Chemistry and Invariant Theory

28 March, 2023

In an alternative history of the world, perhaps quantum mechanics could have been discovered by chemists following up on the theories of two mathematicians from the late 1800s: Sylvester, and Gordan.

Both are famous for their work on invariant theory, which we would now call part of group representation theory. For example, we now use the Clebsch–Gordan coefficients to understand the funny way angular momentum ‘adds’ when we combine two quantum systems. This plays a significant role in physical chemistry, though Gordan never lived to see that.

But Sylvester already wanted to connect chemistry to invariant theory back in 1878! He published a paper on it, in the first issue of a journal he himself founded:

• James Joseph Sylvester, On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, with three appendices, American Journal of Mathematics 1 (1878), 64–104. (Available on JSTOR.)

The title suggests he is applying ideas from chemistry to invariant theory, rather than the other way around! I haven’t absorbed the paper, but this impression is somewhat confirmed by these passages:

To those unacquainted with the laws of atomicity I recommend Dr. Frankland’s Lecture Notes for Chemical Students, vols. 1 and 2, London (Van Voorst), a perfect storehouse of information on the subject arranged in the most handy order and put together and explained with true scientific accuracy and precision.

and then:

The more I study Dr. Frankland’s wonderfully beautiful little treatise the more deeply I become impressed with the harmony or homology (I might call it, rather than analogy) which exists between the chemical and algebraical theories. In travelling my eye up and down the illustrated pages of “the Notes,” I feel as Aladdin might have done in walking in the garden where every tree was laden with precious stones, or as Caspar Hauser when first brought out of his dark cellar to contemplate the glittering heavens on a starry night. There is an untold treasure of hoarded algebraical wealth potentially contained in the results achieved by the patient and long continued labor of our unconscious and unsuspected chemical fellow-workers.

So, he thinks the chemists may have found an ‘untold treasure of algebraical wealth’. What is this?

First he notes that you can use graphs to describe molecules: vertices represent atoms, and edges represent bonds.

This idea, utterly commonplace now, may have been only four years old when Sylvester published his work, since Wikipedia credits the use of graphs for describing molecules to this paper:

• Arthur Cayley, On the mathematical theory of isomers, Philosophical Magazine 47 (1874), 444–446.

Surely it’s not a complete coincidence that Sylvester was friends with Cayley, and that Sylvester was the first to use the term ‘graph’ to mean a bunch of vertices connected by edges!

But Sylvester noticed you can also use graphs to describe ways of building scalars from tensors: a vertex with n edges coming out is a tensor with n indices, and an edge between vertices means you sum over a repeated index, as in the ‘Einstein summation convention’. This idea is often attributed to Penrose, who explained it more clearly much later:

Penrose on spin networks.

Still later, Penrose’s spin networks and the theory of Feynman diagrams were unified via ‘string diagrams’ in category theory. I explain the story and the math here:

• John Baez and Aaron D. Lauda, A prehistory of n-categorical physics, in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson, Cambridge U. Press, Cambridge, 2011, pp. 13–128.

So, we can think of Sylvester’s chemistry-inspired work as another obscure chapter in the prehistory of n-categorical physics!

To be precise, in Sylvester’s setup a vertex with n edges out represents an atom with n bonds coming out, but also a binary quantic, meaning an element of V^{\otimes n} where V is a 2-dimensional vector space with an inner product on it. He notes that hydrogen, chlorine, bromine, and potassium have n = 1, oxygen, zinc, and magnesium have n = 2, and so on.

The inner product on V lets us raise or lower indices in tensors, so we don’t have to worry about which indices are superscripts and which are subscripts, which is usually a major aspect of the Einstein summation convention. In other words, it lets us identify V with its dual V^\ast, so we don’t have to worry about the difference between covariant and contravariant tensors.

It seems that by this method, Sylvester was able to see diagrams of molecules as recipes for building scalars from tensors! Here’s a nice page containing 45 separate figures explained in his paper:

But it wasn’t just Sylvester. Clifford, famous for inventing Clifford algebras, also thought about chemistry and invariant theory. In fact he wrote a letter about it to Sylvester! Sylvester published part of this along with his own work in the first issue of his journal:

• William Kingdon Clifford, Extract of a letter to Mr. Sylvester from Prof. Clifford of University College, London, American Journal of Mathematics 1 (1878), 126–128. (Available on JSTOR.)

Sylvester was so excited that he published this without Clifford’s permission, writing:

The subjoined matter is so exceedingly interesting and throws such a flood of light on the chemico-algebraical theory, that I have been unable to resist the temptation to insert it in the Journal, without waiting to obtain the writer’s permission to do so, for which there is not time available between the date of its receipt and my proximate departure for Europe. It is written from Gibraltar, whither Professor Clifford has been ordered to recruit his health, a treasure which he ought to feel bound to guard as a sacred trust for the benefit of the whole mathematical world.

I have not managed to understand Clifford’s ideas yet, but they may have been better than Sylvester’s—though unfortunately not developed, due to Clifford’s untimely death one year later in 1879. Olver and Shakiban write:

Although Sylvester envisioned his theory as the future of chemistry, it is Clifford’s graph theory that, with one slight but important modification, could have become a useful tool in computational invariant theory. The algebro-chemical theory reduces computations of invariants to methods of graph theory. Our thesis is that the correct framework for the subject is to use digraphs or “directed molecules” as the fundamental objects. One can ascribe both a graph theoretical as well as a chemical interpretation.

