Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Terence Tao: AI turns beautiful mathematics into ‘toxic waste’

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Last Updated

29/10/2026

Terence Tao: AI turns beautiful mathematics into ‘toxic waste’

By: James Tapper
The Observer: 24/10/2026

James Tapper: James Tapper has worked for The Observer since 2015 and was highly commended at the British Journalism Awards in 2021 for his reporting on the Covid pandemic. He has been a journalist and troublemaker since 1999, writing for other London newspapers including the Mail on Sunday, and working as a foreign correspondent in India for the Global Post.

...

The top mathematician says Silicon Valley’s rush to solve long-investigated conundrums excludes ‘elegant solutions’ and reduces maths to raw training material.


Joe Leavenworth

If tech bros don’t understand why Terence Tao, probably the world’s leading mathematician, is horrified that OpenAI published the answers to 722 unsolved maths problems, they should think about Douglas Adams.

In Adams’s book The Hitchhiker’s Guide to the Galaxy, the Deep Thought supercomputer took 7.5m years to answer the ultimate question of life, the universe and everything and came up with 42. Then it asked for another 10m years to work out what the question was.

Adams’s joke has become an accidental satire on AI because Tao and his colleagues are now wading through OpenAI’s answers to work out how it reached them.

At 51, Tao could be forgiven for not wanting to tidy up after Silicon Valley, although he has worked with OpenAI and still has contacts there. The Australian, speaking via video from California where he is professor of mathematics at UCLA, is one of the youngest winners of the Fields Medal (the maths Nobel), and has a slew of other prizes honouring his contributions. He attempts to distil the field in his book, Six Maths Essentials: How to Model the World, which is published on Tuesday.

But perhaps the worst part for Tao is that he believes the OpenAI didn’t even make a new model to solve these difficult problems. Conundrums that people have spent careers addressing were merely raw material for training its models.

AI companies are “racing towards the most powerful model”, Tao says, so they need to “train and benchmark and give the AI lots of tests to see whether what they’re doing is improving their capability”. The practical problem is that the newest AI models are beating every test.

“Maths problems are almost the only thing left for them to train on, so suddenly all the big prizes and goals in our profession are benchmarks for them,” Tao says. “And as a byproduct, they’re generating these terrible proofs. Great results, but they don’t care how elegant the solutions are, or who understands them, or what to do next.”

An updated Deep Thought would probably involve a share offering; Tao says that if OpenAI or its main rival Anthropic, which makes Claude, suspect the other is close to a breakthrough “they just go insane”.

[OpenAI] really, really wanted to beat Anthropic. It’s insane’ - Terence Tao

The dump of mathematical “toxic waste”, as Tao calls it, began when rumours surfaced online that Anthropic researchers were close to solving one of the seven $1m Millennium Prize Problems set by the Clay Mathematics Institute.

“My contacts at OpenAI said this became code red priority number one,” Tao says. “They dumped tens of millions of dollars. They set up this 10,000 AI agent swarm to attack these problems.” OpenAI focused its agents on the problem of the Navier-Stokes equations, which seem to perfectly describe the motion of a fluid, yet might fail in some conditions.

“They even suspended signups, so that for a few days you could not sign up to their premium AI service because they were redirecting all the compute for their flagship services to this one problem,” Tao says. “They really, really wanted to beat Anthropic. It’s insane.”

On 8 September, OpenAI published a proof that seemed to show that Navier-Stokes did fail, although reviews suggest there may be some errors. The backlash from the maths community left OpenAI with a problem, Tao says. “They had been testing their models on these problems for quite a while, and they actually had accumulated a bunch of very badly polished solutions to these problems. They didn’t know what to do with them.

Joe Leavenworth

“[When] you generate toxic waste, you have no good options. You can you can sit on it forever, or you can just dump it out all at once, or you can dole it out in drabs, or you can pay people to process it.”

Anthropic is taking a different approach, by paying mathematicians to write up results to its “trove of questions”, he says.

That creates a different problem. For years maths was a field with equality of arms; practitioners needed only some chalk and a blackboard, unlike their scientific counterparts – think of Cern or the Hubble space telescope. Now doing cutting edge maths research means getting access to a super-AI model. “They get to pick who writes up the paper.”

In the aftermath of the “toxic waste” dump, Tao became a meme: “Put the fries in the bag, Terence”, an allusion to mathematicians’ impending careers at McDonald’s. He laughs when I mention it.

“You know, social media is going to social media,” he says. It’s understandable that people “who had a bad experience with mathematics or intellectualism in general [would experience some] schadenfreude.”

The memesters miss the point though. Tao agrees that maths culture has to change – he hopes it becomes more collaborative, less focused on giving credit for solving a problem and more on explaining it.

“Maths is half science, half humanities,” he says. “We care about the outcome but also the process, how we got there, what we learned on the way.” He still uses theorems discovered by Euclid and Pythagoras 2,500 years ago. The story of solving a problem, the history of the arguments and the elegance of the solution are essential to the beauty of mathematics. Otherwise, “you can go to the gym and get a robot to lift your weights, but what’s the point?”

Six Maths Essentials is a short, pacy story about the evolution of numbers, algebra, geometry, probability, analysis and dynamics. The invention of the infinity symbol by John Wallis in 1657, and the decision to use the letter i to refer to the impossible square root of -1 (an imaginary number) created tools that allowed mathematicians to do things that had been out of reach. Is there a tool he’d like someone to create?

Joe Leavenworth

There is. We can describe shapes well – their volume, their width, their curvature. “But one notion which is still very elusive is how complex is the shape. Intuitively, some shapes look very simple, and some look very complicated.” Maths lacks the language to describe in detail the complexity of, say, a tennis ball compared to a steel wool scourer.

