Nosimpler

Johannes Kepler loved geometry, so of course he was fascinated by Platonic solids. His early work Mysterium Cosmographicum, written in 1596, includes pictures showing how the 5 Platonic solids correspond to the 5 elements:

Five elements? Yes, besides earth, air, water and fire, he includes a fifth element that doesn’t feel the Earth’s gravitational pull: the ‘quintessence’, or ‘aether’, from which heavenly bodies are made.

In the same book he also tried to use the Platonic solids to explain the orbits of the planets:

The six planets are Mercury, Venus, Earth, Mars, Jupiter and Saturn. And the tetrahedron and cube, in case you’re wondering, sit outside the largest sphere shown above. You can see them another picture from Kepler’s book:

These ideas may seem goofy now, but studying the exact radii of the planets’ orbits led him to discover that these orbits aren’t circular: they’re ellipses! By 1619 this led him to what we call Kepler’s laws of planetary motion. And those, in turn, helped Newton verify Hooke’s hunch that the force of gravity goes as the inverse square of the distance between bodies!

In honor of this, the problem of a particle orbiting in an inverse square force law is called the Kepler problem.

So, I’m happy that Greg Egan, Layra Idarani and I have come across a solid mathematical connection between the Platonic solids and the Kepler problem.

But this involves a detour into the 4th dimension!

It’s a remarkable fact that the Kepler problem has not just the expected conserved quantities—energy and the 3 components of angular momentum—but also 3 more: the components of the Runge–Lenz vector. To understand those extra conserved quantities, go here:

• Greg Egan, The ellipse and the atom.

Noether proved that conserved quantities come from symmetries. Energy comes from time translation symmetry. Angular momentum comes from rotation symmetry. Since the group of rotations in 3 dimensions, called SO(3), is itself 3-dimensional, it gives 3 conserved quantities, which are the 3 components of angular momentum.

None of this is really surprising. But if we take the angular momentum together with the Runge–Lenz vector, we get 6 conserved quantities—and these turn out to come from the group of rotations in 4 dimensions, SO(4), which is itself 6-dimensional. The obvious symmetries in this group just rotate a planet’s elliptical orbit, while the unobvious ones can also squash or stretch it, changing the eccentricity of the orbit.

(To be precise, all this is true only for the ‘bound states’ of the Kepler problem: the circular and elliptical orbits, not the parabolic or hyperbolic ones, which work in a somewhat different way. I’ll only be talking about bound states in this post!)

Why should the Kepler problem have symmetries coming from rotations in 4 dimensions? This is a fascinating puzzle—we know a lot about it, but I doubt the last word has been spoken. For an overview, go here:

• John Baez, Mysteries of the gravitational 2-body problem.

This SO(4) symmetry applies not only to the classical mechanics of the inverse square force law, but also the quantum mechanics! Nobody cares much about the quantum mechanics of two particles attracting gravitationally via an inverse square force law—but people care a lot about the quantum mechanics of hydrogen atoms, where the electron and proton attract each other via their electric field, which also obeys an inverse square force law.

So, let’s talk about hydrogen. And to keep things simple, let’s pretend the proton stays fixed while the electron orbits it. This is a pretty good approximation, and experts will know how to do things exactly right. It requires only a slight correction.

It turns out that wavefunctions for bound states of hydrogen can be reinterpreted as functions on the 3-sphere, S³ The sneaky SO(4) symmetry then becomes obvious: it just rotates this sphere! And the Hamiltonian of the hydrogen atom is closely connected to the Laplacian on the 3-sphere. The Laplacian has eigenspaces of dimensions n² where n = 1,2,3,…, and these correspond to the eigenspaces of the hydrogen atom Hamiltonian. The number n is called the principal quantum number, and the hydrogen atom’s energy is proportional to -1/n².

If you don’t know all this jargon, don’t worry! All you need to know is this: if we find an eigenfunction of the Laplacian on the 3-sphere, it will give a state where the hydrogen atom has a definite energy. And if this eigenfunction is invariant under some subgroup of SO(4), so will this state of the hydrogen atom!

The biggest finite subgroup of SO(4) is the rotational symmetry group of the 600-cell, a wonderful 4-dimensional shape with 120 vertices and 600 dodecahedral faces. The rotational symmetry group of this shape has a whopping 7,200 elements! And here is a marvelous moving image, made by Greg Egan, of an eigenfunction of the Laplacian on S³ that’s invariant under this 7,200-element group:

We’re seeing the wavefunction on a moving slice of the 3-sphere, which is a 2-sphere. This wavefunction is actually real-valued. Blue regions are where this function is positive, yellow regions where it’s negative—or maybe the other way around—and black is where it’s almost zero. When the image fades to black, our moving slice is passing through a 2-sphere where the wavefunction is almost zero.

For a full explanation, go here:

• Greg Egan, In the chambers with seven thousand symmetries, 2 January 2018.

Layra Idarani has come up with a complete classification of all eigenfunctions of the Laplacian on S³ that are invariant under this group… or more generally, eigenfunctions of the Laplacian on a sphere of any dimension that are invariant under the even part of any Coxeter group. For the details, go here:

• Layra Idarani, SG-invariant polynomials, 4 January 2018.

