they have found is a 5D de Sitter solution, and we don’t live in 5D,” said Antonio Padilla of the University of Nottingham.

Bruno Bento (left) and Miguel Montero published work that connects the hidden dimensions of string theory to the positive energy of our universe.

Laura Marcos Mateos

Still, the work is expected to launch a new era in matching the mathematical elegance of string theory to the actual world we live in.

“What they have done,” Padilla said, “is open up a new frontier to finding explicit de Sitter solutions in string theory.”

The Cutoff

The new work was inspired by a bizarre feature of quantum theory first predicted over 75 years ago.

In a vacuum, space is never completely empty. Particles pop in and out of existence, and tiny fluctuations cause quantum fields to do the same.

In 1948, the Dutch physicist Hendrik Casimir recognized that in the narrow space between two conducting plates, not all quantum fields can pop into existence. In this region, the long wavelengths get cut off. This leads to a lower energy density inside the plates than outside. The mismatch of energies creates a force that tries to push the plates together.

Mark Belan/Quanta Magazine

Bento and Montero applied this line of thinking to the process of “compactification,” in which the 10-dimensional physics of string theory becomes the four-dimensional realm we inhabit. The basic premise of compactification is that the extra dimensions should shrink down and curl up into a shape so tiny that if you were to travel along one of them, you would almost instantly come back to the starting point. The precise shape of the “manifold” that houses these extra dimensions would dictate the properties of all the particles and forces observed in nature.

In the new scenario, the space enclosed within a six-dimensional manifold takes the place of the space between Casimir’s conducting plates. Inside the manifold’s interior, fluctuations are similarly restricted, which generates a Casimir-like force. “That’s their key ingredient,” said David Andriot of France’s National Center for Scientific Research.

The researchers counterbalanced the Casimir effect with a force generated by a flux. Fluxes are standard elements in string theory compactifications. They’re made up of field lines that wind through string theory’s extra dimensions. Unlike the Casimir force, which works toward reducing the volume of the manifold’s interior, a flux creates a countervailing effect that tries to expand that volume.

Bento and Montero were able to calculate a specific value for dark energy that was both positive and small. The value they arrived at, 10−15 in Planck units, is still far from the actual, even smaller value of 10−120, but it is “going down the right path,” Montero said.

The solution is considered explicit, he explained, which “means we can tell you every detail involved and how it fits together. We can compute a precise value for the dark energy that is close to the exact result.” And if you give your model to other physicists, he said, “they can compute the value of any observable … with precision.”

The original idea to look for a Casimir-like effect came from a 2021 paper by Eva Silverstein of Stanford University and two collaborators. But Bento and Montero’s goal from the outset was to find a simpler recipe for compactification than previous researchers had.

Eva Silverstein, photographed at Stanford University in 2017, co-authored the paper that inspired Bento and Montero’s solution.

Ryan Schude for Quanta Magazine

In selecting a geometry for the compact extra dimensions, for instance, they chose a space that resembles a torus. “It’s a simple shape,” Bento said. A doughnut is an example of a 2D torus; it is considered “flat” because it can be made by rolling a flat sheet into a tube and then fastening the ends. Bento and Montero picked shapes of this general type, called 6D Riemann-flat manifolds, to house the extra dimensions in their model. Using this 6D space for the compactification gave them the physical properties they sought.

In comparison, the Silverstein team selected a much more complicated geometry to work with: negatively curved hyperbolic manifolds. That made their calculations dramatically harder.

Shortly after Bento and Montero published their paper, Gianguido Dall’Agata and Fabio Zwirner of the University of Padua published their own paper, in which they used a similar setup — also involving Riemann-flat manifolds — to compute the strength of the Casimir effect and show how it can be used to produce dark energy. “We use different techniques that are complementary,” Zwirner said.

Bento and Montero took things further than the Padua team, at least in terms of carrying out a full-fledged string compactification. But, Montero said, “it was nice that these two approaches agreed, because that provided a good check on the general idea.”

A Dose of Reality

The work of Bento and Montero comes with some substantial caveats, as the authors acknowledge.

First, their de Sitter solution is unstable; its dark energy, though positive, will diminish over time. A changeable, dynamical dark energy of this sort, Andriot pointed out, “is much easier to get from string theory” than a dark energy that remains fixed — a notion Einstein introduced in 1917 as the “cosmological constant.”