This is from here:

• Peter J. Olver and Cherzhad Shakiban, Graph theory and classical invariant theory, Advances in Mathematics 75 (1989), 212–245.

It sounds like what Clifford realized is that by using a directed graph we get a better theory that lets us drop the inner product on V. Having graphs with directed edges lets the graphical notation distinguish between covariant and contravariant tensors.

Now let’s jump forward a decade or two! At some point Paul Gordan read the work of Clifford and Sylvester and concluded that invariant theory could contribute to the understanding of chemical valence. But his own ideas were somewhat different. In 1900 he and his student W. Alexejeff wrote an article about this:

• Paul Gordan and W. Alexejeff, Übereinstimmung der Formeln der Chemie und der Invariantentheorie, Zeitschrift für Physikalische Chemie, 35 (1900), 610–633.

In 2006, Wormer and Paldus wrote:

The origins of the coupling problem for angular momenta can be traced back to the early—purely mathematical—work on invariant theory by (Rudolf Friedrich) Alfred Clebsch (1833–1872) and Paul (Albert) Gordan (1837–1912), see Section 2.5. Even before the birth of quantum mechanics the formal analogy between chemical valence theory and binary invariant theory was recognized by eminent mathematicians as Sylvester, Clifford, and Gordan and Alexejeff. The analogy, lacking a physical basis at the time, was criticised heavily by the mathematician E. Study and ignored completely by the chemistry community of the 1890s. After the advent of quantum mechanics it became clear, however, that chemical valences arise from electron–spin couplings … and that electron spin functions are, in fact, binary forms of the type studied by Gordan and Clebsch.

I learned of this quote from James Dolan, who happened to be studying the work of Eduard Study, who is mostly famous for his work on the so-called dual numbers, the free algebra on one generator that squares to zero. The paper by Wormer and Paldus is here:

• Paul E. S. Wormer and Josef Paldus, Angular momentum diagrams, Advances in Quantum Chemistry 51 (2006), 51–124.

Here’s another paper I should read:

• Karen Hunger Parshall, Chemistry through invariant theory? James Joseph Sylvester’s mathematization of the atomic theory, in Experiencing Nature: Proceedings of a Conference in Honor of Allen G. Debus, Springer, Berlin, 1997.

Sylvester was a colorful and fascinating character. For example, he entered University College London at the age of 14. But after just five months, he was accused of threatening a fellow student with a knife in the dining hall! His parents took him out of college and waited for him to grow up a bit more.

He began studies in Cambridge at 17. Despite being ill for 2 years, he came in second in the big math exam called the tripos. But he couldn’t get a degree… because he was Jewish.

In 1841, he was awarded a BA and an MA by Trinity College Dublin. In the same year he moved to the United States to become a professor of mathematics at the University of Virginia.

After just a few months, a student reading a newspaper in one of Sylvester’s lectures insulted him. Sylvester struck him with a sword stick. The student collapsed in shock. Sylvester thought he’d killed the guy! He fled to New York where one of his brothers was living.

Later he came back to Virginia. But according an online biography, “the abuse suffered by Sylvester from this student got worse after this”. Soon he quit his job.

He returned to England and took up a job at a life insurance company. He needed a law degree for this job, and in his studies he met another mathematician, five years younger, studying law: Cayley! They worked together on matrices and invariant theory.

Sylvester only got another math job in 1855, at the Royal Military Academy of Woolwich. He was 41. At age 55 they made him retire—that was the rule—but for some reason the school refused to pay his pension!

The Royal Military Academy only relented and paid Sylvester his pension after a prolonged public controversy, during which he took his case to the letters page of The Times.

When he was 58, Cambridge University finally gave him his BA and MA.

At age 62, Sylvester went back to the United States to become the first professor of mathematics at the newly founded Johns Hopkins University in Baltimore, Maryland. His salary was $5,000—quite generous for the time.

He demanded to be paid in gold.

They wouldn’t pay him in gold, but he took the job anyway. At age 64, he founded the American Journal of Mathematics. At 69, he was invited back to England to become a professor at Oxford. He worked there until his death at age 83.

One thing I’ve always liked about Sylvester is that he invented lots of terms for mathematical concepts. Some of them have caught on: matrix, discriminant, invariant, totient, and Jacobian! Others have not: cyclotheme, meicatecticizant, tamisage and dozens more.

But only now am I realizing how Sylvester’s fertile imagination, inspired by chemistry, connected graph theory and invariant theory in ways that would later become crucial for physics.

Four ‘Universes’

26 March, 2023

This chart made by Toby Ord shows four things:

• Everything we can observe now is the ‘observable universe’.

• Everything we can ever observe if we stay here is the ‘eventually observable universe’.

• Everything we can ever observe if we send spacecraft out in every direction at all speeds slower than light is the ‘ultimately observable universe’.

• Everything those spacecraft can ever affect is the ‘affectable universe’.