All that is a continuation of a story whose beginning is summarized here:

• John Baez, Quantum mechanics and the dodecahedron.

So, there’s a lot of serious math under the hood. But right now I just want to marvel at the fact that we’ve found a wavefunction for the hydrogen atom that not only has a well-defined energy, but is also invariant under this 7,200-element group. This group includes the usual 60 rotational symmetries of a dodecahedron, but also other much less obvious symmetries.

I don’t have a good picture of what these less obvious symmetries do to the wavefunction of a hydrogen atom. I understand them a bit better classically—where, as I said, they squash or stretch an elliptical orbit, changing its eccentricity while not changing its energy.

We can have fun with this using the old quantum theory—the approach to quantum mechanics that Bohr developed with his colleague Sommerfeld from 1920 to 1925, before Schrödinger introduced wavefunctions.

In the old Bohr–Sommerfeld approach to the hydrogen atom, the quantum states with specified energy, total angular momentum and angular momentum about a fixed axis were drawn as elliptical orbits. In this approach, the symmetries that squash or stretch elliptical orbits are a bit easier to visualize:

This picture by Pieter Kuiper shows some orbits at the 5th energy level, n = 5: namely, those with different eigenvalues of the total angular momentum, ℓ.

While the old quantum theory was superseded by the approach using wavefunctions, it’s possible to make it mathematically rigorous for the hydrogen atom. So, we can draw elliptical orbits that rigorously correspond to a basis of wavefunctions for the hydrogen atom. So, I believe we can draw the orbits corresponding to the basis elements whose linear combination gives the wavefunction shown as a function on the 3-sphere in Greg’s picture above!

We should get a bunch of ellipses forming a complicated picture with dodecahedral symmetry. This would make Kepler happy.

As a first step in this direction, Greg drew the collection of orbits that results when we take a circle and apply all the symmetries of the 600-cell:

For more details, read this:

• Greg Egan, Kepler orbits with the symmetries of the 600-cell.

Postscript

To do this really right, one should learn a bit about ‘old quantum theory’. I believe people have been getting it a bit wrong for quite a while—starting with Bohr and Sommerfeld!

If you look at the ℓ = 0 orbit in the picture above, it’s a long skinny ellipse. But I believe it really should be a line segment straight through the proton: that’s what’s an orbit with no angular momentum looks like.

There’s a paper about this:

• Manfred Bucher, Rise and fall of the old quantum theory.

Matt McIrvin had some comments on this:

This paper from 2008 is a kind of thing I really like: an exploration of an old, incomplete theory that takes it further than anyone actually did at the time.

It has to do with the Bohr-Sommerfeld “old quantum theory”, in which electrons followed definite orbits in the atom, but these were quantized–not all orbits were permitted. Bohr managed to derive the hydrogen spectrum by assuming circular orbits, then Sommerfeld did much more by extending the theory to elliptical orbits with various shapes and orientations. But there were some problems that proved maddeningly intractable with this analysis, and it eventually led to the abandonment of the “orbit paradigm” in favor of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics, what we know as modern quantum theory.

The paper argues that the old quantum theory was abandoned prematurely. Many of the problems Bohr and Sommerfeld had came not from the orbit paradigm per se, but from a much simpler bug in the theory: namely, their rejection of orbits in which the electron moves entirely radially and goes right through the nucleus! Sommerfeld called these orbits “unphysical”, but they actually correspond to the s orbital states in the full quantum theory, with zero angular momentum. And, of course, in the full theory the electron in these states does have some probability of being inside the nucleus.

So Sommerfeld’s orbital angular momenta were always off by one unit. The hydrogen spectrum came out right anyway because of the happy accident of the energy degeneracy of certain orbits in the Coulomb potential.

I guess the states they really should have been rejecting as “unphysical” were Bohr’s circular orbits: no radial motion would correspond to a certain zero radial momentum in the full theory, and we can’t have that for a confined electron because of the uncertainty principle.

Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.

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It is demonstrated is this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of $O{KN+C_pN\log^2N}$ such that the error vanishes as $O{N^{-p}}$ where $p\in\{1,2,3,4\}$ and $K$ is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula ($C_p=p^3$) and the implicit Adams method ($C_p=(p-1)^3$) of which the latter proves to be the most accurate family of methods for fast NFT.

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Chemistries exhibiting complex dynamics—from inorganic oscillators to gene regulatory networks—have been long known but either cannot be reprogrammed at will or rely on the sophisticated enzyme chemistry underlying the central dogma. Can simpler molecular mechanisms, designed from scratch, exhibit the same range of behaviors? Abstract chemical reaction networks have been proposed as a programming language for complex dynamics, along with their systematic implementation using short synthetic DNA molecules. We developed this technology for dynamical systems by identifying critical design principles and codifying them into a compiler automating the design process. Using this approach, we built an oscillator containing only DNA components, establishing that Watson-Crick base-pairing interactions alone suffice for complex chemical dynamics and that autonomous molecular systems can be designed via molecular programming languages.