“Unstable,” in this case, has a specific meaning to physicists. It indicates that the period of stability, or constancy, of dark energy shouldn’t last much longer than a Hubble time — the estimated age of the universe, or about 14 billion years.

Until recently, most observations have been consistent with a universe containing a constant amount of dark energy. But recent results suggest that dark energy may be changing. In April 2024, the Dark Energy Spectroscopic Instrument presented tentative evidence that dark energy is weakening, and the finding was bolstered a year later. “If those results are here to stay, they are really hinting that the cosmological constant is not a constant,” Montero said.

Related:

1. Is Dark Energy Getting Weaker? New Evidence Strengthens the Case.

2. Dark Energy May Be Weakening, Major Astrophysics Study Finds

3. Why Is M-Theory the Leading Candidate for Theory of Everything?

In their pursuit of a de Sitter solution, Bento and Montero simplified their task by starting from M-theory (sometimes called “the mother of all string theories”). Whereas most versions of string theory require our universe to have six extra dimensions, M-theory requires it to have seven. Despite the larger number of dimensions, M-theory has fewer ingredients than string theory, so starting with M-theory made Bento and Montero’s calculations markedly easier. But subtracting the six extra dimensions curled into their manifold from the 11 total dimensions of M-theory left the theorists with a universe in 5D — one “D” too many.

The issue of landing on a 5D solution in a 4D universe is no small matter, and Bento and Montero consider resolving it a top priority. “If we cannot find the four-dimensional solution,” Bento said, “our work cannot be the final answer.”

“I hope it works, and they manage to get it [to work] in four dimensions,” Andriot said. However, he cautioned, given the myriad challenges string theorists have faced over the past few decades, he wouldn’t be surprised if the de Sitter problem threw at least a few more obstacles in their path.

2026 Jan 12

- Cells Use ‘Bioelectricity’ To Coordinate and Make Group Decisions

CELLULAR COMMUNICATION | VIEW ON WEBSITE

Cells Use ‘Bioelectricity’ To Coordinate and Make Group Decisions

By Elise Cutts

January 12, 2026

The discovery that tissues use electricity to expel unhealthy cells is part of a surge of renewed interest in the currents flowing through our bodies.

A cell shrinks, due to a change in its bioelectrical properties, before it is extruded from tissue. The “lightning” flash represents a change in light refraction as the membrane is depolarized and water exits the cell. Original video quality enhanced using AI with creator’s approval. (Images taken every two minutes.)

King’s College London

Introduction

We’re used to thinking of the brain as an electric organ. The rest of the body? Not so much. But it would be a mistake to dismiss your other tissues as dumb hunks of electrically inert flesh. Even the protective layers of cells that compose your skin and line your organs use electrical signals to make decisions, according to recent research.

Results published in Nature show that cells use bioelectricity to coordinate a complex collective behavior called extrusion, a vital process that ejects sick or struggling individual cells from tissue to maintain health and keep growth in check. Merciless as it might seem, extrusion helps keep you alive. It’s vital for the health of protective epithelial tissues, and when it goes wrong, it can contribute to disease, including cancer and asthma. Until now, it’s been unclear how cells were singled out for this process.

According to the new results, as epithelial tissue grows, cells are packed more tightly together, which increases the electrical current flowing through each cell’s membrane. A weak, old, or energy-starved cell will struggle to compensate, triggering a response that sends water rushing out of the cell, shriveling it up and marking it for death. In this way, electricity acts like a health checkup for the tissue and guides the pruning process.

“This is a very interesting discovery — finding that bioelectricity is the earliest event during this cell-extrusion process,” said the geneticist GuangJun Zhang of Purdue University, who studies bioelectrical signals in zebra fish development and wasn’t involved in the study. “It’s a good example of how a widening electronic-signaling perspective can be used in fundamental biology.”

The new discovery adds to the growing assortment of bioelectrical phenomena that scientists have discovered playing out beyond the nervous system, from bacteria swapping signals within a biofilm to cells following electric fields during embryonic development. Electricity increasingly appears to be one of biology’s go-to tools for coordinating and exchanging information between all kinds of cells.

“People just kind of relegated [bioelectricity] to ‘This is just neurons.’ No — it’s all of our bodies,” said study author Jody Rosenblatt, an epithelial cell biologist at King’s College London and the Francis Crick Institute. “There are electrical currents going through your body all the time, and they’re doing things.”

