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2025 Nov 12

- New Proofs Probe Soap-Film Singularities

GEOMETRY | VIEW ON WEBSITE

New Proofs Probe Soap-Film Singularities

By Steve Nadis

November 12, 2025

Mathematicians have broken through a long-standing barrier in the study of “minimizing surfaces,” which play an important role in both math and physics.

The gyroid is an area-minimizing surface that has been used in materials design and drug delivery.

Paul Nylander

Introduction

In the mid-19th century, the Belgian physicist Joseph Plateau — who had been designing and conducting scientific experiments since he was a child — submerged loops of wire in a soapy solution and studied the films that formed. When he bent his wire into a circular ring, a soap film stretched across it, creating a flat disk. But when he dipped two parallel wire rings into the solution, the soap stretched between them to form an hourglass shape instead — what mathematicians call a catenoid. Different wire frames produced all sorts of different films, some shaped like saddles or spiraling ramps, others so complicated they defied description.

These soap films, Plateau posited, should always take up the smallest area possible. They’re what mathematicians call area-minimizing surfaces.

It would take nearly a century for mathematicians to prove him right. In the early 1930s, Jesse Douglas and Tibor Radó independently showed that the answer to the “Plateau problem” is yes: For any closed curve (your wire frame) in three-dimensional space, you can always find a minimizing two-dimensional surface (your soap film) that has the same boundary. The proof later earned Douglas the first-ever Fields Medal.

Since then, mathematicians have expanded on the Plateau problem in hopes of learning more about minimizing surfaces. These surfaces appear throughout math and science — in proofs of important conjectures in geometry and topology, in the study of cells and black holes, and even in the design of biomolecules. “They’re very beautiful objects to study,” said Otis Chodosh of Stanford University. “Very natural, appealing and intriguing.”

Mathematicians now know that Plateau’s prediction is categorically true up through dimension seven. But in higher dimensions, there’s a caveat: The minimizing surfaces that form might not always be nice and smooth, like the disk or hourglass. Instead, they might fold, pinch or intersect themselves in places, forming what are known as singularities. When minimizing surfaces have singularities, it becomes much harder to understand and work with them.

Mathematicians consequently want to know how common such non-smooth minimizing surfaces are, and what properties they might have. If singularities are rare in a given dimension, appearing only under contrived circumstances, then they’ll disappear if you wiggle your wire frame just right. You’ll be left with a smooth minimizing surface that you can study more easily, which will give you the chance to develop a thorough understanding of such surfaces in that dimension.

Soap films stretch within wire frames to form area-minimizing surfaces.

Gift of Wesleyan University

In 1985, mathematicians proved that in eight-dimensional space, singularities can indeed be wiggled away. But in higher dimensions, “all hell breaks loose,” Chodosh said. The singularities become much more difficult to analyze. For nearly 40 years, no one could make much progress on the problem.

That barrier has finally been broken. In 2023, Chodosh — along with Christos Mantoulidis of Rice University and Felix Schulze of the University of Warwick — showed that in dimensions nine and 10, smooth minimizing surfaces are the norm. And earlier this year, the team, joined by Zhihan Wang of Cornell University, showed that the same is true in dimension 11.

The work marks a major advance toward understanding the strange kinds of minimizing surfaces that can arise in higher and higher dimensions. And mathematicians can now use the result to resolve a host of other math problems that have long been limited in scope to dimension eight or below — making those theorems even more powerful.

A Singular History

In 1962, the mathematician Wendell Fleming proved that all minimizing two-dimensional surfaces — any possible soap film that Plateau might have tried to study — must be smooth. Minimizing surfaces with singularities simply don’t exist.

These 2D surfaces exist in our familiar three-dimensional space. But what happens when we move to higher dimensions, where the problem gets harder to visualize? In four dimensions, for instance, the analogue of our wire frame is a 2D surface, and the Plateau problem asks us to find the 3D shape that fills that surface with the smallest possible volume. What might that shape look like? For all we knew, said Brian White of Stanford, “it could be very horrible — fractal-like or extremely irregular.”

Christos Mantoulidis (left), Felix Schulze (center) and Otis Chodosh showed that in dimensions nine and 10, most minimizing surfaces are smooth.

Alison Law; Courtesy of Christos Mantoulidis; Gregor Fels

In the years that followed Fleming’s proof, mathematicians showed that this never happens in four, five, six or seven dimensions. Minimizing surfaces are always smooth. But in 1968, the mathematician Jim Simons constructed a seven-dimensional shape in eight dimensions that had a singularity at just one point. The following year, mathematicians proved that this shape was a minimizing surface, establishing that minimizing surfaces in eight-dimensional space could, in fact, be singular.

The question then became: Just how bad are these singularities, really? Are they rare or common? And can you get rid of them by changing the shape of your wire frame just a bit, in just the right way? “If you want to figure out things about a surface, singularities make it much harder to analyze,” White said. But if singularities arise only rarely, and you can easily nudge them away to get a smooth surface, life becomes much easier — you can use the tools of calculus, for example.

In 1985, Robert Hardt and Leon Simon proved that minimizing surfaces in eight dimensions have this nice property, which mathematicians call generic regularity. But no one could figure out how to adapt their techniques to show whether it exists in higher dimensions.

That’s where things stood for decades — until Chodosh, Mantoulidis and Schulze stepped in.

Into Unfamiliar Domains

The three mathematicians wanted to explore uncharted higher-dimensional realms and understand the nature of their minimizing surfaces, the way a biologist might seek to understand the flora and fauna of a newly discovered island. And so they set out to see whether they could wiggle these singularities away.

The catenoid (left) and the Costa surface are examples of area-minimizing surfaces.

