Showing posts with label complexity. Show all posts
Showing posts with label complexity. Show all posts

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.

Resources

References

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

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  • Home > Ajabbi Research > Library > Subscriptions > Quanta Magazine
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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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How Does Life Unfold? A Landscape Metaphor Comes Into Its Own

Mike's Notes

Excellent article by Phillip Ball in a recent issue of Quanta Magazine.

Important insights from biology. Highly usable within Pipi for fitness landscapes. Could be combined with Hilbert spaces. To be worked on and added in a future minor release.

I'm fascinated. The image below of mouse embryonic development is very similar to how Pipi grows. Must be on the right track.😎

Resources

References

  • The Strategy of the Genes by C.H. Waddington. Routledge. 1957.  ISBN 9781315765471

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  • Home > Ajabbi Research > Library > Subscriptions > Quanta Magazine
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Last Updated

16/10/2026

How Does Life Unfold? A Landscape Metaphor Comes Into Its Own

By: Phillip Ball
Quanta: 09/01/2026

Phillip Ball: Contributing Writer.

...

C.H. Waddington’s powerful image of embryonic stem cells specializing on a rolling landscape has captivated biologists since the 1950s. Experimental data is finally confirming, and complicating, his visionary ideas.

In this update to Waddington’s image, a ball (representing a cell lineage) multiplies and specializes into cell types (represented by color) as it traverses a dynamic landscape. This animation was generated from Waddington’s original drawing using an AI tool.

Mark Belan/Quanta Magazine

By the middle of the 20th century, it had begun to seem that biology was almost solved. Charles Darwin’s theory of evolution by natural selection explained how organisms change and adapt. The modern science of genetics showed how it works at the molecular scale. Genes governing the traits of organisms get passed down from parent to offspring, and random mutations create the variations from which natural selection picks the “fittest.” Natural selection and genetics fitted together in what the biologist Julian Huxley christened “the modern synthesis” in his 1942 book. Once it was shown in 1953 that the genes are encoded by DNA, the rest of biology looked like little more than piecing together details.

But not everyone was satisfied. The British biologist Conrad Hal Waddington, known to his friends and colleagues as Wad, thought this focus on genes was all very well. But it wasn’t clear to him, or anyone else, exactly how genes shape the forms and features of organisms — how they reliably generate all the different tissues of the body, in all the right places, during embryonic development. Around the same time that Huxley unveiled the modern synthesis, Waddington presented a novel picture of how this might unfold.

He visualized the developmental process as a landscape with hills and valleys that split every so often like those in a river network. At the very top of the highest hill he imagined a ball, which represented a population of cells in their earliest embryonic form — what we now call stem or pluripotent cells, which have the potential to become any cell type. The progression of the cells towards their fates — to become, say, skin cells, muscle cells, or nerve cells — is like the ball rolling downhill through the landscape.

When the ball reaches a branching point (bifurcation), the model’s “gravity” pulls it to one track or the other: In scientific terms, it must differentiate. Through a particular sequence of such branching decisions, Waddington argued, cells specialize into their mature types, or fates, each expressing a particular set of genes. The cells are restricted to a small number of well-defined states because they are “canalized,” in Waddington’s lexicon: trapped and channeled by the sides of the valley. In this way, a limited number of cell types reliably arise from the activity of an army of genes.

The biologist Conrad H. Waddington’s prescient ideas shaped the modern understanding of gene regulation, epigenetics, and evolutionary developmental biology. He died in 1975.

Walter Stoneman, National Portrait Gallery London

Waddington’s landscape has been “remarkably useful as a conceptual scaffold,” said James Briscoe, a developmental biologist at the Francis Crick Institute in London. “The core ideas — that development is progressive, that cell states are distinct, that bifurcations represent [cell] fate decisions, that canalization implies robustness — all survive.”