Life’s Spark

It’s no coincidence that Frankenstein’s monster sprang to life with a spark. In the late 18th century, just a few decades before Mary Shelley wrote her science fiction masterpiece, the Italian surgeon Luigi Galvani jolted the scientific community with experiments that used metal and electricity to compel disembodied frog legs to kick. He became convinced that there was an “animal electricity” running through life.

In the late 1700s, the anatomist Luigi Galvani (left) discovered that dissected frog legs twitched when touched with an electrical spark, setting off the study of what he called “animal electricity.”

Public Domain; ETH-Bibliothek Zurich / Science Source

While Galvani was later proven wrong in the details, he wasn’t totally off. Virtually every cell on every branch of the tree of life expends a hefty chunk of its energy budget — in some cells, more than half — on maintaining a voltage across its membrane. The voltage difference that results, called the membrane potential, stores potential energy that can be released later. It’s like the pressure behind a dam: Gravity tugs water downhill, and dams store energy by holding water at the top of a hill. Like gravity, the electrochemical force tugs charges “downhill” — positive charges stream toward negative charges and vice versa in electrical currents. Blocking that flow, for example with a cell membrane, stores up electrical potential energy.

The electric currents that pour from our wall sockets are streams of electrons. In cells, “it’s quite similar, but not exactly the same,” said Elias Barriga, who studies tissue biophysics at the Dresden University of Technology. “We are fueled by ions.”

Ions are atoms or molecules that carry charge because of extra or missing electrons, which give them negative or positive charges, respectively. They can enter and exit cells only through specialized protein channels and pumps. Just as hydroelectric plants can use surplus energy to pump water back up into the reservoir for later use, cells use their chemical energy to pump ions “uphill” against the electric flow. By controlling the natural current and letting positive or negative charge build up on either side of their membranes, cells maintain their membrane potential. And if this energy is used or leaks away, cells can replenish it by expending more of their chemical energy.

Elias Barriga has shown that frog embryos generate electrical fields to guide cell migration. The study of bioelectricity, formerly stranded in biology’s backwaters, is “coming back like crazy,” he said.

Courtesy of Elias Barriga

“You generate a potential: what’s inside versus what’s outside, a different concentration of ions,” Barriga said. “That is the source of bioelectricity.”

Neurons make use of this biological electricity to share information. By releasing messenger molecules called neurotransmitters that open and close ion channels, neurons can nudge their neighbors’ membrane potentials up or down. If these chemical nudges push a neuron’s membrane potential beyond a threshold, the cell “spikes” — voltage-sensitive ion channels throw open the floodgates for positive sodium ions, which rush into the cell and cause a rapid voltage swing that ripples along the neuron’s length. When that signal reaches the interface between neurons, voltage-sensitive channels open wide, triggering the release of neurotransmitters to more neurons downstream.

Muscle contraction also kicks off with a voltage spike; neurons send electrical signals streaming into muscle fibers, triggering contractions. This is why Galvani’s electrified frog legs twitched, and why a jolt of electricity can jump-start a stopped heart. (Specialized cells in the heart use electricity to set the pace of its regular contractions.) While all tissues maintain membrane potentials, researchers don’t really know what they do. Compared to electrophysiology, which often focuses on electricity in the brain and heart, the field of bioelectricity — a grab-bag term for electrical activity everywhere else in organisms — has lagged behind, Barriga said.

“I think that at some point it got stuck,” he said. “But now I can tell you that that is coming back like crazy.”

A Shocking Discovery

The epithelial tissues that make up skin and line organs, blood vessels, and body cavities quietly burn about 25% of their available energy to maintain membrane voltages between minus 30 and minus 50 millivolts. But researchers interested in these tissues typically study mechanical forces, chemical signaling, and gene expression — not currents and voltage, Rosenblatt said.

Until recently, that included her. Rosenblatt has spent 25 years piecing together the details of epithelial extrusion, a process that keeps tissue growth in check. Because epithelial cells grow quickly, even a slight mismatch between the rates of cell division and cell death can quickly add up to a tumor or injury. Runaway replication can grow into cancer, while overzealous culling — as can happen in asthma — compromises the integrity of tissues. It’s important to get the balance right.