Wikimedia Commons

They started by re-proving Hardt and Simon’s decades-old result in eight dimensions, this time using a different method they hoped to test out. First, they assumed the opposite of what they wanted to show: that when you slightly perturb the wire frame that defines your surface, a singularity (a single point) always persists. Each time you make a perturbation, you get a new minimizing surface that still has a singularity. You can then stack all of these minimal surfaces on top of each other, so that the points where the singularities occur form a line.

But that’s impossible. In 1970, the mathematician Herbert Federer found that any singularity on a minimizing surface in n-dimensional space can have a dimension of at most n − 8. That means that in eight dimensions, any singularity must be zero-dimensional: an isolated point. Lines aren’t allowed. Chodosh, Mantoulidis and Schulze extended Federer’s argument to apply to stacks of surfaces in eight dimensions as well. Yet in their proof, they’d produced a stack of surfaces with just such a line. The contradiction showed that their original assumption was false — meaning that you can perturb the wire frame to get rid of the singularity after all.

Zhihan Wang and his colleagues proved that when singularities form on minimizing surfaces in 11-dimensional space, it’s possible to wiggle them away.

Yueqing Feng

They now felt ready to tackle the problem in nine dimensions. They started their proof in the same way: They assumed the worst, made a series of perturbations, and ended up with an infinite stack of minimizing surfaces that all had singularities. They then introduced a new tool called a separation function, which measures the distance between these singularities. If no perturbation can interfere with the singularity, then this separation function should always stay small. But the trio was able to show that sometimes the function could get large: Some perturbations could make the singularity disappear.

The mathematicians had proved generic regularity for minimizing surfaces in dimension nine. They were able to use the same argument in dimension 10 — but in 11 dimensions, the singularities get even harder to deal with. Their techniques didn’t work for a particular kind of three-dimensional singularity. “There is a zoo of singularity types,” Mantoulidis said. “Any successful argument must be broad enough to handle all of them.”

The team decided to collaborate with Zhihan Wang, who had studied this kind of singularity extensively. Together, they honed their separation function to work in this case, too. They’d solved the problem in dimension 11.

“The fact that they extended [our understanding] by a few dimensions is really fantastic,” White said.

But they’ll likely have to find a different approach to handle higher dimensions. “We need a new ingredient,” Schulze said.

In the meantime, mathematicians expect the new result to help them make progress on other problems in math and physics. The proofs of many conjectures in geometry and topology — about the existence and behavior of shapes with certain curvature properties, for instance — rely on the smoothness of minimizing surfaces. As a result, these conjectures have only been proved up to dimension eight. Now many of them can be extended to dimensions nine, 10 and 11.

Related:

1. Math Duo Maps the Infinite Terrain of Minimal Surfaces

2. Mathematicians Prove Melting Ice Stays Smooth

3. A New Proof Smooths Out the Math of Melting

The same is true for an important statement in general relativity called the positive mass theorem, which claims, loosely speaking, that the total energy of the universe must be positive. In the 1970s, Richard Schoen and Shing-Tung Yau used minimizing surfaces to prove this statement in dimensions seven and below. In 2017, they extended their result to all dimensions. Now, the latest progress on Plateau’s problem offers a new way to confirm the positive mass theorem in dimensions nine, 10 and 11. “They provide another, more intuitive way to do the extension,” White said. “Different proofs give different insights.”

The work could also have plenty of unforeseen consequences. The Plateau problem has been used to study all sorts of other questions, including one related to how ice melts. Mathematicians hope that the team’s new methods will help deepen their understanding of these connections.

As for the Plateau problem itself, there are now two paths forward: Either mathematicians will continue to prove generic regularity in higher and higher dimensions, or they’ll discover that beyond dimension 11, it’s no longer possible to wiggle singularities away. That would be “a bit of a miracle too,” Schulze said — another mystery to unravel. “Either way, it would be very exciting.”

Editor’s Note: Jim Simons founded the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation activities have no influence on our coverage.

2025 Nov 10

- To Have Machines Make Math Proofs, Turn Them Into a Puzzle

Q&A | VIEW ON WEBSITE

To Have Machines Make Math Proofs, Turn Them Into a Puzzle

By John Pavlus

November 10, 2025

Marijn Heule turns mathematical statements into something like Sudoku puzzles, then has computers go to work on them. His proofs have been called “disgusting,” but they go beyond what any human can do.

Marijn Heule stands behind artwork by David Magán at the Opera Gallery in Madrid.

Luis Camacho for Quanta Magazine

Introduction

The mathematical conundrums that Marijn Heule has helped crack in the last decade sound like code names lifted from a sci-fi spy novel: the empty hexagon. Schur Number 5. Keller’s conjecture, dimension seven. In reality, they are (or, more accurately, were) some of the most stubborn problems in geometry and combinatorics, defying solution for 90 years or more. Heule used a computational Swiss Army knife called satisfiability, or SAT, to whittle them into submission. Now, as a member of Carnegie Mellon University’s Institute for Computer-Aided Reasoning in Mathematics, he believes that SAT can be joined with large language models (LLMs) to create tools powerful enough to tame even harder problems in pure math.

“LLMs have won medals in the International Mathematical Olympiad, but these are all problems that humans can also solve,” Heule said. “I really want to see AI solve the first problem that humans cannot. And the cool thing about SAT is that it already has been shown that it was able to solve several problems for which there is no human proof.”

SAT is actually one of the foundations of artificial intelligence itself, although not the kind of AI that makes headlines by mimicking fluent conversation or spooking researchers with supposedly “evil” thoughts. Instead, SAT belongs to the tradition of symbolic artificial intelligence (also known as GOFAI, or “good old-fashioned AI”), which uses hard-coded rules — not the inscrutable interactions within a deep neural network — to produce results. In fact, SAT is about as simple as AI gets, conceptually speaking: It relies on statements that can have only two possible values, true or false, linked together in ironclad chains of logic. If problems can be ground down into these logical “atoms,” computer programs called SAT solvers can often build airtight proofs about them — a process called, appropriately, “automated reasoning.” Those proofs might be long, sometimes too long for humans to ever parse ourselves. But they are sound.