Axo-myelinic neurotransmission: a novel mode of cell signalling in the central nervous system

Axo-myelinic neurotransmission: a novel mode of cell signalling in the central nervous system, Published online: 14 December 2017; doi:10.1038/nrn.2017.166

Axo-myelinic neurotransmission: a novel mode of cell signalling in the central nervous system

Author(s): C. Marletto and V. Vedral

Two proposals describe how to test whether gravity is inherently quantum by measuring the entanglement between two masses.

[Phys. Rev. Lett. 119, 240402] Published Wed Dec 13, 2017

Humans use rules to organize their actions to achieve specific goals. Although simple rules that link a sensory stimulus to one response may suffice in some situations, often, the application of multiple, hierarchically organized rules is required. Recent theories suggest that progressively higher level rules are encoded along an anterior-to-posterior gradient within PFC. Although some evidence supports the existence of such a functional gradient, other studies argue for a lesser degree of specialization within PFC. We used fMRI to investigate whether rules at different hierarchical levels are represented at distinct locations in the brain or encoded by a single system. Thirty-seven male and female participants represented and applied hierarchical rule sets containing one lower-level stimulus–response rule and one higher-level selection rule. We used multivariate pattern analysis to investigate directly the representation of rules at each hierarchical level in absence of information about rules from other levels or other task-related information, thus providing a clear identification of low- and high-level rule representations. We could decode low- and high-level rules from local patterns of brain activity within a wide frontoparietal network. However, no significant difference existed between regions encoding representations of rules from both levels except for precentral gyrus, which represented only low-level rule information. Our findings show that the brain represents conditional rules regardless of their level in the explored hierarchy, so the human control system did not organize task representation according to this dimension. Our paradigm represents a promising approach to identifying critical principles that shape this control system.

SIGNIFICANCE STATEMENT Several recent studies investigating the organization of the human control system propose that rules at different control levels are organized along an anterior-to-posterior gradient within PFC. In this study, we used multivariate pattern analysis to explore independently the representation of formally identical conditional rules belonging to different levels of a cognitive hierarchy and provide for the first time a clear identification of low- and high-level rule representations. We found no major spatial differences between regions encoding rules from different hierarchical levels. This suggests that the human brain does not use levels in the investigated hierarchy as a topographical organization principle to represent rules controlling our behavior. Our paradigm represents a promising approach to identifying which principles are critical.

Not the "national debt," but the debts of individual Americans, as depicted on this map (my embed is a screencap - the interactive version here lets you zoom in on your state and your county for data).

2016 data derived from a random sample of deidentified, consumer-level records from a major credit bureau, as well as estimates from summary tables of the US Census Bureau’s American Community Survey (2015 or 2011–15)... Debt in collections includes past-due credit lines that have been closed and charged-off on their books as well as unpaid bills reported to the credit bureaus that the creditor is attempting to collect.

Links to various commentaries at Digg.  My attention was drawn to Minnesota...

A previous analysis by the Urban Institute focused on medical debt, and found one reason it was so concentrated in the South was because the uninsured rates tended to be higher. While that changed to some extent with the Affordable Care Act, many Southern states chose not to expand Medicaid. On the other hand, Minnesota — which has the lowest rates of debt — has one of the most generous Medicaid programs in the country, and a more inclusive and higher-quality health care system.

...only 3% of Minnesotans have medical debt in collections, and only one county (rural Clearwater County) has medical debt rates over 11%.

Compare that picture to the state of medical debt in the rest of the country. Nationwide, 18% of people have medical debt in collections, and, as CityLab noted, much of that debt is concentrated in states that chose not to expand Medicaid under the ACA.

Most Washington politicians are tone-deaf to the financial crises experienced by so many Americans.  They are busy waging their internecine battles, meeting with lobbyists, and pandering to their donors.

Jellyfish have been referred to as the "cockroaches of the sea," with reference to both species' ability to survive under the harshest conditions.  An article in the newest edition of The Atlantic reviews a new book about jellyfish:

Their delicacy notwithstanding, in recent decades jellyfish species have come to be thought of as the durable and opportunistic inheritors of our imperiled seas. Jellyfish blooms—the intermittent, and now widely reported, flourishing of vast swarms—are held by many to augur the depletion of marine biomes; they are seen as a signal that the oceans have been overwarmed, overfished, acidified, and befouled... The vision—hat tipped to science fiction—is of the planet’s oceans transformed into something like an aspic terrine. In waters thickened by the gummy mucus of living and dead jellyfish, other sea life will be smothered. Because jellyfish recall the capsules of single-celled protozoa, this eventuality invites portrayal as a devolution of the marine world—a reversion to the “primordial soup.”..

Perhaps the most complex issue Berwald takes on is jellyfish blackouts. Sweden, Scotland, the Philippines, Tokyo, California, and Israel have all suffered intermittent electrical outages caused by jellyfish sucked into the intake pipes and cooling systems of coal-fired and nuclear power stations... In Spineless, Berwald travels to Spain’s Murcia region and takes us to the Mar Menor lagoon, which had become so jellified in 2002 that “you couldn’t drive a boat through the water.” Here barrel and fried-egg jellyfish are pernicious—so much so that they’re removed from the sea by the bargeload and dumped into ditches near the airport.