“There’s so much richness to some of the concepts Waddington was trying to share,” said Susanne Rafelski, a biochemist at the Institute for Stem Cell and Regenerative Medicine in Seattle.

But the concept was more of a visual metaphor than a representation of an actual biological process. Or so it seemed. In the past several years, researchers have mapped out the real topography of cell-state spaces — in effect, Waddington’s landscapes — from experimental data collected from many thousands of cells over the course of embryonic development. Those analyses are revealing that development is not a mere readout of a genetic program, but rather a dynamic process in which communities of cells build themselves, and the very landscape they navigate, by mutual negotiation and interaction, into an organism.

This cartography of development is now helping researchers understand how identical cells in the early embryo develop into the distinct tissues and cell types of a complex organism. This matters enormously for regenerative medicine and stem-cell engineering, Briscoe said: “If you understand the landscape topology and know where the bifurcations lie, you can in principle design methods that steer cell populations to desired states with precision, rather than by trial and error.”

Waddington’s metaphor, then, could be the key to understanding and reshaping possibilities for what cells, tissues, and embryos can be.

Charting a Landscape

In the 1940s, genetics and embryology were separate sciences; not everyone even believed that genes played a significant role in development. Yet Waddington’s own experiments in embryology, and those of others, convinced him that the formation and patterning of tissues in an embryo were indeed controlled by genes. The cells of different tissues had different groups of genes activated in their chromosomes, he thought. A process of differentiation gradually specialized cells into skin cells, nerve cells, and so on.

Waddington figured that the developmental pathways — for example, the formation of the neural tube from a layer of embryonic tissue called ectoderm, which ultimately develops into the central nervous system — are inevitable once the process has begun, just as water flowing down a river valley is constrained by its surroundings. The path might meander a little, but the valley keeps the river on course. Differentiation corresponds to the branching of a valley, and the final destinations are the mature cell types of the body.

Waddington’s classic drawing of the epigenetic landscape. A ball, representing a pluripotent cell or population, is poised to roll downhill into valleys, each of which represent a specialized cell type, such as a blood cell, liver cell, or brain cell.

From The Strategy of the Genes by C.H. Waddington

Waddington visualized this as a landscape in drawings, but they had “no grounding in physical reality,” wrote Scott Gilbert, a developmental biologist at Swarthmore College in Pennsylvania, in a 1991 essay in Biology & Philosophy. They were useful schematics for thinking about the problem of development, and nothing more.

“Without mathematical content,” Briscoe said, the metaphor “could not distinguish between alternative mechanisms, make quantitative predictions, or be falsified.” As a result, he said, Waddington’s landscape “has sometimes become a cliché rather than a meaningful explanation.”

What, for example, determines the topography of the hills and valleys? In his 1957 book The Strategy of the Genes, Waddington depicted the landscape as he figured it might look from underneath: as a sheet that is tugged into shape by a network of ropes attached to pegs. The pegs represented individual genes, and the ropes their effects on development. Expressing a particular gene — turning it into its corresponding protein — is like pulling on its ropes to change the shape of the sheet.

What determines the landscape’s shape? Waddington sketched a network of pegs and ropes beneath the rolling hills; the pegs are genes, and the ropes are their interconnections. Together they shape the landscape above and represent what we now call gene regulatory networks.

From The Strategy of the Genes by C.H. Waddington

It’s rather more complex than that, Waddington realized, because the ropes are connected in a network. Any one of them might influence the landscape at several points, and any given point on the landscape might be attached to several ropes.

These hidden connections correspond to what researchers now call gene regulatory networks: interactions among genes whereby an increase in the expression of one might change, or regulate, the activity of others. Herein lies the cryptic complexity of development. A given process, such as formation of the neural tube, might be influenced by many interacting genes; there’s no simple relationship between genes (genotype) and the developmental landscape that determines form (phenotype). Gene regulatory networks mediate between them.