His chart is drawn in funny coordinates where a galaxy at rest moves straight up the page and light moves at 45° angles. The Big Bang is the horizontal line at the bottom, and the infinite future is the horizontal line at top. The expansion of the universe is hidden in these coordinates!

How big are these four things?

• When we observe distant galaxies we see what they were like long ago, when they were closer. Those galaxies now form a ball of radius 46 billion light years in diameter. So people say the radius of the observable universe is 46 billion light years. But beware: we can’t see what those galaxies look like now.

• The galaxies in the eventually observable universe now form a ball of radius 63 billion light years.

• The galaxies in the ultimately observable universe now form a ball of radius 80 billion light years.

• The galaxies in the affectable universe now form a ball of radius 16 billion light years.

These figures change with time. For example, shortly after the Big Bang the radius of the affectable universe was 63 billion light years. It has now shrunk to 16 billion light years. 90% of the galaxies we could in theory once reach—if we could have started right away—are lost to us now!

Of course, all these numbers are based on our current cosmology, which says that as the universe expands and ordinary matter thins out, the effect of dark energy becomes more important, and the universe starts expanding almost exponentially. If our theory of cosmology is wrong then these numbers are wrong!

You might wonder why the affectable universe has a finite radius even though the universe will last forever in our current theory. The reason is that because the universe is expanding faster and faster, it’s impossible to catch up with distant galaxies. So the only galaxies we can reach are those that are less than 16 billion light years away now.

For more, read Toby Ord’s paper:

• Toby Ord, The edges of our Universe.

and read my earlier blog post on this subject:

The expansion of the Universe.

The Galactic Center

24 March, 2023

You’ve probably heard there’s a supermassive black hole at the center of the Milky Way—and also that near the center of our galaxy there are a lot more stars. But did you ever think hard about what the Galactic Center is like?

I didn’t, until recently. As a kid I read about it in science fiction—like Asimov’s Foundation trilogy, where the capital of the Empire is near the Galactic Center on the world of Trantor, with a population of 40 billion. That shaped my impressions.

But now we know more. And it turns out the center of our galaxy is a wild and woolly place! Besides that black hole 4 million times the mass of our Sun, it’s full of young clusters of stars, supernova remnants, molecular clouds, weird filaments of gas, and more.

It’s in the constellation of Sagittarius, abbreviated ‘Sgr’. Let me go through the various features named above and explain them.

Sgr A contains the supermassive black hole called Sgr A*, which is worth a whole article of its own. Surrounding that is the Minispiral: a three-armed spiral of dust and gas falling into the black hole at speeds up to 1000 kilometers per second.

Also in Sgr A, surrounding the Minispiral, there is a torus of cooler molecular gas called the ‘Circumnuclear Disk’:

The inner radius of the Circumnuclear Disk is almost 5 light years. And inside this disk there are over 10 million stars. That’s a lot! Remember, the nearest stars to our Sun are 4 light years away.

Even weirder, among these stars there are lots of old red giants—but also many big, young stars that formed in a single event a few million years ago. These include about 100 OB stars, which are blue-hot, and Wolf-Rayet stars, which have blown off their outer atmosphere and are shining mainly in the ultraviolet.

Nobody knows how so many stars were able to form inside the Circumnuclear Disk espite the gravitational disruption of central black hole, and why so many are young. This is called the ‘paradox of youth’.

Stars don’t seem to be forming now in this region. But some predict that stars will form in the Circumnuclear Disk, perhaps causing a starburst in 200 million years, with many stars forming rapidly, and supernovae going off at a hundred times the current rate! As gas from these falls into the central black hole, life may get very exciting.

As if this weren’t enough, a region of Sgr A called Sgr A East contains a structure is approximately 25 light-years in width that looks like a supernova remnant, perhaps created between 35 and 100 thousand years ago. However, it would take 50 to 100 times more energy than a standard supernova explosion to create a structure of this size and energy. So, it’s a bit mysterious.

Moving further out, let’s turn to the Radio Arc, called simply ‘Arc’ in picture at the top of this article. This is the largest of a thousand mysterious filaments that emit radio waves. It’s obvious that the Galactic Center is wild, but these make it ‘woolly’. Nobody knows what causes them!

Here is the Radio Arc and some filaments:

Behind the Radio Arc is the Quintuplet Cluster, which contains one of the largest stars in the Galaxy—but more about that some other day.

Sgr B1 is a cloud of ionized gas. Nobody knows why it’s ionized. Like the filaments, perhaps it was heated up back when the black hole was eating more stars and emitting more radiation. Sgr B1 is connected to Sgr B2, a giant molecular cloud made of gas and dust, 3 million times the mass of the Sun.

The distance from Sgr A to Sgr B2 is 390 light years. That gives you a sense of the scale here! The whole picture spans a region in the sky 4 times the angular size of the Moon.

The two things called SNR are supernova remnants—hot gas shooting outwards from exploded stars. For example, in the top picture at lower right we see SNR 359.1-0.5, which looks like this close up:

The filament at right is called the Snake, while the Mouse at left is actually supposed to be a runaway pulsar. It looks like the Mouse is running away from the Snake! But that’s probably a coincidence.

Sgr D is another giant molecular cloud, and Sgr C is a group of molecular clouds.