Around 14 years ago, Rosenblatt and colleagues discovered that overcrowded epithelial cells are popped up and out of the tissue alive — extruded — to maintain that tissue balance as new cells grow. That raised a question: How does tissue “choose” which living cells to expel?

In earlier work, Rosenblatt’s team watched as some cells dumped their water and shriveled up like raisins before being extruded; indeed, this shrinkage seemed to kick off the process. But the researchers didn’t know what caused the cells to shrink in the first place. They didn’t work on bioelectricity and were unaware of any effect it might have.

The cell biologist Jody Rosenblatt studies extrusion, the process by which a tissue expels cells to prevent overcrowding. Her lab recently described bioelectricity as helping the tissue “choose” which cells to extrude.

Antonio Tabernero

In further experiments, they were able to prevent the cells from shrinking by blocking a pressure-sensitive ion channel in the cell membrane that opens when squeezed. They decided to see if blocking other ion channels might interfere with extrusion too.

“We got so many hits, we were just like: Jesus, this is crazy,” Rosenblatt recalled. One of those hits was a voltage-gated potassium channel, like those that open up during a neuron’s voltage spike. It struck Rosenblatt as “weird” enough to follow up on. Using special dyes that reveal the voltage across cell membranes, the scientists found that epithelial cells destined for extrusion — and only those cells — lose their membrane potential about five minutes before shrinking and being extruded. The result was clear: Extrusion kicks off with an electrical signal.

Instead of sending neurotransmitters back and forth like neurons, densely packed epithelial cells squeeze each other. As the tissue gets more crowded, the squeezing intensifies. This opens pressure-sensitive ion channels, which allow positive sodium ions to leak across the squashed cells’ membranes and into the cell.

Faced with this challenge, a healthy cell will use its chemical energy to activate pumps to push sodium back out and restore its normal voltage. But stressed or unhealthy cells without energy to spare can’t keep up. Their membrane voltage falters, throwing open those “weird” voltage-sensitive channels. When that happens, water pours out of the cell in a “lightning” flash clearly visible in microscope images, Rosenblatt said. Once a cell loses 17% or more of its volume, it is extruded. Her working hypothesis is that a biochemical cascade set off by the shrinkage contracts motor proteins, which mechanically extrude the cell.

In this way, bioelectrical flow across cell membranes lets tissues test which cells are the least healthy and mark them for extrusion. “They’re always pushing against each other and bullying each other. And what they’re doing is probing each other for which one’s the weakest link,” Rosenblatt said. “It’s a community effect.”

Evolution as Electrician

At the University of California, San Diego, the biophysicist Gürol Süel studies electricity in bacterial biofilms, which are collectives composed of single-celled bacteria that can also survive independently. The signaling that Rosenblatt and her team described in human tissues has several things in common with electrical mechanisms Süel has described in microbes — and which appear again and again across the tree of life.

“It’s a very elegant study, very nice results,” he said of the new research. “And conceptually, it makes sense.”

Gürol Süel studies how bacteria in biofilms use bioelectricity to communicate and coordinate. A membrane potential “tells you, in one glance almost, about the state of the cell,” he said.

Suel Lab

Electricity increasingly appears to be one of evolution’s go-to solutions for integrating multiple streams of information. Epithelial tissues use it to keep tabs on crowding. Neurons compile input signals from multiple sources into a spiking output. A Venus flytrap snaps shut when sensory hairs with touch-sensitive ion channels react to prey. These channels are tuned to trigger a voltage spike and tell the trap to close only if stimulated multiple times in rapid succession.

“The membrane potential is so fundamental, and it is very fast,” Süel said. While switching genes on and off or upping protein production could take minutes or hours, a membrane potential can flip in fractions of a second. “It tells you, in one glance almost, about the state of the cell,” he added.

Ten years ago, Süel and his team showed that microbes in biofilms can spike their membrane potentials to communicate, just as neurons do. Since then, they’ve shown that biofilms use electricity to coordinate tasks, prevent runaway growth, and invite free-swimming bacteria to join the collective. Bioelectricity can even help them avoid falling victim to the tragedy of the commons: Two biofilms sharing scarce food can send electrical signals to each other to take turns eating.