It makes no sense, then, to look for genes dedicated to creating specific tissues, organs, or structures in a whole organism, which is one reason Waddington’s metaphor proved useful. The landscape image was “valuable in providing an alternative to a purely gene-centric view of development,” Briscoe said. By emphasizing the landscape’s shape, it pointed “toward system-level properties that cannot be read off from any single gene.”

"Given that the channels and spheres had no physical reality, what was an embryologist supposed to do with them?" - Scott Gilbert, Swarthmore College

Gene regulation wasn’t understood when Waddington first devised his landscape and rope network. A few years later, in the early 1960s, it became clear that one gene could turn another on or off. Researchers, such as the complexity theorist Stuart Kauffman, began to construct simple mathematical models of the networks of gene interactions. But too little was then known about real gene networks to connect abstract theory to experiment. “Given that the channels and spheres had no physical reality,” Gilbert wrote, “what was an embryologist supposed to do with them?”

Answers have begun to emerge over the past two decades thanks to new experimental tools for characterizing cell states. Biologists are now finally becoming able to map experimental data to a real mathematical landscape for differentiation and show just how prescient Waddington was.

Fated Attractors

One way to define the state of a given cell in an embryo is by the expression levels of its genes. These levels can now be measured simultaneously in many cells using a technique called single-cell RNA sequencing, which supplies a snapshot of all the different RNA molecules each cell contains. An RNA molecule (or transcript) is a kind of copy of a gene used to translate it into a functional protein, and generally, more RNA transcripts means higher expression of that gene. Each cell type is characterized by particular gene-expression settings, which are inherited when a cell divides. This ensures that a liver cell will stay a liver cell as it progresses through an organism’s development process.

In principle, RNA-sequencing data can be modeled as a high-dimensional mathematical space in which each axis denotes the number of RNA transcripts of a single gene. Such a plot, with maybe thousands of dimensions to represent thousands of genes, is impossible to visualize, but it can be mathematically projected onto a space with fewer dimensions, usually just two, to produce something like the shadow of a complex 3D object.

On such a plot, called a UMAP (uniform manifold approximation and projection)(opens a new tab), different cell types appear as clusters of points — blood cells cluster together, say, as do cells of the muscle or forebrain — as they share a gene expression profile. Cell lineages follow paths through this map as development proceeds. When the data is mapped through time — in what’s known as a network flow model — it looks very much like a population of cells rolling through a Waddington-like landscape, being channeled along valleys that lead to basins that correspond to distinct cell types.

A Landscape of Embryonic Development

Mark Belan/Quanta Magazine; Modified from  with permission of the authors

We should be wary of supposing, though, that this gene-expression map is literally Waddington’s landscape. Julie Theriot, a biophysicist at the University of Washington, cautioned that single-cell RNA sequencing is not the only or even the best way to characterize cell states. The technique was “mind-blowing” at first, she said, but RNA transcripts don’t map perfectly to gene expression. Plus, when researchers collapse the data into low-dimensional maps, they often do so in ways that reflect prior assumptions about how cells cluster into different types.

Still, it’s clear that, as Waddington supposed, cells undergo changes in state, defined by which genes are active or which proteins are made, as they mature and develop, and that these changes have a trajectory through time that carries them toward a limited number of final states. So how is that landscape of possibilities shaped? What governs the paths it offers?

"We think this reflects something real about how gene regulatory networks are organized." - James Briscoe, Francis Crick Institute

In 2007, the biologist Sui Huang, now at the Institute for Systems Biology in Seattle, described a bifurcation in cell state of the sort proposed by Waddington. He found that blood-forming cells called myeloid progenitor cells differentiate into two specialized types, setting them on course to become red or white blood cells, in a process dominated by two proteins, GATA1 and PU.1. This landscape of cell differentiation can thus be represented with just two values: the concentrations of these two proteins. More GATA1 promotes the formation of red blood cells, while more PU.1 sends the cell toward a white-blood-cell fate.