So, a lot is going on in our galaxy’s center! Out here in the boondocks it’s more quiet.

Let me show you the first picture in all its glory without the labels. Click to enlarge:

It’s almost impossible to see the Galactic Center in visible light through all the dust, so this is an image in radio waves, made by the MeerKAT array of 64 radio dishes in South Africa. It was made by Ian Heywood with color processing by Juan Carlos Munoz-Mateos.

Here are two other versions of the same image, processed in different ways:

Click to enlarge!

The Vela Pulsar

22 March, 2023

If you could see in X-rays, one of the brightest things you’d see in the night sky is the Vela pulsar. It was formed when a huge star’s core collapsed about 12,000 years ago.

The outer parts of the star shot off into space. Its core collapsed into a neutron star about twice the mass of our Sun—but just 20 kilometers in diameter! Today it’s spinning around 11.195 times every second. As it whips around, it spews out jets of charged particles moving at about 70% of the speed of light. These make X-rays and gamma rays.

The Chandra X-ray telescope made a closeup video of the Vela pulsar! It shows this jet is twisting around.

But the most interesting part of all this, to me, are the ‘glitches’ when the neutron star suddenly spins a bit faster. Let me tell you a bit about those.

First, I can’t resist showing you what happened to the star that exploded. It made this: the Vela Supernova Remnant. It’s so beautiful!

This photo was taken, not by a satellite in space, but by Harel Boren in the Kalahari Desert in Namibia!

Then, I can’t resist showing you a little movie of the Vela pulsar… slowed down:

This was made using the Fermi Gamma-Ray Space Telescope. The image frame is large: 30 degrees across. The background, which shows diffuse gamma-ray emission from the Milky Way, is shown about 15 times brighter than it actually is.

Then I can’t resist showing you a closeup photo of the Vela pulsar, taken in X-rays by the Chandra X-ray Observatory:

The bright dot in the middle is the neutron star itself, and you can see one of the jets poking out to the upper right, while the other is aimed toward us.

Now, about those glitches.

Since it’s putting out powerful jets, which carry angular momentum, we expect the Vela pulsar to slow down—and it does. But it does so in a funny way: every so often there’s a glitch where it speeds up for about 30 seconds! Then it returns to its speed before the glitch—gradually, in about 10 to 100 days.

What’s going on? A neutron star has 3 parts: the outer crust, inner crust, and core. The outer crust is a crystalline solid made of atoms squashed down to a ridiculous density: about 10¹¹ grams per cubic centimeter. But the inner crust contains neutron-rich nuclei floating in a superfluid made of neutrons!

Yes: while helium becomes superfluid and loses all viscosity due to quantum effects only when it’s really cold, highly compressed neutrons can be superfluid even at very high temperatures And the funny thing about a superfluid is that the curl of its flow is zero except along vortices which carry quantized angular momentum, coming in chunks of size ℏ.

Glitches must be caused by how the outer crust interacts with the inner crust. The outer crust slows down. The inner crust, being superfluid, does not. This can’t go on forever, since they rub against each other. So it seems that now and then a kind of crisis occurs: in a chain reaction, vast numbers of superfluid vortices suddenly transfer some angular momentum to the outer crust, speeding it up while reducing their angular momentum. It’s analogous to an avalanche.

So, we are seeing complicated quantum effects in a huge spinning star 1000 light years away!

Scorpius X-1

20 March, 2023

If you could see X-rays, maybe you’d see this.

Near the Galactic Center, the Fermi bubbles would glow bright… but the supernova remnant Vela, the neutron star Scorpius X-1 and a lot of activity in the constellation of Cygnus would stand out.

Scorpius X-1 was the first X-ray source in space to be found after the Sun. It was discovered by accident when a rocket launched to detect X-rays from the Moon went off course!

But why is it making so many X-rays?

Scorpius X-1 is a double star about 9,000 light-years away from us. It’s a blue-hot star orbiting a neutron star that’s three times as heavy. As gas gets stripped off from the lighter star and sucked into the neutron star, it first forms a spinning disk. As it spirals down into the neutron star, it releases a tremendous amount of energy.

This gas is near the ‘Eddington limit’, where the pressure of radiation pushing outward and the gravitational force pulling inward are in balance!

Scorpius X-1 puts out about 23000000000000000000000000000000 watts of power in X-rays! Yes, that’s 2.3 × 10³¹ watts. This is 60,000 times the X-ray power of our Sun.

Scorpius X-1 is considered a low-mass X-ray binary: the neutron star is roughly 1.4 solar masses, while the lighter star is only 0.42 solar masses. These stars were probably not born together: the binary may have been formed by a close encounter inside a globular cluster.

The lighter star orbits about once every 19 days.

Puzzle. Why is such a light star blue-hot, rather than a red dwarf?

I want to read more about Scorpius X-1 and similar X-ray binaries! Besides the Wikipedia article:

• Wikipedia, Scorpius X-1.

I’m finding technical papers like this:

• Danny Steeghs and Jorge Casares, The mass donor of Scorpius X-1 revealed, The Astrophysical Journal 568 (2002), 273.

which gets into details like “The insertion of the calcite slab in the light path results in the projection of two target beams on the detector.” But I’d like to read a synthesis of what we know, like an advanced textbook.