More at that link. Then today I found a report of jellyfish menacing Chinese aircraft carriers:

In 2006, the aircraft carrier USS Ronald Reagan was incapacitated while visiting Brisbane, Australia due to blubber jellyfish swarms. Reportedly, cooling pipes for the ship’s nuclear reactor were clogged with the foot-wide jellies, necessitating an evacuation of the carrier...

Ironically, the jellyfish problem is partially of China’s own doing. As many as one hundred million sharks are killed each year, much of it in the form of bycatch in an attempt to catch other forms of seafood but also for shark’s fin soup, a delicacy in China. Although demand for the soup has declined in recent years, the shark population is still way down. Sharks are a major predator of jellyfish and scientists believe their absence is a major reason why jellyfish populations have exploded.

Related: Your children will eat jellyfish for dinner.

Photo credit: GettyLucia /Terui

Similarity search—for example, identifying similar images in a database or similar documents on the web—is a fundamental computing problem faced by large-scale information retrieval systems. We discovered that the fruit fly olfactory circuit solves this problem with a variant of a computer science algorithm (called locality-sensitive hashing). The fly circuit assigns similar neural activity patterns to similar odors, so that behaviors learned from one odor can be applied when a similar odor is experienced. The fly algorithm, however, uses three computational strategies that depart from traditional approaches. These strategies can be translated to improve the performance of computational similarity searches. This perspective helps illuminate the logic supporting an important sensory function and provides a conceptually new algorithm for solving a fundamental computational problem.

“But justice roll down like waters and righteousness like an ever-flowing stream” Amos 5:24 https://t.co/o89PSY1YBd

— James Comey (@Comey) December 1, 2017

Notwithstanding the usefulness of system dynamics in analyzing complex policy problems, policy design is far from straightforward and in many instances trial-and-error driven. To address this challenge, we propose to combine system dynamics with network controllability, an emerging field in network science, to facilitate the detection of effective leverage points in system dynamics models and thus to support the design of influential policies. We illustrate our approach by analyzing a classic system dynamics model: the World Dynamics model. We show that it is enough to control only 53% of the variables to steer the entire system to an arbitrary final state. We further rank all variables according to their importance in controlling the system and we validate our approach by showing that high ranked variables have a significantly larger impact on the system behavior compared to low ranked variables.

If and when you emerged from your happiness bubble to read the news, you’ll have seen (at least if you live in the US) that the cruel and reckless tax bill has passed the House of Representatives, and remains only to be reconciled with an equally-vicious Senate bill and then voted on by the Republican-controlled Senate.  The bill will add about $1.7 trillion to the national debt and raise taxes for about 47.5 million people, all in order to deliver a massive windfall to corporations, and to wealthy estates that already pay some of the lowest taxes in the developed world.

In a still-functioning democracy, those of us against such a policy would have an intellectual obligation to seek out the strongest arguments in favor of the policy and try to refute them.  By now, though, it seems to me that the Republicans hold the public in such contempt, and are so sure of the power of gerrymandering and voter restrictions to protect themselves from consequences, that they didn’t even bother to bring anything to the debate more substantive than the schoolyard bully’s “stop punching yourself.”  I guess some of them still repeat the fairytale about the purpose of tax cuts for the super-rich being to trickle down and help everyone else—but can even they advance that “theory” anymore without stifling giggles?  Mostly, as far as I can tell, they just brazenly deny that they’re doing what they obviously are doing: i.e., gleefully setting on fire anything that anyone, regardless of their ideology, could recognize as the national interest, in order to enrich a small core of supporters.

But none of that is what interests me in this post—because it’s “merely” as bad as, and no worse than, what one knew to expect when a coalition of thugs, kleptocrats, and white-nationalist demagogues seized control of Hamilton’s and Jefferson’s experiment.  My concern here is only with the “kill shot” that the Republicans have now aimed, with terrifying precision, at the system that’s kept American academic science the envy of the world in spite of the growing dysfunction all around it.

As you’ve probably heard, one of the ways Republicans intend to pay for their tax giveaway, is to change the tax code so that graduate students will now need to pay taxes on “tuition”—a large sum of money (as much as $50,000/year) that PhD students never actually see, that can easily exceed the stipends they do see, and that’s basically just an accounting trick that serves the internal needs of universities and granting agencies.  Again, to eliminate any chance of misunderstanding: PhD students, who are effectively low-wage employees, already pay taxes on their actual stipends.  The new proposal is that they’ll also have to pay taxes on a whopping, make-believe “X” on their payroll sheet that’s always exactly balanced out by “-X.”

For detailed analyses of the impacts, see, e.g. Luca Trevisan’s post or Inside Higher Ed or the Chronicle of Higher Ed or Vox or NPR.  Briefly, though, the proposal would raise taxes by a few thousand dollars per year, or in some cases as much as $10,000 per year (!), on PhD students who already live hand-to-mouth-to-ramen-bowl, with the largest impact falling on students in STEM fields.  For many students who aren’t independently wealthy, this could push a PhD beyond the realm of affordability, and cause them to leave academia or to do their graduate work in other countries.