Heule argues that most mathematicians overvalue understanding and undervalue trust.

Luis Camacho for Quanta Magazine

Heule admits his expertise isn’t necessarily in the actual math underlying such proofs, but rather in the puzzlelike thinking required to translate problems into a format that SAT solvers can attack. And his talent came early. “According to my parents, I was able to solve my first puzzle of 100 pieces when I was 1 year old — so, before I could walk,” Heule said. Heule took his first class in satisfiability as an undergraduate at the Delft University of Technology, built his first SAT solver soon after, and earned a doctorate in computer science under the supervision of Hans van Maaren, “one of the founding fathers of the field” of satisfiability research, Heule said. The two men went on to co-author a definitive 1,500-page textbook on SAT.

Heule says he’s long been intrigued by whether computers can solve problems beyond human reasoning. “That’s the question I’m still asking: How do you automate reasoning? Can it be done the way humans do, or does it need to be something completely different? My conclusion, so far, is the latter. All my successes are based on that insight.”

Quanta spoke to Heule about the difference between machine and human reasoning, how SAT’s simplicity is its secret weapon, and why understanding is overrated in mathematics. The interview has been condensed and edited for clarity.

Heule in front of Órgano, a sculpture by Eusebio Sempere.

Luis Camacho for Quanta Magazine

First things first: What is SAT?

It uses something called a propositional formula, which you can imagine as a very big sudoku board. In every cell, you only have two options: only one or zero, standing for true or false. You also have the rules, or constraints, about how many zeros or ones can be in each row or column. Can you put in all the zeros and ones such that all those constraints are satisfied?

Despite its simplicity, this formulation is remarkably powerful. A wide variety of important problems, including hardware and software verification, scheduling, and even areas of pure mathematics, can be translated into SAT.

That just sounds like binary computation. How is SAT-solving different from anything else a digital computer does?

What SAT tools do is fundamentally different from ordinary computation. A standard program takes input and carries out a sequence of operations to produce output. A SAT tool is not computing with the zeros and ones. Instead, it is searching for a combination of them that satisfies all the constraints.

That makes it more like solving a puzzle: You explore possibilities, rule out large portions of the space using clever reasoning, and keep going until you either find a satisfying assignment or conclude that none exists.

What can generative AI add to the power of SAT solvers?

Once you’ve figured out the right representation of a problem, called the encoding, SAT tools are extremely powerful. One of my skills is that I’m really good at coming up with the right representation. I’ve really studied how these tools reason; I know what is required so that you can get everything out of them. But ideally, you wouldn’t need this knowledge. I would really like to take myself out of the equation — then the technology could really shine.

Heule at the Monument to the Spanish Constitution of 1978.

Luis Camacho for Quanta Magazine

So how can we make these effective translations also automatic? If you feed an LLM with lots of good examples of how you should do this, it will come up with something much better than the average user would do. Of course, the challenge here is how to be sure that the translation is correct.

Mathematicians such as Terence Tao think that generative AI can help the research process itself. Where does SAT fit into that picture?

LLMs can generate a lot of plausible-sounding lemmas [statements that are used to prove larger theorems], and automated reasoning can check whether they’re correct or not. But as soon as something is incorrect, the SAT solver can give counterexamples back — ideally, the smallest counterexample. Because the solver is really good at figuring out: OK, that mistake I just made, what did it depend on?

This matters because, in mathematics, counterexamples are incredibly valuable for developing intuition. If you ask a SAT solver, “Does a counterexample exist?” you don’t want it to return an enormous, incomprehensible object. A small counterexample often reveals immediately why the statement fails. Feeding such counterexamples back to an LLM can guide it in the same way a human would: by helping it to refine its suggestions and to propose better lemmas next.

Heule in Castilla, a sculpture by Gustavo Torner.

Luis Camacho for Quanta Magazine

Kind of like a “targeting computer” for the AI?

Yes. I think LLMs can focus on the big picture of a mathematical statement and suggest how to break it into smaller pieces. Once you have that decomposition, automated reasoning can go through the pieces one by one: It can prove a piece, or it can refute it by giving a counterexample. And just as important, it can also check that the pieces really cover everything, so that nothing slips through the cracks.

To make the whole story trustworthy, you want a system like Lean, [a formal proof checker] that checks all the proofs and all the gluing: that the lemmas are correct and that they really add up to the full statement. So the LLM performs the high-level carving, automated reasoning certifies each piece, and Lean ensures the whole thing is watertight. When all of that lines up, the pieces really do add up.

These automated proofs can end up extremely long — far too long for humans to even read, much less understand. Won’t that become even more of a problem if LLMs, which are already hard to understand themselves, enter the picture?

The initial criticism of my work was mostly about this lack of understanding. [The Fields Medal–winning mathematician] Timothy Gowers called my work on the Pythagorean triples problem “the most disgusting proof ever.” But how I feel about this is that understanding in mathematics is highly overrated.

Luis Camacho for Quanta Magazine

The thing is, there’s no mathematician alive who understands all of mathematics. It’s more that there are reputable mathematicians who can say, for each little piece of the puzzle, “OK, I checked. This is correct.” And now others can build on top of that. They overvalue understanding and undervalue trust. I think we should really embrace trust as the key thing to further mathematics, and this is what automation can give you. LLMs can do all of their bullshitting, but as soon as automated reasoning is able to say, “OK, but this part is actually correct, and here’s a proof,” this is actually more trustworthy than most of the pen-and-paper proofs out there. As long as there’s a good trust story, then you can build on top of that and really help mathematics. In such a way, you can see it as a “co-author” because it’s something that follows along with you and tries to spot the gaps in your reasoning.