X-Ray Chimneys

18 March, 2023

First astronomers discovered enormous gamma-ray-emitting bubbles above and below the galactic plane—the ‘Fermi bubbles’ I wrote about last time.

Then they found ‘X-ray chimneys’ connecting these bubbles to the center of the Milky Way!

These X-ray chimneys are about 500 light-years tall. That’s huge, but tiny compared to the Fermi bubbles, which are 25,000 light years across. They may have been produced by the black hole at the center of the Galaxy. We’re not completely sure yet.

Here’s an X-ray image taken by the satellite XMM-Newton in 2019. It clearly shows the X-ray chimneys:

Sagittarius A* is the black hole at the center of our galaxy. It’s an obvious suspect for what created these chimneys!

Puzzle. What’s the white circle?

For more, try this:

• G. Ponti, F. Hofmann, E. Churazov, M. R. Morris, F. Haberl, K. Nandra, R. Terrier, M. Clavel and A. Goldwurm, The Galactic centre chimney, Nature 567 (2019), 347–350.

Abstract. Evidence has increasingly mounted in recent decades that outflows of matter and energy from the central parsecs of our Galaxy have shaped the observed structure of the Milky Way on a variety of larger scales. On scales of ~15 pc, the Galactic centre has bipolar lobes that can be seen in both X-rays and radio, indicating broadly collimated outflows from the centre, directed perpendicular to the Galactic plane. On far larger scales approaching the size of the Galaxy itself, gamma-ray observations have identified the so-called Fermi Bubble features, implying that our Galactic centre has, or has recently had, a period of active energy release leading to a production of relativistic particles that now populate huge cavities on both sides of the Galactic plane. The X-ray maps from the ROSAT all-sky survey show that the edges of these cavities close to the Galactic plane are bright in X-rays. At intermediate scales (~150 pc), radio astronomers have found the Galactic Centre Lobe, an apparent bubble of emission seen only at positive Galactic latitudes, but again indicative of energy injection from near the Galactic centre. Here we report the discovery of prominent X-ray structures on these intermediate (hundred-parsec) scales above and below the plane, which appear to connect the Galactic centre region to the Fermi bubbles. We propose that these newly-discovered structures, which we term the Galactic Centre Chimneys, constitute a channel through which energy and mass, injected by a quasi-continuous train of episodic events at the Galactic centre, are transported from the central parsecs to the base of the Fermi bubbles.

The Fermi Bubbles

16 March, 2023


How come nobody told me about the ‘Fermi bubbles’? If you could see gamma rays, you’d see enormous faint glowing bubbles extending above and below the plane of the Milky Way.

Even better, nobody is sure what produced them! I love a mystery like this.

The obvious suspect is the black hole at the center of our galaxy. Right now it’s too quiet to make these things. But maybe it shot out powerful jets earlier, as it swallowed some stars.

Another theory is that the Fermi bubbles were made by supernova explosions near the center of the Milky Way.

But active galactic nuclei—where the central black hole is eating a lot of stars—often have jets shooting out in both directions. So I’m hoping something like that made the Fermi bubbles. Computer models say jets lasting about 100,000 years about 2.6 million years ago could have done the job.

The Fermi bubbles were discovered in 2010 by the Fermi satellite: that’s how they got their name. I learned about them by reading this review article:

• Mark R. Morris, The Galactic black hole.

I recommend it! I get happy when I hear there are a lot of overlapping, complex, poorly understood processes going on in space. I get sad when pop media just say “Look! Our new telescope can see a lot of stars! I already knew there are a lot of stars. But the interesting stories tend to be written in a more technical way, like this:

Another cool thing: we may have detected some neutrinos emanating from the Fermi bubbles! These neutrinos have energies between 18 and 1,000 TeV. That’s energetic! Our best particle accelerator, the Large Hadron Collider, collides protons with an energy of about 14 TeV. This suggests that the Fermi bubbles contain a lot of very high-energy protons—so-called ‘cosmic rays’ — which occasionally collide and produce neutrinos.

• Paul Sutter, Something strange is happening in the Fermi bubbles,, September 4, 2019.

See also these:

• Rongmon Bordoloi, Andrew J. Fox, Felix J. Lockman, Bart P. Wakker, Edward B. Jenkins, Blair D. Savage, Svea Hernandez, Jason Tumlinson, Joss Bland-Hawthorn and Tae-Sun Kim, Mapping the nuclear outflow of the Milky Way: studying the kinematics and spatial extent of the Northern Fermi bubble, The Astrophysical Journal 834 (2017) 191.

• P. Predehl, R. A. Sunyaev, W. Becker, H. Brunner, R. Burenin, A. Bykov, A. Cherepashchuk, N. Chugai, E. Churazov, V. Doroshenko, N. Eismont, M. Freyberg, M. Gilfanov, F. Haberl, I. Khabibullin, R. Krivonos, C. Maitra, P. Medvedev, A. Merloni, K. Nandra, V. Nazarov, M. Pavlinsky, G. Ponti, J. S. Sanders, M. Sasaki, S. Sazonov, A. W. Strong, and J. Wilms, Detection of large-scale X-ray bubbles in the Milky Way halo.