Multicellular animals, too, use electrical signaling to organize themselves. Zhang, of Purdue, studies bioelectrical signaling in zebra fish, which develop striking extra-long tails when a certain ion channel is mutated. This suggests that electrical signaling somehow tells tissues in a developing embryo how long to grow. Michael Levin, a researcher at Tufts University, has blocked cell channels to manipulate the membrane potentials of developing worm embryos, causing genetically identical worms to develop different body plans. And last year, Barriga and his colleagues showed that frog embryos generate natural electric fields that guide the migration of specific stem cells to their proper locations in the developing embryo.

Related:

1. How Much Energy Does It Take To Think?

2. Tiny Tweaks to Neurons Can Rewire Animal Motion

3. Brainless Embryos Suggest Bioelectricity Guides Growth

4. How a Biofilm’s Strange Shape Emerges From Cellular Geometry

The failure of bioelectric processes might be an overlooked cause of disease. Cancerous cells tend to have different membrane potentials than healthy ones, and Levin has argued that some cancers might result from a breakdown in multicellularity that happens when cells can no longer use electricity to coordinate. For example, maybe they can no longer communicate the message “I’m struggling and should be extruded,” and the result is the uncontrolled growth and, ultimately, a tumor.

Süel is convinced that bioelectricity is as old as life itself. Indeed, an electric current drives the molecular turbines that synthesize life’s universal energy currency, ATP, in every cell alive today. One leading origin-of-life scenario places the beginning at deep-sea hydrothermal vents. There, natural currents of positively charged protons could have served as a kind of primordial membrane potential and powered prebiotic chemical reactions. But whether life started with such a spark or not, bioelectricity’s ubiquity suggests it has deep evolutionary roots that we’re just beginning to unearth.

“There are a lot of interesting things that cells are probably doing, just like this paper showed, that we just don’t know yet,” Süel said. “We have not uncovered even half of this. … There’s a lot of opportunity for discovery.”

2026 Jan 9

- Using AI, Mathematicians Find Hidden Glitches in Fluid Equations

MATHEMATICAL PHYSICS | VIEW ON WEBSITE

Using AI, Mathematicians Find Hidden Glitches in Fluid Equations

By Charlie Wood

January 9, 2026

A $1 million prize awaits anyone who can show where the math of fluid flow breaks down. With specially trained AI systems, researchers have found a slew of new candidates in simpler versions of the problem.

Samuel Velasco/Quanta Magazine

Introduction

Nearly 200 years ago, the physicists Claude-Louis Navier and George Gabriel Stokes put the finishing touches on a set of equations that describe how fluids swirl. And for nearly 200 years, the Navier-Stokes equations have served as an unimpeachable theory of how fluids in the real world behave — from ocean currents threading their way between the continents to air wrapping around an aircraft’s wings.

Nevertheless, many mathematicians suspect that glitches hide deep within the equations. They have a hunch that in certain situations, the theory fails. In these cases, the equations will predict a fluid moving in some unphysical, incomprehensible way — spinning into an impossibly fast vortex, for instance, or instantly reversing its flow. Some quantity in the equations will grow infinitely large, or “blow up,” as mathematicians put it.

Despite immense effort, no one has been able to come up with a situation where the Navier-Stokes equations falter. Doing so — or, alternatively, proving that the equations never blow up — would come with a $1 million reward. And so, as a prelude to solving the Navier-Stokes problem, mathematicians have searched for blowups (also called singularities) in an assortment of simplified fluid equations, such as those that operate in only one dimension.

They’ve found them. But essentially all the singularities they’ve identified have been “stable,” meaning that they can form in many possible ways. In the most realistic fluid theories, including Navier-Stokes, blowups (if they exist) are likely to be far more delicate, occurring in an unimaginably precise way. These “unstable” blowups have been nearly impossible to find, the ultimate needles in the haystack.

In these realistic theories, “a lot of people believe that there are singularities, but that they are unstable, so we never see them,” said Charlie Fefferman, a mathematician at Princeton University who formulated the million-dollar Navier-Stokes challenge.

Now one group of mathematicians has developed a way of training machines to spot these phantom glitches. In a preprint posted in September, they reexamined simpler fluid equations already known to host a stable singularity. There, they found additional potential blowup scenarios — including unstable ones. It was the first time a possible unstable singularity was uncovered in a fluid of more than one dimension.

The team went on to discover an assortment of unstable singularity candidates in several other fluid equations as well. They have not found any million-dollar singularities. And they still need to rigorously prove that the ones they have found do indeed blow up. But their success in uncovering potential unstable singularities in simple models raises hopes that it will also be possible to find unstable blowups in higher-stakes scenarios.