Related:

1. Mathematical Beauty, Truth and Proof in the Age of AI

2. Why Mathematical Proof Is a Social Compact

3. Proof Assistant Makes Jump to Big-League Math

Suppose that the productive interaction you describe between LLMs and SAT were actually built. What would be left for human mathematicians to do?

When I’ve solved open problems with SAT, it has always been together with mathematicians. They had already spent years thinking hard about these questions. I took their insights and encoded them so that the solver could finish the job. If I’d tried to do it alone, starting from scratch, I would have gotten nowhere.

I expect the future will look similar. My role has been to translate mathematical understanding into a SAT representation, but LLMs could help many more mathematicians learn how to do this themselves, so they don’t need me as the middleman. That’s incredibly exciting. With mathematicians, generative AI and automated reasoning working together, we have a real shot at cracking long-standing open problems. But removing humans from the loop entirely would be a mistake. The creative intuition, the conceptual reframing, that’s still something people are uniquely good at. The magic comes from the collaboration.

2025 Nov 7

- Physicists Take the Imaginary Numbers Out of Quantum Mechanics

QUANTUM PHYSICS | VIEW ON WEBSITE

Physicists Take the Imaginary Numbers Out of Quantum Mechanics

By Daniel Garisto

November 7, 2025

Quantum mechanics has at last been formulated exclusively with real numbers, bringing a mathematical puzzle at the heart of the theory into a new era of inquiry.

The imaginary number i, a mathematical invention that’s defined as the square root of −1, has long been a discomfiting presence in the equations of quantum physics.

Michele Sclafani for Quanta Magazine

Introduction

A century ago, the strange behavior of atoms and elementary particles led physicists to formulate a new theory of nature. That theory, quantum mechanics, found immediate success, proving its worth with accurate calculations of hydrogen’s emission and absorption of light. There was, however, a snag. The central equation of quantum mechanics featured the imaginary number i, the square root of −1.

Physicists knew i was a mathematical fiction. Real physical quantities like mass and momentum never yield a negative amount when squared. Yet this unreal number that behaves as i^2 = −1 seemed to sit at the heart of the quantum world.

After deriving the i-riddled equation — essentially the law of motion for quantum entities — Erwin Schrödinger expressed the hope that it would be replaced by an entirely real version. (“There is undoubtedly a certain crudeness at the moment” in the equation’s form, he wrote in 1926.) Schrödinger’s distaste notwithstanding, i stuck around, and new generations of physicists took up his equation without much concern.

Then, in 2021, the role of imaginary numbers in quantum theory attracted newfound interest. A team of researchers proposed a way to empirically determine whether i is essential to quantum theory or a mere mathematical convenience. Two teams quickly followed up to perform the intricate experiments and found supposedly unequivocal evidence that quantum theory needs i.

This year, however, a series of papers has overturned that conclusion.

Erwin Schrödinger was dissatisfied that his eponymous equation used complex numbers. He hoped to find a replacement for it. But his equation has stuck around.

Public Domain

In March, a group of theorists based in Germany rebutted the 2021 studies, putting forward a real-valued version of quantum theory that’s exactly equivalent to the standard version. Two theorists in France followed up with their own formulation of a real-valued quantum theory. And in September, another researcher approached the question from the perspective of quantum computing and arrived at the same answer: i isn’t necessary for describing quantum reality after all.

Although the real-valued theories avoid explicit use of i, they do retain hallmarks of its distinct arithmetic. This leads some to wonder whether the imaginary aspect of quantum mechanics — or even reality itself — is truly vanquished.

“The mathematical formulation does guide what we infer about the nature of the physical world,” said Jill North, a philosopher of physics at Rutgers University.

Impossible Values

Living in Amsterdam in 1637 at the peak of tulip mania (the Dutch frenzy for flowers which led to impossibly valued tulip bulbs), René Descartes grappled with equations whose solutions also seemed to have impossible values. Using x^3 − 6x^2 + 13x − 10 = 0 as an example, Descartes wrote that its solutions “are not always real; but sometimes only imaginary. … There is sometimes no quantity that corresponds to what one imagines.” The three numbers you can plug in for x are 2, 2 − i and 2 + i. The latter two numbers, each of which has both a real part and an imaginary part in the form a + ib, came to be called complex numbers.

Descartes viewed them with derision, but complex numbers were later adopted for their utility in fields as diverse as geometry, optics and signal analysis.

Schrödinger grudgingly acknowledged their ease of use in quantum theory. His equation governs the evolution of the wave function, an entity representing the possible quantum states of an object. (These states can interfere destructively and constructively like waves.) Schrödinger’s wave function was complex-valued, even though actual measurements of quantum systems always return real values. “Quantum theory really is the first physical theory where the complex numbers seem to be right smack in the middle of the theory,” said Bill Wootters, a quantum information theorist at Williams College.

One way to represent a complex number like a + ib is as a point on a plane, where a is the position on the x-axis (which can be thought of as the real number line) and b is the position on an imaginary y-axis. Each complex number is an arrow, called a vector, pointing from the origin to the complex coordinate (a, b). These complex vectors obey the unusual math of complex numbers: Multiplying by i, for example, rotates the vector 90 degrees.

These properties made them a natural fit for the quantum states of the wave function — also vectors obeying odd combination rules.

In a 2021 paper in Nature, Marc-Olivier Renou (top), Nicolas Gisin and six co-authors devised an experiment that would falsify any quantum theory based on real numbers. The experiment was subsequently performed. But work this year has shown that the experiment rested on an objectionable assumption.