Also try this, for something related but different:

• Jure Japelj, Astonishing radio view of the Milky Way’s Heart, Sky and Telescope, February 3, 2022.

Runaway Supermassive Black Hole

14 March, 2023

Many galaxies have a ‘supermassive black hole’ at their center. These range from hundreds of thousands to billions times the mass of our Sun.

I was surprised to read that astronomers have found evidence for a supermassive black hole shooting out of its host galaxy. They’ve seen a long thin feature—apparently a ‘wake’ of shocked gas and young stars—stretching 200,000 light years from the galaxy’s center and ending in a bright object that’s putting out 100 million times more power than our Sun. It

This is consistent with a supermassive black hole that was thrown out the galactic center at a speed of 1600 kilometers/second, which has been traveling for about 40 million years. This speed is faster than the galactic escape velocity!

But what could be muscular enough to throw a supermassive black hole around?

Only two possibilities are known.

One is another supermassive black hole. When two galaxies collide, their central black holes meet – and may start orbiting each other. If a third galaxy with its own supermassive black hole crashes in, one of the three black holes can get flung out.

That seems quite reasonable to me: galactic collisions are fairly common.

The other possibility is weirder.

When two black holes collide, they can emit gravitational radiation that’s beamed mainly in one direction… and this can give them a ‘kick’ in the opposite direction. I find this surprising in the first place. And it’s more surprising that this effect can be big enough to kick a black hole out of a galaxy! But that’s what some calculations say.

The picture above is from this paper:

• Pieter van Dokkum, Imad Pasha, Maria Luisa Buzzo, Stephanie LaMassa, Zili Shen, Michael A. Keim, Roberto Abraham, Charlie Conroy, Shany Danieli, Kaustav Mitra, Daisuke Nagai, Priyamvada Natarajan, Aaron J. Romanowsky, Grant Tremblay, C. Megan Urry and Frank C. van den Bosch, A candidate runaway supermassive black hole identified by shocks and star formation in its wake.

Abstract. The interaction of a runaway supermassive black hole (SMBH) with the circumgalactic medium (CGM) can lead to the formation of a wake of shocked gas and young stars behind it. Here we report the serendipitous discovery of an extremely narrow linear feature in HST/ACS images that may be an example of such a wake. The feature extends 62 kpc from the nucleus of a compact star-forming galaxy at z=0.964. Keck LRIS spectra show that the [OIII]/Hβ ratio varies from ~1 to ~10 along the feature, indicating a mixture of star formation and fast shocks. The feature terminates in a bright [OIII] knot with a luminosity of 1.9×1041 ergs/s. The stellar continuum colors vary along the feature, and are well-fit by a simple model that has a monotonically increasing age with distance from the tip. The line ratios, colors, and the overall morphology are consistent with an ejected SMBH moving through the CGM at high speed while triggering star formation. The best-fit time since ejection is ~39 Myr and the implied velocity is v~1600 km/s. The feature is not perfectly straight in the HST images, and we show that the amplitude of the observed spatial variations is consistent with the runaway SMBH interpretation. Opposite the primary wake is a fainter and shorter feature, marginally detected in [OIII] and the rest-frame far-ultraviolet. This feature may be shocked gas behind a binary SMBH that was ejected at the same time as the SMBH that produced the primary wake.

For more about the kick caused by gravitational waves, try this:

• Manuela Campanelli, Carlos O. Lousto, Yosef Zlochower and David Merritt, Maximum gravitational recoil, Phys. Rev. Lett. 98 (2007), 231102.

Azimuth Project News

9 March, 2023

It’s time to update you on the Azimuth Project. This project started out in 2010 when I moved to Singapore, had more time to think thanks to a great job at the Centre for Quantum Technologies, and decided to do something about climate change—or more broadly, the Anthropocene.

But do what? You can read my very first thoughts here. I rounded up some interested people, many of them programmers from outside academia, and we started a wiki to compile relevant scientific information. We thought a lot and wrote a lot about the huge problems confronting our civilization. We did some interesting stuff like making simple climate models—purely for educational purposes, not for trying to predict anything! We also recapitulated a network-based attempt to predict El Niños.

But it soon became clear to me that my own strengths lay not in climate science, and certainly not in leading a group of people outside academia trying to accomplish something practical. I got more and more interested in using category theory to study networks—and more generally in getting category theorists interested in practical things. I figured that category theory could really transform how we think about complex systems made of interacting parts.

I understand a bit about what motivates academics, and how to get them working on things. So, once I put my mind to it, I managed to speed up the trend toward applied category theory, which by now has its own annual conference. I’m on the steering committee of that conference, but luckily there are so many energetic people involved that I don’t have to do much. By now I can barely keep up with the progress in applied category theory, which is visible on the Category Theory Community Server, a forum set up by my student Christian Williams.

Indeed, part of how academia works is that if you get really good students, they go off and do things much better than you could do yourself!

For example, my former student Brendan Fong is an order of magnitude better at organizing things than I am. Together with Joshua Tan and Nina Otter he started the journal Compositionality, which has a strong emphasis on applied category theory, though it’s also open to other ways of thinking about compositionality (the study of how complex things can be assembled out of simpler parts). But even more importantly, Brendan now leads the Topos Institute, which brings together applied category theorists and people developing new technologies for the betterment of humanity. I’ll get back to that later.