Huang and his colleagues mapped out the landscape using data from RNA-sequencing experiments and theoretical calculations. At first there is just a single basin, corresponding to the pool of myeloid progenitor cells. But as differentiation proceeds, this basin becomes less stable — less deep — relative to two new adjacent basins, which correspond to the red (high GATA1) and white (high PU.1) blood-cell states. Think of it like a sheet suspended horizontally from its corners, with water pooled in the middle. If you push up on the pool from below, the water flows out sideways toward new, lower pools on either side. These basins are what mathematicians call attractors: states toward which the cells are inevitably drawn, like water following gravity across a landscape.

The biologist Sui Huang has found that blood cell differentiation can be mapped like a real epigenetic landscape, with changes in gene activity altering the shape of the landscape, resulting in an output: red or white blood cell.

Allison Kudla

Researchers have now documented many cell-fate decisions(opens a new tab) that can be described in this way: as changes in gene activity that alter the shape of a landscape. The changes produce new attractors that correspond to stable cell types in the low-lying valleys; unstable gene-activity configurations constitute the high ground.

Briscoe is part of a research group that has pursued the landscape idea using the language of dynamical systems theory, which explores how systems described by sets of interacting variables evolve. “Our work starts from Waddington’s metaphor,” said David Rand), a mathematician at the University of Warwick in the United Kingdom, who collaborates with Briscoe. It’s more than a convenient image, he said: It “reveals something [fundamental] about the structure of developmental systems.”

Combining experiments and theory, they too have found that a small number of key genes drive cell-fate decisions. The rest of the network enacts the decision, as other genes are activated or suppressed farther downstream, but isn’t active in it. “We think this reflects something real about how gene regulatory networks are organized,” Briscoe said.

This makes sense from an evolutionary perspective. For any multicellular organism, the various cell types must be reliably generated and stable. Gene regulatory networks that have deep, well-separated attractors will be selected for, Briscoe said.

These deep wells represent canalization of cell-fate decisions in the way Waddington supposed. The molecular world is noisy, filled with random fluctuations. Two cells of the same type will inevitably have slightly different levels of gene expression at any moment. In the landscape picture, they don’t have to occupy the exact same spot. But so long as a group of cells is in the same basin, they will all follow the same basic trajectory.

“If fate decisions were highly sensitive to any molecular fluctuation,” Briscoe said, “development would not be robust.”

Forces of Nature

Waddington’s alluringly intuitive view left some open questions. What does the height of the landscape actually represent? And what’s the developmental equivalent of time passing as his balls roll down the slopes?

Briscoe and Rand’s dynamical-systems view provides some answers. It suggests that the landscape’s height reflects the stability of the cells’ states, with the least stable states on the highest peaks and the most stable in the lowest valleys. Time, meanwhile, is not literal time, but a kind of “pseudo-time” that reflects how the gene-expression profiles of all the cells change. In theory, all the cells can be ordered along a single pseudo-time axis so that the greater the change in a given cell’s state, the farther along the axis it is. But these changes won’t all be in sync, so there’s no ticking clock that applies to them all at any instant — hence, pseudo-time.

The biologist James Briscoe has found that a small number of key genes determine whether a cell specializes into a certain cell type, such as a brain or liver cell. Those genes then activate or suppress other genes farther downstream in the network.

Dave Guttridge/The Photo Unit

The trouble is that this notion of time doesn’t always work, Rand said. It fits when cells are crossing over high, unstable points in the landscape. But when they are in a low-lying attractor basin, their expression profiles don’t settle into a single, stable pattern but rather seem to evolve more randomly than systematically, as if “time” meanders.