In a 2021 paper in Nature, Marc-Olivier Renou (left), Nicolas Gisin and six co-authors devised an experiment that would falsify any quantum theory based on real numbers. The experiment was subsequently performed. But work this year has shown that the experiment rested on an objectionable assumption.

Julie Dugast; Carole Parodi

Physicists tried now and again to define equivalent vectors with real numbers. In 1960 the Swiss physicist Ernst Stueckelberg developed a real-valued quantum mechanics that mapped the wave function from a complex-valued space to a real one, using a few tricks to get real numbers to mimic the rotations around an imaginary axis. But where complex-valued theory was compact, the real-valued theory was cumbersome. The wave function for two particles involves four complex numbers; extending Stueckelberg’s formulation to two particles increases the description to 16 real numbers.

The clunkiness of real-valued quantum theories notwithstanding, in 2008 and 2009, two groups showed it was possible to use these theories to re-create the standard results of the Bell test — a crucial probe of quantum theory’s properties. “For a lot of things, you actually can get away with the real theory,” Wootters said. But would the real-valued theory always produce the same results?

Key Assumptions

In 2021, a group of researchers including Nicolas Gisin, a physicist at the University of Geneva, realized that they could test the limits of real-valued theories by making a Bell test more complicated.

Canonically, Bell tests involve the creation of a pair of “entangled” particles: particles whose possible states are linked, such as photons with correlated polarizations. The particles are separated and sent to two participants nicknamed Alice and Bob, who measure their polarizations and compare notes.

From top: Michael Epping, Dagmar Bruß, Anton Trushechkin, Pedro Barrios Hita and Hermann Kampermann made the case in a recent paper that “the use of complex numbers is a matter of convenience,” rather than of necessity.

From top: German Aerospace Center; Nicolas Stumpe; Courtesy of Anton Trushechkin; Ghislane Coulter-de Wit

Clockwise from top left: Michael Epping, Dagmar Bruß, Anton Trushechkin, Hermann Kampermann and Pedro Barrios Hita made the case in a recent paper that “the use of complex numbers is a matter of convenience,” rather than of necessity.

From top left: German Aerospace Center; Nicolas Stumpe; Courtesy of Anton Trushechkin; Ghislane Coulter-de Wit

Gisin’s team instead considered a bespoke Bell test with two separate sources of entangled particles and three participants: Alice, Bob and Charlie. Running the numbers, they found that there was a ceiling on how correlated the polarizations of the entangled particles could be for a real-valued quantum theory, and a different, higher ceiling for a complex-valued quantum theory. This was no longer a matter of calculational ease or philosophy: An empirical test existed that could rule out real-valued quantum mechanics.

Soon after, a group at the University of Science and Technology of China (USTC) in Hefei ran the protocol and found that the observed correlations between entangled photons far exceeded the limit for the real-valued theory. Complex numbers seemed essential for describing these quantum states.

But the statistically overwhelming result didn’t quell the questions.

“Complex numbers are just two real numbers with some calculation rules,” said Michael Epping, a physicist at the German Aerospace Center and a co-author on the new German paper. “Why shouldn’t you be able to describe quantum mechanics just using real numbers?”

“Quantum theory does not need complex numbers,” the physicists Timothée Hoffreumon (top) and Mischa Woods argued in the title of a recent paper.

“Quantum theory does not need complex numbers,” the physicists Timothée Hoffreumon (left) and Mischa Woods argued in the title of a recent paper.

Courtesy of Timothée Hoffreumon; Dr. Samuel Edwin Slezak

Mischa Woods of the École Normale Supérieure in Lyon and Timothée Hoffreumon of Paris-Saclay University, co-authors on the new French paper, were also dubious. In the 2021 paper, Gisin and his colleagues made a critical assumption about the “tensor product,” a mathematical operation that wrangles the complex vectors describing Alice’s particle and Charlie’s particle into one entangled state. Gisin and his co-authors assumed that a real-valued version of quantum theory would use the same mathematical formulation of the tensor product to combine states.

But the French and German teams argue that that form of tensor product is the wrong rule for a real-valued theory. By way of analogy, in flat space, the hypotenuse of a right triangle is always a^2 + b^2 = c^2. But that rule doesn’t hold for a triangle in curved space, like one that’s stamped on the surface of a sphere. A recent argument, adopted by the two teams, is that that the standard tensor product is a specific case of a more general class of vector-combination rules. They developed different combination rules to create real-valued quantum theories that give exactly the same predictions as a complex quantum theory.

A new development in quantum computing also shows how to avoid complex numbers. Quantum computers use “logic gates” to manipulate quantum bits. One common logic gate, called a T gate, rotates the vector representing the quantum bit’s state around the complex plane. In September, Craig Gidney, a quantum computing expert at Google Quantum AI, found a way to eliminate T gates from any quantum algorithm — numerically proving that quantum computing doesn’t require complex numbers.

What Comes Naturally

Jill North, a philosopher of physics at Rutgers University, asks why complex numbers are so well-suited to quantum mechanics.

Tori Repp

The feasibility of real-valued quantum theory raises provocative questions. Foremost among them: Why is it so much more complicated? The question has been with us since the birth of quantum mechanics; Schrödinger attempted to work with a real-valued wave equation but  turned to a complex one because it was “extraordinarily much simpler for computational purposes,” as he described it in his notes.

Today it seems that quantum theory does not explicitly need i, but there may still be something natural about the simplicity Schrödinger found. “Complex quantum theory, with its natural tensor product, remains far more concise, elegant and mathematically straightforward,” said Chao-Yang Lu, an experimental physicist at USTC who was part of the team that carried out the bespoke Bell test proposed by Gisin’s team.

“Even when you translate quantum theory into real numbers, you still see the hallmark of complex-number arithmetic,” Wootters said.