“But isn’t there some workaround?”  Indeed, financial ignoramus that I am, my first reaction was to ask: if PhD tuition is basically an accounting fiction anyway, then why can’t the universities just declare that the tuition in question no longer exists, or is now zero dollars?  Feel free to explain further in the comments if you understand this stuff, but as far as I can tell, the answer is: because PhD tuition is used to calculate how much “tax” the universities can take from professors’ grant money.  If universities could no longer take that tax, and they had no other way to make up for it, then except for the richest few universities, they’d have to scale back research and teaching pretty drastically.  To avoid that outcome, the universities would be relying on the granting agencies to let them keep taking the overhead they needed to operate, even though the “PhD tuition” no longer existed.  But the granting agencies aren’t set up for this: you can’t just throw a bomb into one part of a complicated bureaucratic machine built up over decades, and expect the machine to continue working with no disruption to science.

But more ominously: as my friend Daniel Harlow and many others pointed out, it’s hard to look at the indefensible, laser-specific meanness of this policy, without suspecting that for many in Congress, the destruction of American higher education isn’t a regrettable byproduct, but the goal—just another piece of red meat to throw to the base.  If so, then we’d expect Congress to direct federal granting agencies not to loosen their rules about overhead, thereby forcing the students to pay the tax, and achieving the desired destruction.  (Note that the Trump administration has already made tightening overhead rules—i.e., doing the exact opposite of what would be needed to counteract the new tax—a central focus of its attempt to cut federal research funding.)

OK, two concluding thoughts:

1. When Republicans in Congress defended Trump’s travel ban, they at least had the craven excuse that they were only following the lead of the populist strongman who’d taken over their party.  Here they don’t even have that.  As far as I know, this targeted destruction of American higher education was Congress’s initiative, not Trump’s—which to me, underscores again the feather-thinness of any moral distinction between the Vichy GOP leadership and the administration with which it collaborates.  Trump didn’t emerge from nowhere.  It took decades of effort—George W. Bush, Sarah Palin, Karl Rove, Rush Limbaugh, Mitch McConnell, and all the rest—to transform the GOP into the pure seething cauldron of anti-intellectual resentment and hatred that we know today.
2. Given the existential risk to American higher education, why didn’t I blog about this earlier?  The answer is embarrassing to admit, and reflects no credit on me.  It’s simply that I didn’t believe it—even given all the other stuff that could “never happen in the US,” until it happened this past year.  I didn’t believe it, not because it was too far from me but because it was too close—because if true, it would mean the crippling of the research world in which I’ve spent most of my life since age 15, so therefore it couldn’t be true.  Surely even the House Republicans would realize they’d screwed up this time, and would take out this crazy provision before the full bill was voted on?  Or surely there’s some workaround that makes the whole thing less awful than it sounds?  There has to be … right?

Anyway, what else is there to say, except to call your representative, if you’re American and still have the faith in the system that such an act implies.

Since returning from a vacation partly spent isolated from the internet, I’ve been catching up and noticed that some of the most prominent sources of funding for math and physics research have been making the news:

• The New York Times and other sources have extensive reports based on leaked records from an offshore law firm that specializes in helping you avoid inconvenient US tax and reporting requirements. The story starts out with the example of Jim Simons, who has become the largest non-governmental funder of math and physics research. His Simons Foundation has been doing an excellent job of providing such funding. They have about \$3 billion in assets, annual income of around \$500 million. The Times reports that Simons (with a net worth of about \$18.5 billion) has an offshore version of the Foundation, the Simons Foundation International, with assets of \$8 billion, dwarfing the onshore version.
• The assets of these Foundations are presumably largely invested in the secretive and extremely successful Renaissance Technologies hedge fund, which also is the employer of quite a few physicists and mathematicians. I’ve asked many people over the years, but have never found anyone who knows (or will admit to knowing) what it is that RenTech does that is so successful. A peculiar aspect of the coming age of private math/physics research funding is that no one getting this funding really knows where the money comes from.

  In other news while I was away the CEO of RenTech, Robert Mercer, was finally induced to leave. Mercer had drawn a lot of attention recently since he in recent years has been taking the opposite tack to Simons, funding institutions devoted to promoting untruth over truth (e.g. Breitbart News), achieving fantastic success last year. He also has branched out from doing whatever secretive things RenTech does to make mountains of money using computers and data, starting up a firm called Cambridge Analytica, a firm involved in secretively using computers and data to undermine democracy in the US and elsewhere. I had been wondering for quite a while what Simons thought of Mercer’s activities. My understanding of highly-paid finance jobs was that your employer pays you a lot of money in return for having your full attention and devotion to not having negative stories about them come to public attention, so Mercer’s continued employment was surprising. It seems that Simons finally had enough, after realizing how much damage Mercer was doing to his firm, in particular by creating a situation that would discourage many people from wanting to work there (there also was a campaign underway to get institutions to divest from investments with RenTech).