Large language models have thrown this problem into focus because they represent words as coordinates in space. The corpus of English becomes a vast cloud of points, plotted in thousands of dimensions, an “insanely complicated shape that there’s no way you could look at”.

“It turns out that this weird object has a simpler shape than we realised,” Tao says. “Not simple to humans, but AI machine learning algorithms can detect the shape, and they can use it to predict, given a portion of a sentence, what the next word is, and the next word after that, next word after that, and suddenly they can start speaking in fluent English.”

English has a similar shape to French, but quite a different one to Finnish. “There’s all these recipes in there that we do not know, in part because mathematicians have not actually developed a theory of complexity of shapes that would explain this. If we had, maybe we would understand [why] there are some languages that AIs pick up better and faster than others.”

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The Riemann Hypothesis, Explained

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Very cool.

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Last Updated

24/10/2026

The Riemann Hypothesis, Explained

By: Alex Kontorovich
Quanta Magazine: 05/01/2021

Alex Kontorovich: A professor of mathematics at Rutgers University.

...

The Riemann Hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. The person who solves it will win a $1 million prize from the Clay Institute of Mathematics. So, what is the Riemann Hypothesis? Why is it so important? What can it tell us about the chaotic universe of prime numbers? And why is its proof so elusive? Alex Kontorovich, professor of mathematics at Rutgers University, breaks it all down in this comprehensive explainer. 

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Graduate Student Proves a Quantum Uncertainty Principle for Fractals

Mike's Notes

This Quanta Magazine article about Alex Cohen's discovery is also available as a podcast. His work builds on the work of many who came before him.

The discovery connects to porosity (percolation?), the Uncertainty Principle via Fourier Transform, etc., and might be very useful to add to Pipi. I need to investigate it further.

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  • Fractal uncertainty in higher dimensions v2, by Alex Cohen, 5 Oct 2024. arxiv.org/abs/2305.05022.

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Last Updated

18/10/2026

Graduate Student Proves a Quantum Uncertainty Principle for Fractals

By: Shalma Wegsman
Quanta Magazine: 12/08/2026

Shalma Wegsman: Staff Writer.

...

The math, which combines chaos, quantum theory, and infinitely complex fractal structures, has been called a “foundational result.”

Introduction

At the quantum scale, tiny particles behave in bizarre ways. One reason for this is the uncertainty principle, which says that the more you know about where a quantum particle is, the less you can know about how fast it’s moving, and vice versa. Recently, this rule got a rare upgrade.

The new uncertainty principle relates to fractals, shapes that remain equally complex no matter how much you zoom in on them.

Around a decade ago, Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, was studying whether quantum particles behave differently than ordinary particles when put into the same chaotic situations. Sometimes, an object moving chaotically can become trapped into following a fractal-like path forever. Could quantum particles do the same?

The Quanta Podcast

The uncertainty principle tells us that the more you know about where a quantum particle is, the less you can know about how fast it’s moving, and vice versa. Recently, this rule got a rare upgrade.

Quantum particles tend to spread out like waves, which blurs their exact location. To figure out whether quantum particles blur too much to take on these intricate trapped paths, Dyatlov needed a new uncertainty principle — one that could tackle fractals.

In 2016, with key ideas from Jean Bourgain — a renowned mathematician who died shortly after this work — Dyatlov proved the fractal uncertainty principle for one-dimensional fractals, which look like jagged lines. These lines can represent the paths taken by objects moving in two dimensions, like balls traveling around a billiard table. That fall, Dyatlov and Bourgain gathered mathematicians from around the world in New Jersey for a workshop, hoping to extend the proof to higher dimensions. An extended proof could be used to study the three-dimensional world and would become a universal mathematical tool in its own right.

But the task proved too difficult. By the end of the workshop, “nobody really believed that it could be done,” said one of the attendees, Frédéric Naud, a mathematician from Sorbonne University.


Alex Cohen’s proof of the fractal uncertainty principle has already proven influential across math.

Hertz Foundation

It wasn’t until years later that Alex Cohen, while a doctoral student at MIT, finally made a breakthrough. In a paper published in 2025 in the Annals of Mathematics, widely considered to be the field’s top journal, he extended the fractal uncertainty principle to all higher dimensions. The result became Cohen’s thesis and earned him an assistant professorship at New York University at the age of 25.

The fractal uncertainty principle is “a foundational result,” said Peter Sarnak of the Institute for Advanced Study — “a pretty remarkable achievement for a guy in his thesis.”

Already, this principle has revealed a new deep way that quantum particles differ from classical ones.

Pinball Wizard

Though the uncertainty principle may seem strange in the context of particles, it can also crop up in less mysterious forms. The briefer a sound, the less sure you can be about the tones that make it up. A short radar pulse can accurately locate a submarine, but it takes a longer signal to determine where it’s moving.

All these uncertainty principles, including the quantum one, arise from the same mathematical source. This deeper mathematical uncertainty principle applies broadly to any function — or any curve, roughly speaking, no matter how bumpy and wild it looks. It comes from an equation invented in the 19th century called the Fourier transform. Named for the Frenchman Joseph Fourier, the Fourier transform decomposes any function into a set of simple waves, each with a different frequency, or tone. Add those simple waves together, and you’ll get back your original function.

Mark Belan/Quanta Magazine

The uncertainty principle comes built in. A simple sine wave, which spreads infinitely throughout space, has one definite frequency, so its Fourier transform is a single peak. But the snap of a snare drum looks like a pulse as it travels through the air. Its Fourier transform has a wide spread of frequencies — a sound with no discernible pitch.

In general, the narrower a function, the more spread out its Fourier transform must be, and vice versa. Or, in other words, the more certain you are about where a function peaks in space, the less certain you can be about its frequency, or how it’s changing in time.