And what plays the role of gravity in this picture? What is the force that moves cells along their development pathway? “In developmental biology, gravity is [equivalent to] the inevitable progression of differentiation in multi-celled organisms, starting from a small number of cell types,” Rafelski said. This process seems to have an inexorable direction and impetus. “Most of the time, cells follow a predictable order in which they differentiate,” Theriot said. But that order is not programmed into the genome(opens a new tab); after all, the cells in an organism all have the same genome. Rather, the cell-fate decisions are made collectively, as cells interact with and communicate with one another through chemical signals that are read at the cell surface, and through the tugging and squeezing on cell membranes that takes place as tissues grow and fold.

“In development, there’s always feedback,” Theriot said, “and things [in the cell] can adjust depending on inputs from the outside.”

"It is as though the landscape is constantly being reshaped by underlying tectonic forces".

So while Waddington’s landscape looked like a static territory waiting to be traversed, the real landscape changes as cells develop. “In our framework, the landscape is a feature of each cell and is not fixed,” Rand said. “Attractors appear and disappear through bifurcations, and the routes connecting them change.” It is as though the landscape is constantly being reshaped by underlying tectonic forces. Waddington’s rope network, you might say, is itself in constant motion, pulling the sheet this way and that.

In this respect, Rand said, Waddington’s picture was incomplete. It was never really a kind of universal gravity that made the ball roll; rather, cell lineages all move through a constantly shifting landscape in the process of life’s unfolding.

In such ways, modern data has both confirmed and complicated Waddington’s powerful metaphor. Combined theoretical and experimental approaches have helped to elucidate the complex but vital process through which the cells of multicellular organisms like us navigate a path from all-purpose, pluripotent stem cells to well-defined and specific tissues. Briscoe, for instance, has found that in neural-tube patterning there are two ways of reaching the same place(opens a new tab) in the landscape — a finding that reconciled previously contradictory observations. His team could not have made sense of their results without the landscape framework, he said.

Huang is now applying the landscape picture to the development of cancers. The traditional perspective is that cancer cells are malfunctioning, generally because gene mutations have damaged cell division so that it proceeds unchecked. It’s a picture that sees cancer as an aberrant state. But Huang regards it as just another attractor in the landscape — albeit a potentially catastrophic one — that can be all too easily accessed by many different routes. In this view, becoming cancerous is just one of the possible things that our cells do: an inevitable consequence, you might say, of being a multi-tissue organism.

This doesn’t mean, however, that there’s nothing we can do about it. Once we view cancer as an attractor state, a new option becomes possible: to guide those cells back out of the cancer basin and into another that is more benign — one where the cells do nothing, undergo spontaneous cell death, or even regain a healthy state. “Rather than just saying we need to kill them all,” Theriot said, “the idea is: Can we drive them to differentiate?”

Such control of landscape trajectories might also be used for tissue regeneration, or for transforming one cell type into another — for example, to regrow damaged nerves. There could be uses in agriculture as well, for selectively enhancing edible parts of plants, Theriot said.

Briscoe said that “the landscape framework provides a rational basis” for this kind of developmental design, opening up the promise of a “generative” biology in which tissues and possibly organisms can be sculpted to order. In that case, Waddington’s landscape model is far more than metaphor — it could be an atlas of a hitherto uncharted terrain.

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Sizing Up the Ghosts in the Machines

Mike's Notes

Very true.

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15/09/2026

Sizing Up the Ghosts in the Machines

By: David Krakauer
Santa Fe Institute: 01/09/2026

David Krakauer: David Krakauer, President of the Santa Fe Institute.

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A BEYOND BORDERS column by David Krakauer, President of the Santa Fe Institute.

...


Diagram of the Human Mind , Robert Fludd, 1617, via Public Domain Review

All of science is in the business of exorcism. When philosopher Gilbert Ryle coined “the ghost in the machine” in The Concept of Mind (1949), he was identifying an unnecessary explanatory entity in Cartesian dualism and expelling it. Descartes had posited a res cogitans, a thinking substance categorically distinct from the mechanical body. Ryle diagnosed this as a “category mistake,” and explained it away in the manner an Oxford professor does any superstition: like being shown the colleges, library, and playing fields of the university, and still asking the benighted question, “But where is the University?”