Even those who emancipated the theory from complex numbers admit that the latter are a natural fit. The real-valued theories do not contain i, but they copy its ability to rotate vectors. We “simulate complex numbers by means of real numbers,” Anton Trushechkin, a physicist at Heinrich Heine University Düsseldorf and co-author on the German paper.

Related:

1. ‘It’s a Mess’: A Brain-Bending Trip to Quantum Theory’s 100th Birthday Party

2. ‘Metaphysical Experiments’ Probe Our Hidden Assumptions About Reality

3. How Bell’s Theorem Proved ‘Spooky Action at a Distance’ Is Real

North, the philosopher of physics, agrees with Lu. “Even if complex numbers aren’t truly necessary, they do give rise to a formulation that seems particularly well suited to quantum mechanics,” she said. Her goal is to “pinpoint something peculiarly quantum mechanical” that contributes to that good fit. One possibility might be spin, a property of quantum particles that has no classical counterpart.

The lingering essence of complex numbers in the real-valued theories gives some researchers pause; reports of i’s demise may be somewhat exaggerated. “You can write them down whichever way you like, but it’s unavoidable that they have to multiply exactly as though they were complex numbers,” said Vlatko Vedral, a physicist at the University of Oxford. His preference would be to find simpler axioms for quantum mechanics — intuitive principles that would let theorists re-derive the theory in a new form altogether.

“We really don’t have a single alternative to how quantum mechanics was already done 100 years ago,” he said. “And the question is, why? Why can’t we go beyond this?”

2025 Nov 5

- How Your Brain Creates ‘Aha’ Moments and Why They Stick

NEUROSCIENCE | VIEW ON WEBSITE

How Your Brain Creates ‘Aha’ Moments and Why They Stick

By Nora Bradford

November 5, 2025

A sudden flash of insight is a product of your brain. Neuroscientists track the neural activity underlying an “aha” and how it might boost memory.

Irene Pérez for Quanta Magazine

Introduction

Here are three words: pine, crab, sauce. There’s a fourth word that combines with each of the others to create another common word. What is it?

When the answer finally comes to you, it’ll likely feel instantaneous. You might even say “Aha!” This kind of sudden realization is known as insight, and a research team recently uncovered how the brain produces it, which suggests why insightful ideas tend to stick in our memory.

Maxi Becker, a cognitive neuroscientist at Duke University, first got interested in insight after reading the landmark 1962 book The Structure of Scientific Revolutions by the historian and philosopher of science Thomas Kuhn. “He describes how some ideas are so powerful that they can completely shift the way an entire field thinks,” she said. “That got me wondering: How does the brain come up with those kinds of ideas? How can a single thought change how we see the world?”

Such moments of insight are written across history. According to the Roman architect and engineer Vitruvius, in the third century BCE the Greek mathematician Archimedes suddenly exclaimed “Eureka!” after he slid into a bathtub and saw the water level rise by an amount equal to his submerged volume (although this tale may be apocryphal). In the 17th century, according to lore, Sir Isaac Newton had a breakthrough in understanding gravity after an apple fell on his head. In the early 1900s, Einstein came to a sudden realization that “if a man falls freely, he would not feel his weight,” which led him to his theory of relativity, as he later described in a lecture.

Insights are not limited to geniuses: We have these cognitive experiences all the time when solving riddles or dealing with social or intellectual problems. They are distinct from analytical problem-solving, such as the process of doing formulaic algebra, in which you arrive at a solution slowly and gradually as if you’re getting warmer. Instead, insights often follow periods of confusion. You never feel as if you’re getting warmer; rather, you go from cold to hot, seemingly in an instant. Or, as the neuropsychologist Donald Hebb, known for his work building neurobiological models of learning, wrote in the 1940s, sometimes “learning occurs as a single jump, an all-or-none affair.”

According to legend, the Greek mathematician Archimedes exclaimed “Eureka!” after his body caused the water level to rise in his bath, suggesting a way to calculate volume.

Ann Rosan Picture Library

An abrupt cognitive shift in how the mind understands information is known as a representational change. Although researchers have inferred sudden shifts in understanding from the behavior of subjects, they have not pinned down how the brain supports representational change.

During moments of insight, representational change typically occurs, said John Kounios, a cognitive neuroscientist at Drexel University and co-author of the book The Eureka Factor: Aha Moments, Creative Insight, and the Brain. “The question is: How is it occurring?”

Insightful Activity

While at Humboldt University of Berlin, Becker set out to uncover this neural signature of insight. Given that it’s nearly impossible to fabricate life-changing, field-altering insights in the lab, her team needed to identify a simple task that could produce a sudden feeling of understanding rather than a slowly unfolding solution.

They turned to abstracted black-and-white pictures called Mooney images, which are made by cranking up the contrast on a photograph all the way so that the subjects — a dog or a coffee mug, for example — are unrecognizable at first. The pictures pose a challenge for human brains, which typically identify objects by piecing together their different parts. But if given enough time with a Mooney image, even a few seconds, the brain can rearrange the contours to recognize the pictured object — and trigger the insightful “aha” feeling, a representational change.

Click to reveal

To induce moments of insight in the lab, researchers used Mooney images, which have their contrast turned up to obscure the pictured object. Can you identify the object in this Mooney image? Click on the right square to reveal the answer.

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Courtesy of Maxi Becker

Click to reveal

To induce moments of insight in the lab, researchers used Mooney images, which have their contrast turned up to obscure the pictured object. Can you identify the object in this Mooney image? Tap on the bottom square to reveal the answer.