In quantum mechanics, a particle is described mathematically as a wave, with peaks in places where it’s most likely to be found. The Fourier transform of that wave describes the particle’s motion. This is why a quantum particle’s position and momentum can’t both be precisely known at once: They are related by a Fourier transform.

But what if you take the Fourier transform of a function that looks like a fractal?

To make sense of the question, it helps to imagine a simple kind of fractal: Start with a line and cut it into three segments. Then delete the center segment. Repeat the process: Cut each remaining segment into three, remove the center, and so on. The result is a fractal set called the Cantor set. The set has the feature of being what mathematicians call porous — it’s full of holes, like a sponge, at every scale.

Now imagine that this Cantor set lives on the x-axis, and at each point on this set is a peak representing a frequency.

The fractal uncertainty principle says that if you add together waves of that fractal-like set of frequencies, the resulting curve cannot also look like a fractal — it cannot be porous. The reverse also holds: If you take the Fourier transform of a fractal-like function, the result cannot be a fractal-like set of frequencies.

Such fractal-like functions might sound hard to come by. But they pop up when mathematicians study what happens to quantum particles — or waves more generally — in chaotic situations.

Like a ball bouncing around a pinball machine, an object experiencing chaos will often travel erratically around the entire space. “If you look at your path, it’s going to look like a random scribble,” said Elena Kim, a mathematician at Harvard University.

But in rare instances, an object can find stability in the chaos. In a flat pinball machine, a ball could stay trapped bouncing between three bumpers forever. The ball wouldn’t repeat the exact same path, but it would stay confined between the bumpers, bouncing off a slightly different spot each time. And if you marked each spot where the ball hits each bumper, you’d find that the marks make up something like the Cantor set.

This collection of marks is also called fractal dust. Most balls that hit the bumpers will fly off. “This is the dust that’s left,” said Maciej Zworski of the University of California, Berkeley.


Elena Kim, a mathematician at Harvard University, has used the fractal uncertainty principle to understand chaotic systems in unconventional spaces.

Steph Stevens

Unlike a pinball, however, a wave cannot be confined to a fractal-like path, according to the fractal uncertainty principle. If you tried to trap a wave between three bumpers, it would leak out and escape.

“So there’s something different about quantum and classical [chaos],” Dyatlov said. “And one ingredient that you can try to use for that would be uncertainty principles.”

Unfinished Knowledge

Cohen arrived at MIT in 2021 a self-described “young, energetic harmonic analyst” — harmonic analysis being a field of mathematics dedicated to studying functions and their Fourier transforms. As a doctoral student, he sat in Dyatlov’s talks about the fractal uncertainty principle. “He would end every talk being like, ‘We don’t have a higher-dimensional fractal uncertainty principle. I wish we had that!’” Cohen said. “I was like, OK, this seems like a fun thing to work on.”

His enthusiasm was soon checked. For one thing, mathematicians already knew of many situations where the fractal uncertainty principle would fail in higher dimensions. Cohen’s first challenge was to come up with a clean way to avoid these cases.

The typical requirement of being a fractal is to be porous — meaning that there are holes everywhere you look. The two-dimensional fractal called the Sierpiński carpet is an example; it’s built by dividing a square into a grid of smaller squares and removing the middle square, repeatedly. But while this shape has many holes, it’s also possible to draw a straight line that is fully contained within it.

Lines like these spell trouble for the fractal uncertainty principle. The Fourier transform of a vertical line returns a horizontal line, and vice versa. In two or more dimensions, both of these lines count as fractals — that’s because a line takes up no area and leaves most of the surrounding space empty, which satisfies the condition of having many holes. So any fractal that contains uninterrupted lines breaks the fractal uncertainty principle.

Cohen needed a new, more stringent kind of porosity. He came up with what he called “line porosity” — any line you draw on the fractal should encounter many holes. His proof excludes any fractals that don’t have this condition, including the typical Sierpiński carpet, but a modified Sierpiński carpet with much more empty space satisfies the rule.

With his assumptions in place, Cohen moved on to the actual proof. He quickly realized that this was unlike any Fourier-related problem he had worked on before. “I tried using all my tools to prove the fractal uncertainty principle, and none of them even came remotely close to working,” he said.

Feeling stuck, he went back to Dyatlov and Bourgain’s proof of the principle in one dimension and sought to understand exactly how it worked.

Dyatlov and Bourgain used an uncommon method in their proof. It involved isolating one peak from a fractal-like function at a time and showing that the Fourier transform of that peak would spread out. Doing this for all peaks, and considering how the Fourier transforms would add together, they proved that the total Fourier transform could never equal zero often enough to form a fractal — there wouldn’t be enough holes.


Jean Bourgain, seen here in 2012, was an author on more than 500 papers spanning much of mathematics.

George M. Bergman

Isolating each peak required constructing a very specific function that, when multiplied by the original fractal-like function, would pull out just the peak and be close to zero everywhere else. This is called a damping function, and it needs to be perfectly tailor-made to work. “This is a challenging thing to construct,” Cohen said. But he knew that if he could do it in higher dimensions, he could unlock the entire proof.

Cohen consulted Dyatlov about his plan to construct this special function. Before Bourgain died in late 2018, he too struggled with this problem, and he shared his unpublished notes with Dyatlov. Now, Dyatlov shared them with Cohen. “Bourgain was a legendary analyst,” Cohen said. Reading the note felt like “receiving this unfinished knowledge from him.”

The notes contained exactly the hint Cohen needed. “It just blew my mind,” Cohen said. “It really unlocked the problem for me.”

Before reading Bourgain’s note, Cohen had a few ideas for how to construct the damping function, but they were highly complicated and precise, like the designs for building a house brick by brick. The note revealed an unexpected way to do it. It involved taking a detour into complex analysis — the study of functions of imaginary numbers, which include the square root of negative 1. This detour allowed Cohen to build a much more flexible object, which could then be used to construct the damping function indirectly.

Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, proved the one-dimensional fractal uncertainty principle in 2016.

Xuwen Zhu

Armed with this insight, Cohen then needed to find a way to create just the right version of this flexible object to produce a proper damping function. “To construct something like this that has very specific properties is highly nontrivial. It’s delicate,” said Wilhelm Schlag(opens a new tab) of Yale University, with whom Cohen studied as an undergraduate. “In two dimensions, nobody knew how to do that, and Alex came up with a brilliant construction of such a thing.”

Cohen stunned the math world when he posted the proof online in May 2023.

“His paper is very beautiful, and it made a huge impression,” Schlag said.

Later, Cohen found out that the trick revealed to him in Bourgain’s note wasn’t actually a secret. The method came from a well-known theorem from the 1960s called the Beurling-Malliavin theorem. “I thought that I had this special inside knowledge,” Cohen said. “I found out later that everyone in the field already knew about this strategy.”

Had he known that his insider tip was no secret, Cohen might have given up too soon. “I think I had a lot of confidence because I didn’t know other people had tried it,” he said.

Funhouse Chaos

Soon after Cohen shared his result, other mathematicians started using it to unlock new proofs about how waves behave in chaotic situations.

In nature, chaos appears in systems like turbulent water and the weather — situations where objects that start close together quickly end up in drastically different places. These systems are too complex to describe mathematically. Instead, mathematicians seeking to study chaos often turn to an odd kind of space that has chaos built in, called hyperbolic space.

In hyperbolic space, parallel lines diverge dramatically, getting farther from each other as you follow their paths. (It’s the opposite of a sphere, where parallel lines converge.) This means that small separations between objects can become huge down the line — the telltale sign of chaos.

“Of course, this looks nothing like predicting weather,” Dyatlov said. But hyperbolic spaces offer mathematicians a simple chaotic system to explore. “We study what we can handle, and we isolate phenomena. That’s a common thing to do in math.”

In 2017, Dyatlov and Long Jin from Tsinghua University in Beijing used the one-dimensional fractal uncertainty principle to prove that you can never trap a wave on a hyperbolic surface; it will always spread out until it touches every corner.

To do so, they imagined a region on the surface that a wave never enters, even after having infinite time to spread out. When they removed all the trajectories that entered that region, what remained was the same sort of fractal dust that appeared in the pinball example. Since the fractal uncertainty principle forbids a wave from being trapped on a fractal, no such region can exist — the wave must spread everywhere.

In 2025, Kim, along with Nicholas Miller(opens a new tab) of the University of Oklahoma, used Cohen’s higher-dimensional fractal uncertainty principle to extend the results to hyperbolic spaces of higher dimensions. “That’s the most spectacular application of the higher-dimensional one so far,” Sarnak said.

Many mathematicians are hoping to prove that waves experiencing chaos not only spread out completely but also spread out exactly evenly over the entire space. This is the statement of a famous 1994 conjecture by Sarnak and Zeév Rudnick, who were interested in the problem’s connections to number theory. “This is a huge open conjecture,” Kim said. “It’s a really difficult problem that a lot of people care about.”

Proving this conjecture would show that waves experiencing chaos move in ways that look surprising and random at small scales but are simple at the macroscopic level.

“It’s like those Monet paintings,” Dyatlov said. “When you go very close, they have a lot of microscopic features. There are a lot of brushstrokes.” But if you stand back and squint, “it just looks uniformly colored.” This is different from classical chaos, where objects can get stuck in complicated and detailed paths.

The impact of the fractal uncertainty principle won’t stop with quantum chaos. The tools of Fourier analysis are used in many fields of mathematics and form the backbone of any industry that relies on signal processing. “It’s a foundational fact about Fourier analysis,” Sarnak said. “We haven’t seen all the applications yet.”

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Mentorship Cafe [GFS Rising Founders]

Mike's Notes

I had to get up yesterday morning at 3am (NZ Time) to attend the Mentorship Cafe organised by Google. I learned a lot from the mentors. Thanks for the help, guys.

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16/10/2026

Mentorship Cafe [GFS Rising Founders]

By: Mike Peters
On a Sandy Beach: 15/10/2026

Mike Peters: Mike invented and designed Pipi and founded Ajabbi.

...

Google for Startups (GFS) Rising Founders Community is a global initiative Google created to support high-potential tech startup founders.

I have been attending some of it from New Zealand, as long as the hours aren't in the middle of the night. It has been useful.

Yesterday, there was a 90-minute Mentorship Cafe. I had applied to attend and was successful. There were two 30-minute one-on-one sessions with a mentor. It was excellent; this is what I learned from Georg, Nsikak and Kennedy.

Context

Ajabbi is mission-led, bootstrapped, with no investor funding planned. All profits go to a non-profit foundation. Its SaaS product, Pipi, solves a very hard, expensive problem. There is no competition. The algorithms and maths involved require top-notch talent.

My questions

How do I recruit a remote workforce for the mission? How do I know that I have found the right people? How do I connect directly with Google DeepMind scientists, since connecting Pipi to Gemini will be a 2-way experiment by both teams?

Suggestions from mentors

  • Build a community first.
  • Grad students want a project anyway.
  • Creds on their CV for career building.
  • Internship program.
  • Be associated with a big name (University, etc.).
  • Need to connect to a senior Google scientist who understands what Mike is talking about.

Discussion

  • Ajabbi is using "Honey Traps" to attract people with the right mindset/talent. They all have to prove themselves; all the work is frontier and hard.
  • Pipi doesn't use LLM vectors, skills, transformers, or harnesses. It is based on how a biological cell works, so it uses different algorithms. High-level mathematicians are needed. MCP may not work here.