Another amazingly successful student of mine is Nina Otter, now at Queen Mary University. At least I’ll gladly count her as a student, because she did a master’s thesis with me, on operads and the tree of life. But then she switched to topological data analysis, and she’s now using that to study weather regimes.

A big part of the Azimuth project’s focus on networks has always been studying Petri nets: a general formalism for studying chemical reactions, population biology and many other things.

A bunch of blog articles on Petri nets, written at the Centre for Quantum Technologies with Jacob Biamonte, eventually turned into our book Quantum Techniques for Stochastic Mechanics. But a new direction came when Brendan Fong developed decorated cospans, a general technique for studying open systems. My student Blake Pollard and I used these to study ‘open Petri nets’, which we called open reaction networks.

Later, my student Jade Master made the theory of open Petri nets really beautiful using structured cospans, a simplified version of Brendan’s decorated cospans developed by my student Kenny Courser.

Meanwhile something big was brewing. Two fresh PhDs named James Fairbanks and Evan Patterson came up with AlgebraicJulia, a software system that aims to “create novel approaches to scientific computing based on applied category theory”. And among many other things, they grabbed ahold of structured cospans and turned them into something you could write programs with!

In October 2020, together with Micah Halter, they used AlgebraicJulia to redo part of the UK’s main COVID model using open Petri nets. At the time I wrote:

This is a wonderful development! Micah Halter and Evan Patterson have taken my work on structured cospans with Kenny Courser and open Petri nets with Jade Master, together with Joachim Kock’s whole-grain Petri nets, and turned them into a practical software tool!

Then they used that to build a tool for ‘compositional’ modeling of the spread of infectious disease. By ‘compositional’, I mean that they make it easy to build more complex models by sticking together smaller, simpler models.

Even better, they’ve illustrated the use of this tool by rebuilding part of the model that the UK has been using to make policy decisions about COVID19.

All this software was written in the programming language Julia.

I had expected structured cospans to be useful in programming and modeling, but I didn’t expect it to happen so fast!

Here’s a video about these ideas, from 2020:

Later Evan got a job at the Topos Institute and this work blossomed into the following paper:

• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James Fairbanks, An algebraic framework for structured epidemic modeling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.

I should have blogged about this, but things are happening so fast I never got around to it! This illustrates why I’ve lost interest in the Azimuth Project as originally formulated, with this blog as the main communication hub and the wiki as the information depot: academics with their own modes of communication have been pushing things forward in their own ways too fast for me to blog about it all!

Another example: last summer in Buffalo I helped mentor a bunch of students at a program on applied category theory run by the American Mathematical Society. This led to two very nice papers on open Petri nets and related open networks:

• Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo S. Ocal, Evan Patterson and Brandon T. Shapiro, A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks.

• Benjamin Merlin Bumpus, Sophie Libkind, Jordy Lopez Garcia, Layla Sorkatti and Samuel Tenka, Additive invariants of open Petri nets.

I want to blog about these, and I will soon!

But at the same time, the use of category theory in epidemiological modeling keeps growing. The early work attracted the attention of a bunch of actual epidemiologists, notably my old grad school pal Nate Osgood, who now works at the University of Saskatchewan, both in computer science and also the department of community health and epidemiology. He helps the government of Canada run its main COVID models! This was a wonderful coincidence, made even sweeter by the fact that Nate was hankering to apply category theory to these tasks.

Nate explained that for modeling disease, Petri nets are less popular than another style of diagram, called ‘stock-flow diagrams’. But one can deal with open stock-flow diagrams using the same category-theoretic tricks that work for Petri nets: decorated or structured cospans. We worked this out together with Evan Patterson, Nate’s grad student Xiaoyan Li, and Sophie Libkind at the Topos Institute. And these folks—not me—converted these ideas into AlgebraicJulia code for making big models of epidemic disease out of smaller parts!

We wrote about it here:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams, to appear in the proceedings of Applied Category Theory 2022.

Alas, I’ve been too busy to properly blog about this paper, but I’ve given a bunch of talks about it, and you can see some on YouTube. The easiest is probably this one:

Since then we’ve made a huge amount of progress, due largely to Nate and Xiaoyan’s enthusiasm for converting abstract ideas into practical tools for epidemiologists. The current state of the art is pretty well reflected in this paper:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams.

In particular, Nate’s student Eric Redekopp built a graphical user interface for the software, so epidemiologists knowing nothing of category theory or the language Julia can collaboratively build disease models on their web browsers!

So, a lot of my energy that originally went into the Azimuth Project has, by a series of unpredictable events, become focused on the project of applied category theory, with the most practical application for me currently being disease models.

What happened to climate change? Well, a lot of these modeling methodologies could be applied to power grids or world economic models. In fact stock-flow diagrams were first developed for economics and business in James Forrester’s book Industrial Dynamics, and they were later used in the famous Limits to Growth model of the world economy and ecology, called World3. So there is a lot to do in this direction. But—I’ve realized—it would require finding an energetic expert who is willing to learn some category theory and teach me (or some other applied category theorist) what they know.