What Ryle did for mind, other sciences have done for their own spectral inhabitants. But the ghosts differ in kind, and so do their rites of exorcism. Physics exorcises supernumerary particles — unnecessary free parameters in pursuit of universality. Biology exorcises redundant processes — divine or teleological agencies. And the sciences of intelligence exorcise imposed patterns — transcendent sources of cognition. Each tradition seeks minimality, yet each means something profoundly different by it.

The deepest ambition of physics has always been to reduce the number of independent facts the universe requires us to accept. Theoretical physicist Eugene Wigner observed that symmetry and invariance principles act as filters on possible theories, eliminating free parameters with a scandalous efficiency. And Einstein’s general relativity derived gravitation from the minimal geometric postulate that spacetime curvature is gravity, eliminating Newton’s action-at-a-distance.

Biology’s ghost was never a free parameter but an agent. Before Darwin, the complexity of organisms demanded an intelligent designer the way a watch implied a watchmaker. On the Origin of Species (1859) performed an exorcism on the demiurge itself and replaced a transcendent creative process with a mechanical one. The ghost to be exorcised is the unnecessary rule, an act of God or Lamarckian inheritance. What remains after exorcism is not the universal parameterization of physics, but a universal mechanism (selection) that operates locally and iteratively.

Alan Turing showed in 1936 that a device of absurd simplicity consisting of a tape, a head, and a finite set of states could compute anything computable. Warren McCulloch and Walter Pitts demonstrated in 1943 that networks of binary threshold neurons could instantiate logical operations. Both of these exorcisms expelled patterns of architectural extravagance that Descartes and his followers had imposed on the mind. David Marr, in his 1969 model of cerebellar cortex, proposed a single, repeating architecture consisting of Purkinje cells trained by climbing-fiber error signals, that could learn arbitrary input–output mappings. Computation of near arbitrary complication could be solved through scale.

Computer scientists build architectures that seek to be spirit-proof. When physicists criticize biology for having too many parameters, they are applying parametric parsimony where processual parsimony is the goal. And when biologists accuse neuroscience of ignoring evolutionary constraint, they are importing processual parsimony into a domain governed by architectural parsimony. Each field is haunted by a different ghost, exists in a different intellectual bardo, each governed by its own rituals.

The old disciplinary ghosts aspired to be ontological, frightening entities posited to occupy ordered states of reality, from which ensuing and more skeptical generations banished them. We now have epistemic ghosts, patterns in large datasets, manifesting as regularities, correlations, and structures, that no human theorist seems to have previously named or noticed. LLMs suggest that we have been haunted by vast domains of regularity, in language, protein folding, weather, and genomics, that are simply too high-dimensional for human intuition to observe but not too complex for a sufficiently scaled statistical model to discern.

The ghosts are multiplying, many of them hallucinations, and scientific exorcists must learn new rites.

— David Krakauer

President, Santa Fe Institute

From the Summer 2026 edition of the SFI Parallax newsletter. Subscribe for our monthly e-Parallax, or email “news at santafe.edu” to request quarterly home delivery in print.

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The curious life of a clever slime mold

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Slime Moulds are one cell, yet perform tricks of memory. From Knowable Magazine.

This example from nature is one of the many reasons why Pipi is modelled on how biological cells work.

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10/09/2026

The curious life of a clever slime mold

By: Tim Vernimmen
Knowable Magazine: 11/02/2026

Tim Vernimmen: Tim Vernimmen is a freelance science writer based near Antwerp, Belgium. For this article, he was hoping to interview Physarum itself, but scientists are still working out how to do that.

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In its quest to feed, avoid nasty substances and just generally live its life, the brainless, one-celled Physarum polycephalum performs some impressive tricks of learning and memory

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Sixteen years ago, a brainless, unicellular organism blew our human minds. And it continues to fascinate and surprise researchers to this day.