Actions

  • Update Stealth Mode plan

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The Martians

Mike's Notes

The Martians is the working title for a long-term publishing and associated screen project currently in Development.

With help from Google Search AI Mode, here is some historical background on some of the main characters (some of whom appeared in Christopher Nolan's film Oppenheimer). This project will be self-funded by the future Ajabbi Foundation.

This project exists because Pipi uses much of the maths and science these people developed.

I read Ananyo Bhattacharya's excellent book, The Man from the Future: The Visionary Life of John von Neumann, last month while researching Hilbert Spaces to add to Pipi in a future release.

The top secret sneaky plan

As part of the forthcoming Pipi documentation going on the Ajabbi websites, these scientists and mathematicians will be featured alongside their insights, including references to read, embedded video, and Jupyter Notebooks. This will happen early on.

As resources allow, this effort will expand to include other scientists of the period, including Einstein, Boltzmann, Planck, Noether, etc., historical reenactments with actors, and the factual historical factors at play.

This science communication resource will be made freely available to all, and can also be used for teaching, exhibitions, documentaries and historical drama for TV, etc.

Redworks

This project is well underway. Redworks will be an initial development vehicle once we quickly sort out some minor legal and tax issues. Initially, the focus will be on "world-building", leading into production design. Scripts will come later.

Update 25/10/2026

The Martians will use the Workspaces for Screen. It will be a good dogfooding test of its capabilities and drive additional useful features.

Update 27/10/2026

Include Euler, Gauss, and Riemann from The Martians' perspective to show how these personalities and continuity shaped the development of maths and physics and the emergence of The Martians.

Euler–Riemann Zeta Function

"If Euler was the master of manipulating infinite series, algebra, and the properties of real numbers, Riemann was the architect who took those exact same equations and projected them into multi-dimensional, complex space." - Wikipedia

Leonard Euler (15 April 1707 – 18 September 1783) was the greatest mathematician who has ever lived and was revered throughout Europe at the time. Bernhard Riemann (17 September 1826 – 20 July 1866) was a student of Carl Gauss, who was deeply influenced by Euler.

Resources

References

  • The Man from the Future: The Visionary Life of John von Neumann by Ananyo Bhattacharya. Penguin (2022).
  • Macrae, Norman (1992). John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Pantheon Press. p. 33. ISBN 0-679-41308-1.
  • Marx, George (2001). The Voice of the Martians: Hungarian Scientists Who Shaped the 20th Century in the West. Budapest: Akadémiai Kiadó. ISBN 963-05-7830-1. OCLC 48621487.

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Last Updated

25/10/2026

The Martians

By: Mike Peters
Redworks: 13/10/2026

Mike Peters: Mike invented and designed Pipi and founded Ajabbi.

...

The Martians is a working title for a publishing and associated screen project under development. It is about how a group of Hungarian Jews revolutionised mathematics and physics between the French Revolution and WW2. After the revolutions of 1848, the slow decay of the Austro-Hungarian Empire led to WWI, then to the Russian Revolution, the Stalinist counter-revolution, anti-semitism and the rise of Hitler. The project ends with the start of World War II.

The sudden explosion of genius from late 19th and early 20th-century Budapest is one of the greatest intellectual mysteries of modern history. Historians generally agree that it wasn't a coincidence, but rather the result of a highly specific "perfect storm" of political, educational, and cultural factors unique to the Austro-Hungarian Empire.

Here are the primary reasons why this specific environment bred such extraordinary minds:

1. The Legendary High Schools (Gimnáziums)

The Hungarian secondary education system at the time is widely considered one of the best in human history. Following reforms in the late 19th century, the state established gimnáziums (high schools) modelled after the rigorous German classical system, but with a heavier emphasis on mathematics and hard sciences.

  • Elite Teachers: Because university positions were limited, many world-class scientists and mathematicians taught at the high school level.
  • The "Fasori" Phenomenon: Schools like the Fasori Evangélikus Gimnázium produced John von Neumann and Eugene Wigner in the exact same generation.

2. A Hyper-Competitive Culture of Math

Hungary turned mathematics into a highly prestigious, nationwide competitive sport.

  • The KöMaL Journal: Founded in 1894, this monthly high school mathematics and physics journal published incredibly difficult, creative problems. Students from all over the country mailed in their solutions, which were graded and published. This created a national, collaborative network of young geniuses who pushed each other to their absolute limits before ever reaching university.

  • The Kürschák Competition: This annual, blind-graded mathematics competition identified the country's top talent, turning young mathematicians into national celebrities within academic circles.

3. Jewish Emancipation and the "Golden Age" of Budapest

Following the Austro-Hungarian Compromise of 1867, Jews in the empire were granted full civil rights. This triggered a massive wave of urbanisation, and Budapest quickly grew into a booming, wealthy metropolis.

  • The New Middle Class: Barred from traditional aristocratic roles like landownership or politics, the newly emancipated Jewish population poured their wealth and energy into education, medicine, science, and industry. Most of the "Martians" came from this highly driven, affluent Budapest Jewish bourgeoisie, where intellectual achievement was viewed as the ultimate status symbol.

4. The "Budapest Coffee House" Culture

Much like Vienna, Budapest developed a vibrant coffee house culture. Intellectuals, writers, artists, and scientists did not lock themselves in laboratories; they spent their afternoons in cafes like the New York Café or Café Centrál. Ideas from physics, psychology, literature, and economics were fiercely debated across disciplines, fostering lateral thinking, intense debate, and intellectual agility.

5. A State of Creative Anxiety

The "Martians" grew up in a world of profound instability. They witnessed the collapse of the Austro-Hungarian Empire, a communist revolution, a Stalinist counter-revolution, hyperinflation, and the rapid rise of virulent anti-Semitism and Fascism.