• Another high profile source of funding for math and physics, in this case for cash prizes to mathematicians and physicists, has been venture capitalist Yuri Milner, with his Breakthrough Prize organization. New prizes will be announced in three weeks at a December 3 prize ceremony (I also believe there will be an associated Breakthrough Prize symposium held at Stanford shortly thereafter). It has always been well-known that much of Milner’s wealth derived from investments in Facebook and Twitter. Less well-known and recently revealed was that a major source of the funds for these investments was Russian state organizations closely tied to Vladimir Putin.
• Turning to sources of public funding, there’s not very positive news about a possible ILC collider in Japan, with reports of a cutback of the proposal from a 500 GeV to a 250 GeV machine (which would still cost about $7 billion).
• Foreign policy magazine has an article discussing the proposal for a huge new collider in China (discussed here). The point of view of the article is quite critical of the idea of locating a huge new project in a country with an increasingly authoritarian regime:
  

  China’s next-generation supercollider will unlock secrets of the universe — and destroy the ideals of the scientists running it.

  Luckily, for another more local prominent large country with an increasingly authoritarian and xenophobic regime, the issue of a possible problem with locating an international collider project there isn’t likely to come up since its leaders have no interest in funding such projects.

Scientists strive to understand how functionalities, such as conservation laws, emerge in complex systems. Living complex systems in particular create high-ordered functionalities by pairing up low-ordered complementary processes, e.g., one process to build and the other to correct. We propose a network mechanism that demonstrates how collective statistical laws can...

Humans seem to decide for themselves what to do, and when to do it. This distinctive capacity may emerge from an ability, shared with other animals, to make decisions for action that are related to future goals, or at least free from the constraints of immediate environmental inputs. Studying such volitional acts proves a major challenge for neuroscience. This review highlights key mechanisms in the generation of voluntary, as opposed to stimulus-driven actions, and highlights three issues. The first part focuses on the apparent spontaneity of voluntary action. The second part focuses on one of the most distinctive, but elusive, features of volition, namely, its link to conscious experience, and reviews stimulation and patient studies of the cortical basis of conscious volition down to the single-neuron level. Finally, we consider the goal-directedness of voluntary action, and discuss how internal generation of action can be linked to goals and reasons.

In preparation for the Applied Category Theory special session at U.C. Riverside this weekend, my crew dropped three papers on the arXiv.

My student Adam Yassine has been working on Hamiltonian and Lagrangian mechanics from an ‘open systems’ point of view:

• Adam Yassine, Open systems in classical mechanics.

  Abstract. Using the framework of category theory, we formalize the heuristic principles that physicists employ in constructing the Hamiltonians for open classical systems as sums of Hamiltonians of subsystems. First we construct a category where the objects are symplectic manifolds and the morphisms are spans whose legs are surjective Poisson maps. Using a slight variant of Fong’s theory of decorated cospans, we then decorate the apices of our spans with Hamiltonians. This gives a category where morphisms are open classical systems, and composition allows us to build these systems from smaller pieces.

He also gets a functor from a category of Lagrangian open systems to this category of Hamiltonian systems.

Kenny Courser and I have been continuing my work with Blake Pollard and Brendan Fong on open Markov processes, bringing 2-morphisms into the game. It seems easiest to use a double category:

• John Baez and Kenny Courser, Coarse-graining open Markov processes.

Abstract. Coarse-graining is a standard method of extracting a simple Markov process from a more complicated one by identifying states. Here we extend coarse-graining to open Markov processes. An ‘open’ Markov process is one where probability can flow in or out of certain states called ‘inputs’ and ‘outputs’. One can build up an ordinary Markov process from smaller open pieces in two basic ways: composition, where we identify the outputs of one open Markov process with the inputs of another, and tensoring, where we set two open Markov processes side by side. In previous work, Fong, Pollard and the first author showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category. Here we go further by constructing a symmetric monoidal double category where the 2-morphisms are ways of coarse-graining open Markov processes. We also extend the already known ‘black-boxing’ functor from the category of open Markov processes to our double category. Black-boxing sends any open Markov process to the linear relation between input and output data that holds in steady states, including nonequilibrium steady states where there is a nonzero flow of probability through the process. To extend black-boxing to a functor between double categories, we need to prove that black-boxing is compatible with coarse-graining.

Finally, the Complex Adaptive Systems Composition and Design Environment project with John Foley of Metron Scientific Solutions and my students Joseph Moeller and Blake Pollard has finally given birth to a paper! I hope this is just the first; it starts laying down the theoretical groundwork for designing networked systems. John is here now and we’re coming up with a bunch of new ideas:

• John Baez, John Foley, Joseph Moeller and Blake Pollard, Network models.

Abstract. Networks can be combined in many ways, such as overlaying one on top of another or setting two side by side. We introduce network models to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to Cat, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.

I blogged about this last one here:

• Complex adaptive systems (part 6), Azimuth.