“The idea of an unstable singularity no longer prevents the discovery of the singularity,” said Fefferman, who was not involved in the new research.

Singularity Hunting

A solution to the Navier-Stokes equations captures a slice of eternity. Solving the equations for some initial state of the fluid will tell you the fluid’s velocity at each point in space and at every moment in time. In one simple solution, a fluid might start calm and remain calm forever. In a more complicated setup, gentle currents might merge into whirlpools and eddies. The great mystery is whether every solution — every single possible fluid history that satisfies the Navier-Stokes equations — makes sense everywhere and always.

But tackling the Navier-Stokes equations for fluids in three dimensions is unspeakably difficult, so mathematicians have started with easier versions of the problem. For instance, the Euler equations assume that fluids flow with no internal friction, or viscosity. Energy doesn’t dissipate in these frictionless fluids, so they should blow up more easily than viscous ones.

But even in this simpler scenario, finding a blowup solution is hard. Fluid equations are generally too complicated to solve directly with pencil and paper. So a common approach is to use a computer to simulate the fluid’s motion and get an approximate sense of the conditions that seem to produce a blowup. If you can narrowly identify the blowup-producing conditions, you might be able to use that knowledge to rigorously prove that a blowup truly exists.

That’s the approach that Thomas Hou and Guo Luo took in 2013, when they simulated a digital liquid in a can. They set the top half of the liquid spinning in one direction and the bottom half in the other, then evolved this scenario through time using the Euler equations. Eventually, at points where the opposing flows met along the can’s boundary, the vorticity (a measure of how much the liquid spins around a point) got big — bigger than their computer could handle.

Mark Belan/Quanta Magazine

This was a hint that a similar set of conditions would lead to a blowup. But it was not a guarantee. “The graveyards are strewn with alleged singular solutions of 3D Euler,” Fefferman said.

It took Hou and another collaborator, Jiajie Chen, nearly a decade to remove the “alleged.” In 2022, they used a computer to prove that the singularity candidate implied the existence of a true singularity. It was a landmark proof, and it got mathematicians hungry to push the frontier even further.

The research depended on computer simulations, which meant that tiny adjustments to the initial state of the digital fluid (or any digital rounding errors) wouldn’t affect the fluid’s fate. A singularity would still occur at the can’s boundary even if things played out a little bit differently.

Because of this, the singularity was stable. But a singularity need not be stable. A blowup might occur only when the fluid is set up in the most delicate of ways. In such a case, any adjustment to that initial arrangement, no matter how small, would prevent the fluid from blowing up.

Many mathematicians conjecture that if singularities do lurk in more realistic fluid equations, they’ll be unstable like this, springing up without warning.

They’ll also be far harder to find.

Going Finite

It’s essentially impossible to track down an unstable singularity candidate with a computer simulation. First you’d need a cosmic stroke of luck to land on exactly the right initial configuration for your fluid — akin to trying to balance a pen perfectly on its tip, said Tristan Buckmaster, a mathematician at New York University. Then, to keep it balanced, you’d also have to evolve the fluid flawlessly from one moment to the next, since even the smallest deviation will tip it onto a path that doesn’t blow up.

Computers aren’t capable of infinite precision. They’ll inevitably introduce numerical errors that, though tiny, will stop the unstable singularity from forming. “It’s like the wind blowing on your pen,” Buckmaster said.

A frictionless fluid spinning against a cylinder can “blow up,” according to a landmark proof from Thomas Hou (left) and Jiajie Chen in 2022.

Vicki Chiu; Courtesy of Jiajie Chen

As a result, almost all blowup candidates have been stable.

So Buckmaster and his colleague Ching-Yao Lai, now at Stanford University, began to work out a potentially wind-proof way of finding unstable ones.

They didn’t set out to do so. In 2021, they used a neural network as a new way to search indiscriminately for singularity candidates of any kind. A neural network is, in general, a function defined by a vast array of numbers. These numbers get carefully adjusted through a highly efficient “training” process of guessing, checking, and refining until the function can perform some desired task. For instance, if you calibrate a neural network using thousands of labeled photos of cats and dogs, it “learns” to take in unlabeled images it’s never seen before and label them “cat” or “dog.”