Vienna in the 1930s. The sign, belonging to a gambling parlour, reads: “Don’t let luck pass you by.” A horseshoe and chimney sweep, superstitious symbols of good fortune, appear above it. (Photograph: Roman Vishniac)

  • This created what Eugene Wigner called a "feeling of extreme insecurity" and an awareness that everything they owned could be taken away—except what was inside their minds.
  • This background forced them to be adaptable, highly driven, and comfortable with chaos, traits that proved invaluable when they later had to solve unprecedented problems in wartime America.

"The Martians" (Hungarian: A marslakók) was a nickname for a group of exceptionally brilliant Hungarian physicists and mathematicians who emigrated to the United States in the early 20th century. Most were secular Jewish refugees fleeing Nazi expansion and rising anti-Semitism in Europe.

The name originated as a joke among colleagues like Enrico Fermi and Leo Szilárd. When discussing the Fermi Paradox—asking why there is no evidence of intelligent extraterrestrial life—Szilárd quipped that aliens were already here, but they simply disguised themselves as Hungarians. The moniker stuck because these scientists possessed seemingly superhuman intellects, all grew up within walking distance of each other in Budapest, and spoke English with a thick, matching accent.

Core Members of "The Martians"

While the wider group included up to 23 thinkers, the core group of five distinct "Martians" reshaped modern physics, defence, and computing:

  • John von Neumann (1903–1957): Widely considered one of the greatest mathematicians of the 20th century.
  • Leo Szilárd (1898–1964): The visionary physicist who conceived the nuclear chain reaction.
  • Edward Teller (1908–2003): Theoretical physicist known historically as the "father of the hydrogen bomb".
  • Eugene Wigner (1902–1995): Theoretical physicist who won the Nobel Prize in Physics in 1963.
  • Theodore von Kármán (1881–1963): A pioneer in aerodynamics and mathematics.
Others included:
  • Denis Gabor: A physicist who won the 1971 Nobel Prize in Physics for inventing holography.
  • John G. Kemeny: A mathematician and computer scientist who co-developed the BASIC programming language.
  • George Pólya: Mathematician
  • Paul Erdős: One of the most prolific mathematicians in human history, known for his eccentric lifestyle and work in combinatorics and number theory.
  • Cornelius Lanczos:
  • Egon Orowan:
  • Franz Alexander:
  • George de Hevesy:
  • George Olah:
  • John Harsanyi:
  • John Polanyi:
  • Paul Halmos:
  • Paul Neményi:
  • Peter Carl Goldmark:
  • Peter Lax:
  • Valentine Telegdi:

Inside Café Josephinum, the convening place of the Vienna Circle

Impact on Science and Mathematics

The Martians did not just contribute to their fields; they laid the bedrock for the technology, national security, and computational infrastructure of the modern world.

  • The Von Neumann Architecture: John von Neumann designed the framework for modern digital computers. He pioneered the system in which data and program instructions are stored in the same memory space—a fundamental structure that nearly all desktop computers, smartphones, and servers still use.
  • Cellular Automata: Von Neumann created the concept of cellular automata, modelling self-reproducing systems on computer screens before the biological role of DNA was even fully understood.
  • The Nuclear Chain Reaction: In 1933, Leo Szilárd conceived the idea of a neutron-driven nuclear chain reaction and patented it.
  • The Manhattan Project: Recognising the threat of Nazi Germany, Szilárd drafted the famous Einstein–Szilárd letter to President Franklin D. Roosevelt, which sparked the creation of the Manhattan Project. Von Neumann, Wigner, and Teller all performed core calculations that made the atomic bomb a reality.
  • Thermonuclear Weapons: Post-WWII, Edward Teller drove the physics and design behind the first hydrogen bomb, fundamentally changing Cold War geopolitics.
  • Aviation & Space Flight: Theodore von Kármán provided the mathematical and scientific foundation for modern aerodynamics. He discovered the Kármán vortex street (fluid dynamics) and helped establish the U.S. Air Force’s research wing.
  • JPL Founding: Von Kármán later co-founded NASA's legendary Jet Propulsion Laboratory (JPL).
  • Game Theory: John von Neumann co-created mathematical Game Theory, publishing Theory of Games and Economic Behaviour. This framework mathematically analyses strategy, risk, and decision-making, transforming modern economics, evolutionary biology, and military strategy.
  • Quantum Mechanics: Eugene Wigner introduced Group Theory into quantum mechanics. His work on the symmetries of atomic nuclei earned him a Nobel Prize.
  • Heuristics and Education: George Pólya wrote How to Solve It, a foundational text that systematically defined how to approach and teach mathematical problem-solving.
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‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions

Mike's Notes

Important insights. Highly usable within Pipi for phase transitions and to be added in a future minor release.

Resources

References

  • Supercritical sharpness of percolation, by Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion. arXiv:2603.03257 [math.PR] submitted 03/04/2026. https://doi.org/10.48550/arXiv.2603.03257

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Last Updated

08/10/2026

‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions

By: Leila Sloman
Quanta Magazine: 31/08/2026

Leila Sloman:  Contributing Correspondent.

...

Mathematicians found that a broad class of networks will abruptly shift behavior past a critical point.

Introduction

The week before Christmas 2025, five mathematicians were holed up in a classroom at ETH Zurich. The mood was electric: They were this close to a career-defining breakthrough.

The group — consisting of then-postdocs Sahar Diskin(opens a new tab) and Philip Easo(opens a new tab), graduate student Ritvik Ramanan Radhakrishnan(opens a new tab), Benny Sudakov(opens a new tab), and Vincent Tassion(opens a new tab) — was perfecting a solution to one of the biggest open problems in percolation theory, the study of flow in a network.