Network control principles predict neuron function in the Caenorhabditis elegans connectome

Nature 550, 7677 (2017). doi:10.1038/nature24056

Authors: Gang Yan, Petra E. Vértes, Emma K. Towlson, Yee Lian Chew, Denise S. Walker, William R. Schafer & Albert-László Barabási

Recent studies on the controllability of complex systems offer a powerful mathematical framework to systematically explore the structure–function relationship in biological, social, and technological networks. Despite theoretical advances, we lack direct experimental proof of the validity of these widely used control principles. Here we fill this gap by applying a control framework to the connectome of the nematode Caenorhabditis elegans, allowing us to predict the involvement of each C. elegans neuron in locomotor behaviours. We predict that control of the muscles or motor neurons requires 12 neuronal classes, which include neuronal groups previously implicated in locomotion by laser ablation, as well as one previously uncharacterized neuron, PDB. We validate this prediction experimentally, finding that the ablation of PDB leads to a significant loss of dorsoventral polarity in large body bends. Importantly, control principles also allow us to investigate the involvement of individual neurons within each neuronal class. For example, we predict that, within the class of DD motor neurons, only three (DD04, DD05, or DD06) should affect locomotion when ablated individually. This prediction is also confirmed; single cell ablations of DD04 or DD05 specifically affect posterior body movements, whereas ablations of DD02 or DD03 do not. Our predictions are robust to deletions of weak connections, missing connections, and rewired connections in the current connectome, indicating the potential applicability of this analytical framework to larger and less well-characterized connectomes.

The mammalian neocortex contains many cell types, but whether they organize into repeated structures has been unclear. We discovered that major cell types in neocortical layer 5 form a lattice structure in many brain areas. Large-scale three-dimensional imaging revealed that distinct types of excitatory and inhibitory neurons form cell type–specific radial clusters termed microcolumns. Thousands of microcolumns, in turn, are patterned into a hexagonal mosaic tessellating diverse regions of the neocortex. Microcolumn neurons demonstrate synchronized in vivo activity and visual responses with similar orientation preference and ocular dominance. In early postnatal development, microcolumns are coupled by cell type–specific gap junctions and later serve as hubs for convergent synaptic inputs. Thus, layer 5 neurons organize into a brainwide modular system, providing a template for cortical processing.

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I’ve been slacking off on writing this series of posts… but for a good reason: I’ve been busy writing a paper on the same topic! In the process I caught a couple of mistakes in what I’ve said so far. But more importantly, there’s a version out now, that you can read:

• John Baez, John Foley, Blake Pollard and Joseph Moeller, Network models.

There will be two talks about this at the AMS special session on Applied Category Theory this weekend at U. C. Riverside: one by John Foley of Metron Inc., and one by my grad student Joseph Moeller. I’ll try to get their talk slides someday. But for now, here’s the basic idea.

Our goal is to build operads suited for designing networks. These could be networks where the vertices represent fixed or moving agents and the edges represent communication channels. More generally, they could be networks where the vertices represent entities of various types, and the edges represent relationships between these entities—for example, that one agent is committed to take some action involving the other. This paper arose from an example where the vertices represent planes, boats and drones involved in a search and rescue mission in the Caribbean. However, even for this one example, we wanted a flexible formalism that can handle networks of many kinds, described at a level of detail that the user is free to adjust.

To achieve this flexibility, we introduced a general concept of ‘network model’. Simply put, a network model is a kind of network. Any network model gives an operad whose operations are ways to build larger networks of this kind by gluing smaller ones. This operad has a ‘canonical’ algebra where the operations act to assemble networks of the given kind. But it also has other algebras, where it acts to assemble networks of this kind equipped with extra structure and properties. This flexibility is important in applications.

What exactly is a ‘kind of network’? That’s the question we had to answer. We started with some examples, At the crudest level, we can model networks as simple graphs. If the vertices are agents of some sort and the edges represent communication channels, this means we allow at most one channel between any pair of agents.

However, simple graphs are too restrictive for many applications. If we allow multiple communication channels between a pair of agents, we should replace simple graphs with ‘multigraphs’. Alternatively, we may wish to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. This requires replacing simple graphs with ‘directed graphs’. To combine these features we could use ‘directed multigraphs’.

But none of these are sufficiently general. It’s also important to consider graphs with colored vertices, to specify different types of agents, and colored edges, to specify different types of channels. This leads us to ‘colored directed multigraphs’.

All these are examples of what we mean by a ‘kind of network’, but none is sufficiently general. More complicated kinds, such as hypergraphs or Petri nets, are likely to become important as we proceed.

Thus, instead of separately studying all these kinds of networks, we introduced a unified notion that subsumes all these variants: a ‘network model’. Namely, given a set of ‘vertex colors’, a network model is a lax symmetric monoidal functor

where is the free strict symmetric monoidal category on and is the category of small categories.

Unpacking this somewhat terrifying definition takes a little work. It simplifies in the special case where takes values in the category of monoids. It simplifies further when is a singleton, since then is the groupoid where objects are natural numbers and morphisms from to are bijections

If we impose both these simplifying assumptions, we have what we call a one-colored network model: a lax symmetric monoidal functor

As we shall see, the network model of simple graphs is a one-colored network model, and so are many other motivating examples. If you like André Joyal’s theory of ‘species’, then one-colored network models should be pretty fun, since they’re species with some extra bells and whistles.