Buckmaster and Lai turned to what’s known as a physics-informed neural network, or PINN. Unlike an image-classifying neural network, a PINN doesn’t learn by studying external data. Instead, it takes a partial differential equation — an equation that describes how a system changes over time — and adjusts itself until it can represent a function that solves that equation. It can, for instance, take fluid equations and train itself to home in on a function that captures a valid history of a fluid, possibly one that contains a singularity.

There’s no time, so you don’t care that it’s unstable.

Tristan Buckmaster, New York University

But no computer technique can directly render the infinite nature of a singularity. Imagine playing the simulation of your fluid and watching it move forward in time. You might represent some quantity, such as the velocity at different points in the fluid, as a curve on a graph. As the fluid changes over time, you’ll see that curve change as well, like a movie. If the curve gets much steeper from one frame to the next, the fluid might be approaching a singularity. The simulation, however, can’t reach that final destination. The computer will run out of memory before the curve gets infinitely steep, crashing the program. Then you can’t know for sure what was going to happen — if you were truly headed for a blowup or not.

To sidestep the inconvenience of infinity, mathematicians have recently focused their search on singularities with a special property called self-similarity. This means that there is a way of stretching the velocity curve in one frame to match the steeper velocity curve in a later frame. And so, if you want to catch a potential singularity, you no longer need to try to watch the curve get infinitely steep. Instead, you can zoom in on the steepening section of the curve while the movie plays in a way that neutralizes the steepening. From this new, dynamic perspective, the curve gets closer and closer to a frozen curve of finite steepness instead. This transformation renders the target — the frozen limit — a finite object that a finite computer can handle.

Buckmaster and Lai realized that PINNs could be an extremely efficient way of finding these frozen solutions to fluid equations. Moreover, these neural networks could also determine the unique zoom rate that makes a singular solution appear frozen and finite.

At first, their PINN turned up only known candidates. In 2022, for instance, Buckmaster and Lai, along with Lai’s postdoctoral researcher Yongji Wang and Javier Gómez-Serrano of Brown University, used a PINN to home in on the stable blowup that Hou and Luo had found in 2013. (Hou and Chen would prove its existence later that year.)

Tristan Buckmaster (left) and Javier Gómez-Serrano are using neural networks to look for extremely subtle ways of making fluid equations glitch.

Dan Komoda/ Institute for Advanced Study; Jason Rossi/Brown University

They also rediscovered a known singularity candidate in the Córdoba-Córdoba-Fontelos (CCF) equations, which describe a simpler one-dimensional fluid. That singularity candidate was especially notable — it was unstable. It had been found in 2019 because the CCF equations happen to be closely related to an even simpler fluid model that’s well understood. But the PINN could find this solution in a more general way, and much more precisely. That’s because it wasn’t a simulation in the traditional sense, stepping a fluid forward in time. Rather, it went after the frozen limit directly.

“There’s no time, so you don’t care that it’s unstable,” Buckmaster said. “You just try to solve the [equation] itself.”

A New Stable of Instability

The mathematicians were excited by the prospect of using their PINN to find new unstable singularity candidates. They teamed up with Google DeepMind and spent the next few years fine-tuning the neural network approach to look for unstable blowups in a few different classical fluid theories. Wang, now a researcher at DeepMind, led the team in switching from off-the-shelf PINNs to bespoke neural networks tailored to fit the specific fluid equations they were trying to solve. The researchers also further tuned the structure of the PINNs to guide them toward solutions with features they knew the singularities should have.

As they did so, their PINNs got better at spotting singularity candidates. A lot better.

These ‘unstable’ blowups have been nearly impossible to find, the ultimate needles in the haystack.

In September, their collaboration of more than 20 researchers unveiled a host of singularities that had never been seen before, most of them unstable.

Revisiting the spinning fluid in a can, they described a collection of four new unstable singularity candidates in the Euler equations. These were still broadly similar to Hou and Luo’s known stable singularity, though the initial spinning conditions differed slightly in intensity and other variables. Each candidate they found was more unstable than the last — disappearing even more easily when the setup was tweaked in subtle ways.