Percolation captures a vast array of phenomena, but the prototypical examples involve fluids, like hot water seeping through a bed of coffee grounds. Diskin, Easo, Radhakrishnan, Sudakov, and Tassion were trying to work out something fundamental about how graphs — networks of points connected by lines, or edges — can be taken over by large connected areas, the equivalent of pools of fluid. The group had glimpsed a simple argument that could deal with a huge variety of graphs at once.

“We almost didn’t believe it at first,” Radhakrishnan said.

They raced to confirm each detail, eager to get their idea down before it shimmered away — and heedless of the holiday. “I’m not sure the girlfriends and the families were as happy as we were. But we were all very happy at that moment,” Diskin said. “It’s really rare that you’re able to hit something that feels so big and so meaningful.”

They worked through the night. By the morning of December 17, exhilarated from the effort, they were convinced their idea was correct. By Christmas, they’d nailed down a proof.

They had answered a decades-old question about how fast a percolation network floods as you open it up to fluid flow. “I find great joy in this proof,” said Asaf Nachmias(opens a new tab) of Tel Aviv University, who studies percolation theory and probability. “It’s stunning.”


From left: Sahar Diskin, Ritvik Ramanan Radhakrishnan, Philip Easo, Vincent Tassion, and Benny Sudakov take a group selfie after completing their paper on supercritical sharpness. / Benjamin Sudakov

Franklin’s Fluids

Percolation can describe many kinds of flow: the spread of a virus through a city, gas passing through a filter, or the propagation of a wildfire. But its original inspiration was coal.

In the 1940s, the scientist Rosalind Franklin — now famous for her work on the structure of DNA — was employed(opens a new tab) at the British Coal Utilization Research Association (BCURA), trying to understand the intricate properties of coal, charcoals, and graphite. Scientists knew that coal was studded with tiny holes, but they didn’t know why some types of coal allowed fluids to pass through them, while others were impermeable.

By submerging coal in a variety of fluids, Franklin was able to measure the typical size of its holes, as well as the amount of variation. About a decade later, the researchers Simon Broadbent and John Hammersley — wanting to understand the carbon filters in gas masks — developed a mathematical model(opens a new tab).

Their idea was simple. Take a grid of evenly spaced points (also called a lattice) and a coin. For each pair of neighboring points, flip the coin. If it lands on heads, connect the points with an edge. Fluid can flow between these points. If the coin lands on tails, the flow is blocked. Repeat this procedure for every pair of points. How far does the fluid go?

Mark Belan/Quanta Magazine

The answer depends on the probability that your coin lands on heads, which can range from 0% to 100%. When the probability is low, fluid can flow through only a few channels, and so it collects in small, isolated puddles.

But once the probability passes a threshold called the critical probability, the lattice suddenly opens up. Fluid can travel extensively through the system.

The exact value of the critical probability changes depending on the shape of the lattice — a square lattice has a different critical probability than a triangular one, and a 3D lattice has a different critical probability than a 2D one. But as you move above that critical value, you’ll see a phase transition, like liquid water turning to ice. On a finite graph, crossing the critical probability means the network will become dominated by one large sea of fluid. On an infinite graph — an abstraction where the graph extends forever in all directions — one or several infinite seas will dominate. Physicists quickly realized that through percolation, they could learn about melting and freezing, as well as other phase transitions like magnetization.

“Phase transitions in physics are very hard to study rigorously,” Easo said. “Percolation is like the caricature. So people often try to study that first, and then tools trickle down.”

For scientists who had long been stymied by complicated real-world phase transitions, “it was catching the essence in a very simple setup,” said Itai Benjamini(opens a new tab) of the Weizmann Institute for Science. “A lot of things that could cloud the issue were removed.”

Itai Benjamini, along with his collaborator Oded Schramm, made early progress studying the percolation of transitive graphs.

Courtesy of Itai Benjamini

For decades, researchers worked to quantify the percolation phase transition. They wanted to know exactly how quickly pools of fluid can grow as you increase the probability that your coin lands on heads. Many predicted that the pools grow very fast — that below the critical probability, puddles are tiny, and above it, a single ocean covers almost everything. This prediction is called the sharpness conjecture.

When sharpness was proved on lattices in the 1980s — by two independent groups, one in New Jersey(opens a new tab) and one in Moscow(opens a new tab) — it was “foundational,” said Tom Hutchcroft(opens a new tab) of Princeton University and the California Institute of Technology, who was Easo’s doctoral adviser. Knowing sharpness, mathematicians can deduce a lot about the structure of the flooded portion of the network — in particular, that it looks very similar to the underlying lattice.

So when Benjamini and his colleague Oded Schramm plotted an expedition to bring percolation to new types of networks, it seemed natural to wonder if sharpness would go with them.

Off the Grid

In 1996, Benjamini and Schramm wanted to study percolation in a much larger class of graphs, called transitive graphs. To understand what a transitive graph is, imagine the graph as a network of roads on a flat, desolate landscape. If you want to know where you are on these roads, the only landmarks are the intersections. But if the graph is transitive, all the intersections look similar — to figure out where you are, you’ll need GPS or a compass.

A square lattice is one example of a transitive graph: Every intersection consists of four edges meeting at right angles. But there are many kinds of transitive graphs — simple loops (below left) and infinitely expanding “trees” (below right), as well as ones that are almost impossible to visualize.

Many transitive graphs represent objects from other mathematical subfields, like algebra or geometry. Benjamini was intrigued by these interdisciplinary possibilities — he hoped the percolation process would reveal insights into the graph itself. “You have a stage, which is geometry, and a dancer, which is the random process,” he said. By watching the dancer, he hoped to learn more about the stage.