For now, a more instantly attractive option is working with someone I’ve known since I was a postdoc: Minhyong Kim. He’s now head of the International Center of Mathematical Sciences, and he’s dreamt up a project called Mathematics for Humanity. This will fund research workshops, conferences and courses in these areas:

A. Integrating the global research community

B. Mathematical challenges for humanity

C. Global history of mathematics

I’m hoping to coax people to run a workshop on mathematical epidemiology, but also get people together to tackle many other mathematical challenges for humanity. Minhyong has listed some examples:

The deadline to apply for funding is now June 1st, so if you know anyone who might be interested, please tell them about this—and tell me about them!

So, a lot is going on. But I’ve had very little time to do anything with the Azimuth Wiki or the Azimuth Forum (an online discussion forum for the Azimuth Project). Indeed I’ve largely ignored them for years now. David Tanzer has nobly been providing tech support for these sites. But after many conversations with him about this, I’ve decided that it’s time to close down those sites. So that’s what we plan to do on May 1st.

It’s a bit sad, but as I hope I’ve explained, the spirit of the Azimuth Project lives on. And at this moment I want to thank everyone who has been in involved with it in any way. There are many of you not mentioned above. If I tried to list all of you I’d leave some out, so please accept these collective thanks—and good luck with all your projects!

Dividing a Square into 7 Similar Rectangles

6 March, 2023

This is a continuation of my post Dividing a square into similar rectangles, in which I discussed this puzzle: if you partition a square into n similar rectangles, what proportions can these rectangles have?

Some people on Mathstodon put a lot of work into this and made some nice progress. But there’s been a surprising new twist! I’m not talking about how the answer to this puzzle is now listed on the listed on the Online Encyclopedia of Integer Sequences as sequence A359146. I’m not even talking about the fact that the New York Times ran an article about this puzzle:

• Siobhan Roberts, The quest to find rectangles in a square, New York Times, February 7, 2023. Open-access version here.

Remember the story so far. When n = 3 there are just 3 options for what the proportions of the rectangles can be:

I asked folks on Mathstodon about the case n = 4 and they found 11 options, as drawn here by Dan Piker:

From then on the number increases steeply. The biggest case anyone has been able to handle is n = 7. Ian Henderson found 1371 options in that case:

And that was great… but now the story has gotten more interesting, because Daniel Gerbet seems to have found one more! Yes: not 1371, but 1372.

He did the computation twice, in two different ways. He sent me a table listing all 1372 allowed proportions in increasing order, together with the polynomial equations they obey and pictures of the subdivided square. He wrote:

After running the computation again, I was able to identify the possibly missing guy exactly: the 1055th ratio. The pictures of the partitions look all the same for smaller and larger ratios in the tables from Ian Henderson and the one I computed. You find it at page 106 in the tabletable for partitions with 7 rectangles.

The ratio is the only real root of the polynomial

x^7 - 3x^6 + 9x^5 - 10x^4 + 12x^3 - 7x^2 + 2x - 1

You can find a picture of a partition with this ratio in the files attached. Note that Ian Henderson has computed partitions with other ratios, but the same topology as this one, e.g. the 1022th ratio.

Indeed it’s an interesting fact, first noted by Lisanne Taams in a simpler example, that we can get two partitions with the same topology where the rectangles have different proportions!

Here is a picture of the apparently new partition that Daniel Gerbet has found:

You can also see it as the 1055th partition in this list, where the 1372 partitions are conveniently grouped into 25 groups of 54, plus 22 more:

I said “conveniently”, but of course I was kidding—it’s easier to deal with these pictures in a PDF file. Daniel Gerbet has kindly allowed me to share his PDFs for all the interesting cases up to n = 7:

All 3 allowed proportions for 3 similar rectangles that subdivide the square.

All 11 allowed proportions for 4 similar rectangles that subdivide the square.

All 51 allowed proportions for 5 similar rectangles that subdivide the square.

All 245 allowed proportions for 6 similar rectangles that subdivide the square.

All 1372 allowed proportions for 7 similar rectangles that subdivide the square. Also, don’t forget the more detailed table listing the polynomials obeyed by these proportions.

You can also get a copy of the Python program that Daniel Gerbet used to obtain his results.

By the way, when I mentioned this news to my friend Todd Trimble he instantly noticed that 1372 is divisible by 7. Coincidence? Probably.

Some good news here is that Daniel Gerbet’s work confirms the previous calculations for n ≤ 6. And that’s especially good because previously only one person, Ian Henderson, had done the n = 6 case. See my previous article for details).

But we need some people to help settle the case n = 7!

Addendum: Greg Egan has confirmed this new solution is real. Ian Henderson has confirmed that it’s new, and he’s found the bug in his program that made him overlook it:

It is real! I made a mistaken assumption in my code to try to save computation time — instead of trying every way of orienting the N rectangles, it always orients the last rectangle the same way, under the assumption that the same arrangement but flipped will be generated anyway.

The issue is that “last” is in the order the rectangles are visited, which may change when the square is flipped! (See attached image for the arrangement here—the last rectangle is #6:

Either changing it to always orient the first rectangle in the same way (since it’s always in the top left) or just disabling that code entirely does give a total count of 1372.