But if you don’t, there’s still no reason to panic. In relatively down-to-earth terms, a one-colored network model amounts to roughly this. If we call elements of ‘networks with vertices’, then:

• Since is a monoid, we can overlay two networks with the same number of vertices and get a new one. We call this operation

• Since is a functor, the symmetric group acts on the monoid Thus, for each , we have a monoid automorphism that we call simply

• Since is lax monoidal, we also have an operation

We call this operation the disjoint union of networks. In examples like simple graphs, it looks just like what it sounds like.

Unpacking the abstract definition further, we see that these operations obey some equations, which we list in Theorem 11 of our paper. They’re all obvious if you draw pictures of examples… and don’t worry, our paper has a few pictures. (We plan to add more.) For example, the ‘interchange law’

holds whenever and This is a nice relationship between overlaying networks and taking their disjoint union.

In Section 2 of our apper we study one-colored network models, and give lots of examples. In Section 3 we describe a systematic procedure for getting one-colored network models from monoids. In Section 4 we study general network models and give examples of these. In Section 5 we describe a category of network models, and show that the procedure for getting network models from monoids is functorial. We also make into a symmetric monoidal category, and give examples of how to build new networks models by tensoring old ones.

Our main result is that any network model gives a typed operad, also known as a ‘colored operad’. This operad has operations that describe how to stick networks of the given kind together to form larger networks of this kind. This operad has a ‘canonical algebra’, where it acts on networks of the given kind—but the real point is that it has lots of other algebra, where it acts on networks of the given kind equipped with extra structure and properties.

The technical heart of our paper is Section 6, mainly written by Joseph Moeller. This provides the machinery to construct operads from network models in a functorial way. Category theorists should find this section interesting, because because it describes enhancements of the well-known ‘Grothendieck construction’ of the category of elements of a functor

where is any small category. For example, if is symmetric monoidal and is lax symmetric monoidal, then we show is symmetric monoidal. Moreover, we show that the construction sending the lax symmetric monoidal functor to the symmetric monoidal category is functorial.

In Section 7 we apply this machinery to build operads from network models. In Section 8 we describe some algebras of these operads, including an algebra whose elements are networks of range-limited communication channels. In future work we plan to give many more detailed examples, and to explain how these algebras, and the homomorphisms between them, can be used to design and optimize networks.

I want to explain all this in more detail—this is a pretty hasty summary, since I’m busy this week. But for now you can read the paper!

Complex networks are the subject of fundamental interest from the scientific community at large. Several metrics have been introduced to characterize the structure of these networks, such as the degree distribution, degree correlation, path length, clustering coefficient, centrality measures etc. Another important feature is the presence of network symmetries. In particular, the effect of these symmetries has been studied in the context of network synchronization, where they have been used to predict the emergence and stability of cluster synchronous states. Here we provide theoretical, numerical, and experimental evidence that network symmetries play a role in a substantially broader class of dynamical models on networks, including epidemics, game theory, communication, and coupled excitable systems. Namely, we see that in all these models, nodes that are related by a symmetry relation show the same time-averaged dynamical properties. This discovery leads us to propose reduction techniques for exact, yet minimal, simulation of complex networks dynamics, which we show are effective in order to optimize the use of computational resources, such as computation time and memory.

We present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two populations of (at least) two elements using a general method based on delayed interactions with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model-based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge chimera dynamics in phase models and real-world oscillator networks.

In network theory, a question of prime importance is how to assess network vulnerability in a fast and reliable manner. With this issue in mind, we investigate the response to parameter changes of coupled dynamical systems on complex networks. We find that for specific, non-averaged perturbations, the response of synchronous states critically depends on the overlap between the perturbation vector and the eigenmodes of the stability matrix of the unperturbed dynamics. Once averaged over properly defined ensembles of such perturbations, the response is given by new graph topological indices, which we introduce as generalized Kirchhoff indices. These findings allow for a fast and reliable method for assessing the specific or average vulnerability of a network against changing operational conditions, faults or external attacks.

We present a simple and computationally efficient coarse-grained and solvent-free model for simulating lipid bilayer membranes. In order to be used in concert with particle-based reaction-diffusion simulations, the model is purely based on interacting and reacting particles, each representing a coarse patch of a lipid monolayer. Particle interactions include nearest-neighbor bond-stretching and angle-bending, and are parameterized so as to reproduce the local membrane mechanics given by the Helfrich energy density over a range of relevant curvatures. In-plane fluidity is implemented with Monte Carlo bond-flipping moves. The physical accuracy of the model is verified by five tests: (i) Power spectrum analysis of equilibrium thermal undulations is used to verify that the particle-based representation correctly captures the dynamics predicted by the continuum model of fluid membranes. (ii) It is verified that the input bending stiffness, against which the potential parameters are optimized, is accurately recovered. (iii) Isothermal area compressibility modulus of the membrane is calculated and is shown to be tunable to reproduce available values for different lipid bilayers, independent of the bending rigidity. (iv) Simulation of two-dimensional shear flow under a gravity force is employed to measure the effective in-plane viscosity of the membrane model, and show the possibility of modeling membranes with specified viscosities. (v) Interaction of the bilayer membrane with a spherical nanoparticle is modeled as a test case for large membrane deformations and budding involved in cellular processes such as endocytosis...