Over the next decade, Benjamini, Schramm, and their colleagues published a flurry of results on the percolation of transitive graphs. They proved that, for a class of infinite transitive graphs, percolation exhibits a phase transition as you open up edges to flow: Small, isolated pools of fluid suddenly coalesce into an infinite web of connected rivers.

But they still didn’t know how fast that transition happened. Below the critical point, how big and how numerous were the pools? Above it, was the infinite web a meadow crisscrossed with streams — or was it more like an ocean, swamping the entire graph?

Benjamini and Schramm suspected that a version of the sharpness conjecture was true on all infinite transitive graphs. That conjecture could be broken down into two separate problems. The “subcritical” half — addressing what happens below the critical point — was completed in 2007, by Tonći Antunović(opens a new tab) and Ivan Veselić(opens a new tab). Their work(opens a new tab) showed that here, pools of fluid are tiny and far apart. Even a hair below the critical point, the system looks more like Arizona than Minnesota.

"You have a stage, which is geometry, and a dancer, which is the random process." - Itai Benjamini, Weizmann Institute for Science

The “supercritical” half of the conjecture — which deals with probabilities above the critical threshold — seemed harder. Here, the landscape should be made up of possibly many seas, each infinitely large. In this scenario, large pools that are not connected to the infinite seas become exceedingly rare. That’s because a large, isolated pool can only stay separate if there is a lot of dry land — or closed edges — around it.

But a proof of supercritical sharpness seemed unattainable. For one thing, the previous work was no help: A proof(opens a new tab) of supercritical sharpness on lattices was long and complicated, and it couldn’t be adapted to the more general case. While other foundational results were simplified in the last decade, “this was the one remaining fortress,” Nachmias said.

Mathematicians working on this problem “did some very beautiful things, initiated the theory, picked all the low-hanging fruit,” Benjamini said. “And then we started hitting the wall.”

In 2008, as progress on non-lattice percolation slowed, Schramm died(opens a new tab) at age 46 in a fall while hiking. “We lost a genius, Oded Schramm, to a tragic accident,” Benjamini said. “And then we needed to wait for some new geniuses to come.”

About a decade ago, the field began to accelerate again. But proving supercritical sharpness remained difficult.

Then, the team in Zurich produced a simple proof.

A Sharp Turn

Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn’t intend to prove supercritical sharpness. For most of fall 2025, they were trying to understand how critical probability scales with the number of edges in graphs.

But the five mathematicians wanted results by the end of the semester. As that deadline neared, they still had nothing resembling a proof. So Easo suggested pivoting to sharpness. He, Diskin, and Radhakrishnan made some progress and brought their results to Sudakov and Tassion. As Tassion took in their work, an idea — perhaps an outrageous one — formed in his mind.

He thought that, with some tweaks, their strategy might be strong enough to prove sharpness for all infinite transitive graphs. “From there, it was in my head day and night,” Tassion said.

“Vincent went crazy with it,” Diskin said. “I think he didn’t sleep for two weeks at least.”

It wasn’t only Tassion. Over those weeks, the collaboration became frenzied. The mathematicians traded ideas constantly, often texting late at night. “We really all had this hunch that there might be something to it,” Diskin said. “We were half joking at the beginning … maybe the same idea could resolve this huge conjecture. We were all laughing at each other, but what if, what if?”

Radical Simplicity

Brimming with excitement, and with the holidays looming, they decided it was time to get serious and write their paper.

To prove that a large isolated pool of fluid is unlikely above the critical probability, the mathematicians assumed they had such a pool and studied the surrounding shoreline. Along that shoreline, there were streams emptying into the pool, but there were also streams that linked back to one of the infinite seas. If those streams coincided anywhere, the mathematicians would have a contradiction — their so-called finite pool of fluid would actually be part of an infinite sea.

"Maybe the same idea could resolve this huge conjecture. We were all laughing at each other, but what if, what if?" - Sahar Diskin, ETH Zurich

If the pool was big, the shoreline was long — meaning a larger area where the pool might connect to one of the infinite seas. The fivesome showed that this made it nearly impossible to avoid the contradiction.

As they hammered out the last details of their paper, they suddenly saw that with a simple change, their argument could be drastically improved.

They had been using a common technique in probability theory called sprinkling: They set aside a few of their open edges, corresponding to a slight lowering of the critical probability. They then looked for a large pool among the rest of the edges and analyzed the open paths around it. Since the set-aside edges had nothing to do with the pool, they could be analyzed independently. That made it easier to prove that, once combined with the rest of the graph, they almost always created a path to one of the infinite seas.

But as they talked, they hit upon an unorthodox improvement to this strategy. If they analyzed the sprinkles first, the proof got a lot simpler. What’s more, it strengthened the argument enough that it worked for all infinite transitive graphs. “We had this ping-pong of ideas,” Diskin said. “Every time you throw ideas one at another, suddenly this wall becomes more blurry, until it vanishes completely. Then it’s a bit scary, because you might actually have it.”

Finally, they were sure they had proved it: If the probability is anywhere above the critical threshold, even just a smidge, then fluid covers nearly the entire transitive graph.

Two months later, they posted a paper(opens a new tab). Their argument applies to percolation on any infinite transitive graph. “If you zoom into every sentence in the proof, it feels very familiar and simple, but the way they put it all together is genuinely novel,” Nachmias said.

There is no shortage of unstudied percolation systems that their technique could apply to — like graphs where the nodes don’t all look identical, or more complicated models that describe freezing water or quantum materials.

A major question remains, though: On three-dimensional lattices — the graphs that most closely mirror physical systems — what happens exactly at the critical probability? Is there an infinite sea?

The progress on the problem is especially significant to Benjamini, who waited a decade for his expedition to start up again. “For the community, for us, it’s a very deep and meaningful theorem, and it’s a part of the puzzle,” he said.

Of the proof, Benjamini said, “it’s a gem. It’s a gem.”

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