They also looked at equations describing how a fluid filters through an incompressible porous medium, such as soil or rock, in two dimensions. No one had ever found singularity candidates in this setup. They found four — one stable, three unstable. All involved a similar setup that can be visualized in a thought experiment, although in reality no scientist would be able to adjust the fluid with the endless precision necessary to make the experiment a reality. Imagine an ant farm filled with a sand layer and a rock layer (but no ants). Now add a blob of water, wetting some of the sand. Over time, gravity pulls the water down through the sand, and the blob flattens as it drops. Eventually it smacks into the rock layer, and a property related to the fluid’s density seems to blow up.

Finally, the team returned to the one-dimensional CCF equations, this time finding an even more unstable singularity than before. One way of visualizing this model is to imagine an expansive puddle with two opposing currents. The CCF equations describe the interface between the two currents. If you put a carefully shaped kink into this interface, it sharpens into a singular cusp.

Notably, like the Navier-Stokes equations (and unlike the other two kinds of equations the researchers studied), the CCF equations describe fluids that have a dissipation property akin to viscosity. Thus, each model they studied shows that the PINN method can handle some challenging aspect of the full Navier-Stokes equations, such as higher dimensions and dissipation.

“We are trying to isolate the technical difficulties one by one,” Gómez-Serrano said.

Crucially, none of these new singularity candidates has been proved. But Gómez-Serrano expects that they can be, because the PINN’s approximations are so precise. And the more precise a candidate is, the easier it is to prove that it’s a true singularity. Compared to when the group first unleashed their PINN a few years ago, they’ve gotten about a billion times more accurate.

“The precision is remarkable,” said Eva Miranda, a mathematician at the Polytechnic University of Catalonia in Spain. “The residual errors are so small that these solutions could realistically be used as seeds for future computer-assisted proofs.”

Race To Escape the Boundary

Yongji Wang, a researcher at Google Deepmind, developed many of the technical innovations that made the discovery of new unstable singularity candidates possible.

Sang He

The million-dollar question, or technically the warm-up question to the million-dollar question, is whether the DeepMind collaboration can now use their PINN machinery to find a singularity in the Euler equations — for a fluid that isn’t trapped in a can, a much harder problem. The mathematicians say they will need to upgrade their techniques once again for this wilder and more complicated fluid, but they’re optimistic.

“You’re building a robust tool to find things that are very hard to find,” Buckmaster said.

Other mathematicians, however, point out that past performance does not guarantee future returns, because an unbounded fluid is nothing like a bounded one. “It’s a completely different animal,” said Diego Córdoba, a mathematician at the Institute of Mathematical Sciences in Spain and one of the Córdobas of the CCF model. (His father is the other.)

And so the competition is heating up as researchers hunt for “boundary-free” singularities in the Euler equations and beyond. Córdoba and his collaborator, Luis Martínez-Zoroa of CUNEF University in Spain, have used pencil-and-paper techniques to discover stable singularities in a handful of different fluid setups. They believe they’re on the verge of getting their approach to work for a boundary-free Euler fluid. (Córdoba had fretted that the DeepMind collaboration was about to beat them to that goal, but to his relief, their PINNs aren’t yet powerful enough to crack the problem. The solutions they’ve found, he said, “are unstable, but not that unstable.”)

Related:

1. Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations

2. ‘Next-Level’ Chaos Traces the True Limit of Predictability

3. Mathematicians Coax Fluid Equations Into Nonphysical Solutions

Another competitor, Tarek Elgindi of the University of California, San Diego, has already had success working in boundary-free settings (with other caveats) and is intent on extending his strategy’s reach as well.

It’s not clear which technique, if any, will reach the finish line. “I’ll be very proud and very happy if Javi manages to do it,” said Córdoba, who was Gómez-Serrano’s doctoral adviser. “But I’ll be even happier if we manage to do it.”

If someone can, then it will be on to Navier-Stokes. But despite the recent rash of progress in finding new fluid glitches, mathematicians hesitate to raise their hopes too high.

“You may daydream, but only for a day or two,” Gómez-Serrano said. “You don’t have good enough ideas. Then the daydream stops.”

Clarification: January 13, 2026

This article has been updated to reflect Ching-Yao Lai’s contribution to the development of the PINN method for finding blowup candidates.

Table of Contents

Quanta Magazine: 2026 Jan 20

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

2026 Jan 14

String Theory Can Now Describe a Universe That Has Dark Energy

2026 Jan 12

Cells Use ‘Bioelectricity’ To Coordinate and Make Group Decisions

2026 Jan 9

Using AI, Mathematicians Find Hidden Glitches in Fluid Equations