Scientists had known that some slime molds of the species Physarum polycephalum consist of one giant, pulsating cell that keeps changing shape as it moves around and branches out to access food and avoid unpleasant things like salt or light. But it took a 2010 experiment led by Japanese biologist Toshiyuki Nakagaki of Hokkaido University to reveal the depths of its sophistication. When Nakagaki placed the oat flakes that Physarum likes in a pattern mimicking the cities surrounding Tokyo, the slime mold’s branches almost exactly reproduced the efficient transport connections between them that humans had taken years to develop.

From the center of a petri dish, the slime mold Physarum polycephalum extends branches to find food in this time-lapse video.

CREDIT: © DUSSUTOUR / CNRS

To Karen Alim, a theoretical physicist starting a postdoctoral project at Harvard University at the time, that study was a revelation. “I was like, ‘Wow, this is so crazy.’ A single cell that solves complex tasks appealed very much to me as a physicist.” Perhaps, Alim thought, she could apply her training to make sense of this clever creature with its network of contractile tubes and constantly pulsating currents.

So Alim and colleagues grew Physarum on a jelly-like substance called agar and carefully watched and recorded its behavior under the microscope. They measured the strength and direction of the fluid flow in its network of tubes. Then they simulated what they’d seen in mathematical models.

The result? Through studies like this, as Alim recounts in the Annual Review of Condensed Matter Physics, she has become convinced that the flow of fluid can be a way of transmitting information, and she’s working to understand the underlying mechanisms. Other researchers, meanwhile, are continuing to uncover new, intriguing behaviors in Physarum, a creature that appears able to learn, remember and make decisions — all without a brain.

Photograph of Karen Alim holding up a petri dish on which a slime mold is growing.

Researcher Karen Alim (right) and her colleagues grow Physarum polycephalum in petri dishes containing a nutrient-rich agar medium.

CREDIT: STEFAN WOIDIG / TUM

The flow of memory

Though Physarum is a single cell, the large body it forms can often be easily seen by the naked eye, growing to more than a foot in diameter under favorable conditions. It looks like a central blob from which a network of vein-like tubes emanates — larger tubes, then smaller tubes that fan out from them. Inside those tubes, cytoplasmic fluid is rhythmically flowing back and forth, supplying all parts of the cell with what they need. In nature, the slime mold is found in damp, dark spots like forest floors and decaying logs.

Many of the studies revealing Physarum’s unexpected skills revolve around its most important concern: finding food. Whenever Physarum encounters something edible, the outer wall of tubes near the food become soft. As a result — due to the pressure of the constantly moving fluid inside its tubes — that part of the body spreads out like a fan. This fan then slowly morphs into a network of even tinier tubes. Under the microscope, this looks like a river delta network of yellow slime feeding into the larger tubes of Physarum’s body.

How does it happen? Alim, who now works at the Technical University of Munich in Germany, figured out that encounters with food lead to an increase in local fluid flow within the tubes. This exerts greater shear force on the tube walls. The walls in that region grow thinner, allowing the tubes to expand.

Inside the slime mold Physarum polycephalum’s branching, pulsating body, currents of fluid flow back and forth to transport food and other molecules. These currents are key to the organism’s ability to move, change its shape and learn.

CREDIT: DELESCLUSE & DUSSUTOUR / CNRS

The opposite happens when Physarum encounters something awful like salt or light that it wants to get away from. In response to a repellent, the tube walls stiffen and contract, which redirects fluid flow elsewhere.

In sum, researchers now understand that where the flow is stronger, the tubes expand, and where the flow is weaker, they shrink. This interplay of shrinking and expansion in different areas of the body is what reorganizes the shape of Physarum’s tube network, causing it to approach food and avoid things like salt.