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Courtesy of Maxi Becker

Over the course of two days, Becker had study participants lie in a functional magnetic resonance imaging (fMRI) scanner, which detects blood flow in the brain as a proxy for neural activity, and view a series of 120 Mooney images. After 10 seconds of viewing a single image, the participant would indicate whether they recognized the pictured object. If they did, they would then answer a series of questions about the suddenness, positive emotion and certainty associated with their experience — three measures that have been linked to moments of insight.

Becker and her team then used neural networks to parse the fMRI data, looking to identify consistent changes in brain activity shared by participants when they correctly recognized Mooney images. They observed that when a participant noticed a hidden object, brain activity increased in the ventral occipitotemporal cortex (VOTC), a region responsible for recognizing visual patterns in the environment; the amygdala, which processes both positive and negative emotions; and the hippocampus, a deep-brain structure involved in handling memories. This activity was greater for experiences rated more certain and emotionally positive — in other words, more insightful ones.

The hippocampus is sometimes known as the brain’s “mismatch detector,” Becker said, because it reacts when an input doesn’t align with expectations. In this case, insight leads a once-meaningless image to gain meaning, going against the brain’s predictions.

Maxi Becker, a cognitive neuroscientist at Duke University, explores what insight looks like in the brain.

Courtesy of Maxi Becker

These regions — the hippocampus, amygdala and VOTC — create “a plausible network of brain areas” behind representational change, said Kounios, who was not involved in the study. These findings finally “connect the psychological theory with the neural mechanism,” said Yuhua Yu, a postdoctoral researcher in neuroscience at the University of Arizona, who was also not involved with the study.

Becker and her team likely found representational change in the VOTC because of the visual nature of their stimuli. If they had chosen another type of stimulus, like words, the change would probably have appeared in language-processing areas of the brain.

Once the team had figured out which brain areas support insight, they wanted to probe whether these regions might be working together to create a lasting memory.

A Memory Boost

Since they began investigating insight, researchers have suspected that such experiences might boost memory. In his 1949 book The Organization of Behavior, Hebb wrote that “whatever insight is, we now know that it continually affects the learning of the adult mammal.” Insight not only feels notable or salient in the moment but also helps us retain new information as memory.

This memory boost, which became known as the insight-memory advantage, has since been studied in many types of problem-solving, including the unraveling of magic tricks and puzzles. “When you have an insight, you tend to be better able to remember the solution,” Becker said, compared to when you resolve a problem more gradually. She wanted to understand why.

Click to reveal

Can you identify the object in this Mooney image? Click to reveal the answer.

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Courtesy of Maxi Becker

Click to reveal

Can you identify the object in this Mooney image? Tap to reveal the answer.

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Courtesy of Maxi Becker

A few days after the initial experiment, the team tested participants’ memory by having them look at more Mooney images online, including some they had seen before. Participants were better able to remember prior images that they had rated highly on the three aspects of insight. This suggested that the insight-memory advantage was real, but the team wanted to see what was going on under the hood. Did brain activity during insight predict better memory five days later?

The researchers found that the larger the activity boost in both the VOTC and the hippocampus during the initial insight, the better participants remembered the Mooney images. The big change in brain activity likely makes the experience more salient, Becker said, and salient experiences are known to better encode long-term memories.

While insight creates stronger memories of an idea, it doesn’t mean the idea is correct. Previous work has shown that the quicker, more certain and more pleasurable a solution feels, the more likely it is to be correct — but false insights can and do exist. In Becker’s study, participants wrongly identified the subjects of more than half the Mooney images they saw. Of those incorrect trials (which the researchers excluded from the analysis), the participants reported experiencing insight 40% of the time. In comparison, correct trials were accompanied by feelings of insight 65% of the time.

These kinds of studies of insight in the lab will set researchers up to look at how it functions in the real world. Once we decompose insight into “very simple tasks that we already understand well,” Becker said, we can “move on to more complex, truly creative tasks.”

Insight Into the Future

Yuhua Yu, a postdoctoral researcher in neuroscience at the University of Arizona, is fascinated by the role that insight plays in the creative process.

Courtesy of Yuhua Yu

As a self-described uncreative person, Yu has been particularly fascinated by insight’s role in the creative process. Creativity is “like a magic power,” she said. “A really big creative idea is [often] associated with insight because a creative idea is in some way a leap in your cognitive world, and a leap will often elicit an insight or ‘aha’ feeling.”

However, Yu is finding that insight’s role in creativity might depend on the kind of problem a person is solving. In a recent study, she asked participants to come up with metaphors for scientific concepts and asked whether they used insight as they did so. The insight-driven metaphors weren’t more or less creative than those created through analytic thinking, she found — and the participants were more likely to remember the science concepts behind the latter.

This may be because, unlike the task of seeing a hidden object in a Mooney image, creating a metaphor tends to rely on slower cognitive problem-solving rather than sudden moments of insight, Becker suggested. The effects of insight therefore likely depend on the context.

Next, Yu wants to investigate insight in more contexts. “Most of the insight research is looking at insight in the problem-solving context and in the lab setting,” Yu said. She hopes that researchers will begin investigating “insight within many other domains, like in psychotherapy, in meditation, even in psychedelic experiences.”

Related:

1. What Your Brain Is Doing When You’re Not Doing Anything

2. How ‘Event Scripts’ Structure Our Personal Memories

3. The Secret Math Behind Mind-Reading Magic Tricks

Beyond offering a better understanding of how the human brain learns, these findings could have applications in classrooms. Kounios believes that applying insight-boosting strategies to teaching could lead to better learning outcomes for students. Insight seems to be a powerful and positive experience that generates accurate solutions, confidence in our answers and strong memories.

“It’s very intensive for a teacher to do this, but a lot of really good teachers try to get the students to have the insights themselves about how something works, and that will burn it into their memories,” Kounios said. “Another aspect of that [is], it’s very motivating, too.”