More recently, Alim and colleagues discovered just what creates a tube network as efficient as Tokyo’s transport links. It ties into something crucial about the fluid flow in Physarum’s body: Tube walls respond to changes in flow with some delay. First the flow increases. Then, slightly later, the tube expands in response, causing the flow to decrease. This causes the tube to shrink, again with some delay.

The result, Alim found, is that the best-positioned tubes will grow larger and larger and receive more and more flow, while others will fade away — and hence, over time, a super-efficient network of links will form.

Put another way, Alim adds, you could say that the shape of the network (and its underlying fluid flow dynamics) helps Physarum to remember. Appropriately sizing its branches in accordance with food sources it has encountered recently makes for a simple but effective way to recall where food can be found.

Recent work in Alim’s lab suggests it’s not just the shape of the network that helps Physarum remember, but also how, when and where it contracts the walls of its branches. For example, she says, putting food in a certain location leads the slime mold to respond with a certain pattern of contractions and move toward the food. Between experiments, the contraction pattern diminishes, but if food is put in the same spot, it reappears, more quickly than before. In one experiment, a persistent wave of contractions caused a slime mold to keep crawling in one direction for hours to get away from a light source, long after the light had been switched off, which suggests it still remembered it.

Like all of us, Physarum polycephalum has its food preferences, as this time-lapse video shows.

CREDIT: © DUSSUTOUR / CNRS

Salt and slime

The finding is reminiscent of a study by Audrey Dussutour, a biologist working on the same species, now at the National Centre for Scientific Research in France. In 2019, Dussutour had reported something interesting: After a few attempts, Physarum took less time to cross a nasty salty patch inside its dish while moving toward a food source — implying some kind of learning. The effect remained even after the slime mold had spent time in the dormant state it adopts under stressful conditions.

Since the slime mold absorbed and retained some salt, it’s possible that this lowered the shock of encountering it again and allowed it to move faster, says Dussutour. So in a recent experiment, as yet unpublished, she used light as the repellent instead. Physarum still shaved down its travel time with repeated exposures, even though light isn’t retained the same way salt is. Dussutour suspects that contraction patterns may persist and store information “similar to the way waves of activity in our brain can store information,” she says.

In addition to the information stored in the layout and contractions of its tubes, says Dussutour, Physarum has another way to “remember”: the trail of slime it leaves wherever it goes. “Avoiding slime is a convenient way to make sure it’s exploring new places rather than retracing its own trail,” she says.

In the wild, Physarum polycephalum is often found on dead plant material, as shown in this time-lapse video.

CREDIT: © DUSSUTOUR / CNRS

Dussutour was also able to show that slime molds can learn about each other: They are sensitive to aspects of each other’s behavior. “They will approach others that have access to food, and avoid ones that are starving or stressed,” she says. “Remarkably, they also prefer to approach young individuals.”

Older slime molds become very slow and fragile, she adds. “We have some in the lab that are now 5½ years old…. We hardly dare to use them anymore. But intriguingly, when they go through dormancy, or fuse with a younger individual, it’s as if they’re young again.”

For a better understanding of the slime molds’ mysterious ways, it would be very helpful to find out what exactly they’re up to in the wild, says behavioral ecologist Tanya Latty of the University of Sydney. “Almost everything we know about their behavior is based on experiments done in a lab,” she says — experiments that focus on the large, multi-nucleus forms of the creatures, not the single-nucleus, microscopic amoeba state in which they spend most of their lives.

So Latty’s research group has been collecting wild slime molds to investigate. “They’re often found on rotting logs, but we’ve also found them in the leaf litter just in front of our building. People have also found them on house plants. They’re everywhere,” she says.

Latty suspects that the large form, with its various ways of learning, allows Physarum to consume a lot more food in preparation for making the spores it needs to spread. Physarum may be far less clever in its microscopic form, she adds, if its surprising behavior really depends on its ability to branch and contract its tubes. “But who knows? We didn’t think they were capable of anything when we started to study them.”

10.1146/knowable-021126-1

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