It’s a nice feeling when your brain suddenly comes up with an answer. Perhaps you’ve even experienced that feeling since reading this piece’s first sentence. Maybe it even hit you like an apple on the head.

2025 Nov 3

- What Is a Manifold?

TOPOLOGY | VIEW ON WEBSITE

What Is a Manifold?

By Paulina Rowińska

November 3, 2025

In the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics.

Mark Belan/Quanta Magazine

Introduction

Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.

The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.

This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.

Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”

So what are manifolds, and what kind of vocabulary do they provide?

Ideas Taking Shape

For millennia, geometry meant the study of objects in Euclidean space, the flat space we see around us. “Until the 1800s, ‘space’ meant ‘physical space,’” said José Ferreirós, a philosopher of science at the University of Seville in Spain — the analogue of a line in one dimension, or a flat plane in two dimensions.

In Euclidean space, things behave as expected: The shortest distance between any two points is a straight line. A triangle’s angles add up to 180 degrees. The tools of calculus are reliable and well defined.

But by the early 19th century, some mathematicians had started exploring other kinds of geometric spaces — ones that aren’t flat but rather curved like a sphere or saddle. In these spaces, parallel lines might eventually intersect. A triangle’s angles might add up to more or less than 180 degrees. And doing calculus can become a lot less straightforward.

The mathematical community struggled to accept (or even understand) this shift in geometric thinking.

But some mathematicians wanted to push these ideas even further. One of them was Bernhard Riemann, a shy young man who had originally planned to study theology — his father was a pastor — before being drawn to mathematics. In 1849, he decided to pursue his doctorate under the tutelage of Carl Friedrich Gauss, who had been studying the intrinsic properties of curves and surfaces, independent of the space surrounding them.

Bernhard Riemann is widely considered one of the greatest mathematicians in history. His work revolutionized geometry, topology, number theory and more.

Public Domain

In 1854, Riemann was required to deliver a lecture to secure a teaching position at the University of Göttingen. His assigned topic: the foundations of geometry. On June 10, despite a fear of public speaking, he described a new theory in which he generalized Gauss’ ideas about the geometry of surfaces to an arbitrary number of dimensions (and even to infinite dimensions).

Gauss was immediately impressed with the lecture, which involved not just math but also philosophy and physics. But most mathematicians found Riemann’s ideas too vague and abstract to be of much use. “Many scientists and philosophers were saying, ‘This is nonsense,’” Ferreirós said. And so, for decades, the work was largely ignored. Riemann’s lecture didn’t appear in print until 1868, two years after his death.

But by the end of the 19th century, mathematical greats like Henri Poincaré had recognized the importance of Riemann’s ideas. And in 1915, Albert Einstein used them in his general theory of relativity, bringing them out of the realm of philosophical abstraction and into the real world. By the middle of the 20th century, they had become a mathematical staple.

Riemann had introduced a concept that could encompass all possible geometries, in any number of dimensions. A concept that would change how mathematicians view space.

A manifold.

Charted Territory

The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”

A manifold is a space that looks Euclidean when you zoom in on any one of its points. For instance, a circle is a one-dimensional manifold. Zoom in anywhere on it, and it will look like a straight line. An ant living on the circle will never know that it’s actually round. But zoom in on a figure eight, right at the point where it crosses itself, and it will never look like a straight line. The ant will realize at that intersection point that it’s not in a Euclidean space. A figure eight is therefore not a manifold.

Similarly, in two dimensions, the surface of the Earth is a manifold; zoom in far enough anywhere on it, and it’ll look like a flat 2D plane. But the surface of a double cone — a shape consisting of two cones connected at their tips — is not a manifold.

Mark Belan/Quanta Magazine

Manifolds address a problem that mathematicians would otherwise have to deal with: A shape’s properties can change depending on the nature and dimension of the space it lives in (and how it sits in that space). For instance, lay a piece of string on a table, and connect its ends without lifting it. You’ll get a simple loop. Now hold the string in the air and tie its ends together. By considering the string in three dimensions, you can pass it over and under itself before you connect the ends, creating all sorts of knots beyond the simple loop. They all represent the same one-dimensional manifold — the looped string — but they have different properties when considered in two versus three dimensions.

Mathematicians avoid such ambiguities by focusing on the manifold’s intrinsic properties. The defining property of manifolds — that at any point, they look Euclidean — is immensely helpful on that front. Because it’s possible to think about any small patch of the manifold in terms of Euclidean space, mathematicians can use traditional calculus techniques to, say, compute its area or volume, or describe movement on it.

If I know Cyrillic, do I know Russian? No. But try to learn Russian without learning Cyrillic.

Fabrizio Bianchi, University of Pisa

To do this, mathematicians divide a given manifold into several overlapping patches and represent each with a “chart” — a set of some number of coordinates (equal to the manifold’s dimension) that tell you where you are on the manifold. Crucially, you also need to write down rules that describe how the coordinates of overlapping charts relate to one another. The collection of all these charts is called an atlas.

You can then use this atlas — whose charts translate smaller regions of your potentially complicated manifold into familiar Euclidean space — to measure and explore the manifold one patch at a time. If you want to understand how a function behaves on a manifold, or get a sense of its global structure, you can break the problem up into pieces, solve each piece on a different chart, in Euclidean space, and then stitch together the results from all the charts in the atlas to get the full answer you’re seeking.

Today, this approach is ubiquitous throughout math and physics.

Manifold Uses

Manifolds are crucial to our understanding of the universe, for one. In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.

Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”

Related:

1. In Topology, When Are Two Shapes the Same?

2. Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions

3. Dimension 126 Contains Strangely Twisted Shapes, Mathematicians Prove

Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.

Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.

Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”

Table of Contents

Quanta Magazine: 2025 Nov 12

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2025 Nov 3

What Is a Manifold?