Musk, on the other hand, is all in.
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The oligarchs are partying. They feel they have to.
Not long ago Trump threatened to send Zuckerberg to prison for life. Last night Zuckerberg, crushed and beaten, went to the big party in honor of Trump at Peter Thiel's house in DC, wearing a goofy bow tie. The Washington Post reports:
"Billionaires seen around Washington over the weekend included Miriam Adelson, the casino magnate and widow of Sheldon Adelson; Paul Singer, the hedge fund titan who is among the most influential Republican donors in the country; Mark Zuckerberg, the chief executive of Meta, who spent days party-hopping as part of his attempt to win a place in Mr. Trump’s orbit; and Sergey Brin, the co-founder of Google who eight years ago around this time was unexpectedly showing up at protests against Mr. Trump’s travel ban on some Muslim countries."
It looks like Trump is following Putin's playbook: threaten and coerce the oligarchs until they pay obeisance and support him - then reward them and make them dependent on him.
Want to do something about this? See my list of Trump resistance organizations:
https://mathstodon.xyz/@johncarlosbaez/113485662024271142
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Have scientists discovered a new organ that accounts for about 20% of your body weight?
It's called the 'interstitium'. It's a network of channels between the other organs. The fluid in these channels is called 'interstitial fluid', and it transports nutrients and molecules that carry communications between cells, especially involving the immune system. The interstitium drains into the lymph system, but it's not the same thing.
Here's an interesting show about why it remained unnoticed for so long. Basically the problem was that scientists were studying dried-up tissue samples instead of living tissue!
Thanks to @rustoleumlove for alerting me to the interstitium here on Mastodon before I heard about it in on the radio.
https://radiolab.org/podcast/interstitium
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A viroid is a little circle of RNA that manages to reproduce by infecting plants. Here's how it works.
When an infected leaf touches another leaf, a viroid may land on the other leaf. If it manages to get inside a cell of the other leaf, it's in luck. The plant's cell contains a protein called RNA polymerase, which it uses to copy RNA. The viroid takes advantage of this and gets duplicated!
How does the viroid get into the plant's cell? Simple: through the 'plasmodesma', which is like a little channel that the plant uses for transporting chemicals and communicating between cells.
Viroids can get transmitted with the help of aphids, which chew on plant leaves. That should make it easy for the viroids to get in and out of damaged cells.
Here's more detail on how viroids get replicated inside the cell - it's called 'rolling circle replication':
https://en.wikipedia.org/wiki/Rolling_circle_replication
And here's more about viroids:
https://en.wikipedia.org/wiki/Viroid
By the way, the argument about whether viruses and the even simpler viroids are 'life' seems silly to me. It's fun and even a bit useful to think about the different features that make something count as alive. But trying to define a sharp boundary between 'life' and 'non-life' seems like a waste of time. Some things are better at reproducing themselves than others, and viroids are pretty damn good at it.
Sure, they do so in a 'purely passive' way, and don't have any metabolism. But hey: if you don't need a metabolism to reproduce, why bother having one? 😏
(Hey! An emoji just reproduced itself!)
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2024-02-26
https://www.youtube.com/watch?v=7LGvunt8I8Q
The clock arithmetic of slopes: fixing ℤ/n by introducing formal reciprocals for all elements that don't have them, or better yet: introducing formal reciprocals for all elements, but identifying them with the actual reciprocals when those exist. We don't get a ring, just a set with 2n - ϕ(n) elements, where where ϕ(n) is the number of elements of ℤ/n that don't have reciprocals, called 'Euler's totient function':
https://en.wikipedia.org/wiki/Euler%27s_totient_function
The resulting set acts like a projective line for ℤ/n: indeed it's the set of ℤ/n-points for the scheme P¹. Apparently this set can also be seen as SL(2,ℤ)/Γ₀(n) where Γ₀(n), the so-called 'Hecke congruence subgroup of level n', consists of 2×2 integer matrices where the lower left entry equals 0 mod n. For more on congruence subgroups, see:
https://math.berkeley.edu/~sander/speaking/24June2015%20UMS%20Talk.pdf
Next, compare SL(2,ℤ)/Γ₀(n) to the modular curve X₀(n) = H/Γ₀(n) where H is the hyperbolic plane. This sets up connections to the dessin d'enfant for X₀(n):
https://en.wikipedia.org/wiki/Dessin_d%27enfant
and also the Coxeter group
o--3--o--∞--o
namely PGL(2,Z). Drawing a picture of X₀(11). Kostant on Galois' last letter and PSL(2,11):
Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, http://www.ams.org/notices/199509/kostant.pdf
Erratum: 24 is not the largest number such that every number smaller than it and relatively prime to it is prime; that honor goes to 30:
https://mathstodon.xyz/@johncarlosbaez/112017268404654926
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Wow! Biologists seem to have discovered an entirely new kind of life form. They're called 'obelisks', and you probably have some in you.
They were discovered in 2024 - not by somebody actually seeing one, but by analyzing huge amounts of genetic data from the human gut. This search found 29,959 new RNA sequences, completely different from any known. Thus, we don't know where these things fit into the tree of life!
Biologists found them when they were trying to solve a puzzle. Even smaller than viruses, there exist 'viroids' that are just loops of RNA that cleverly manage to reproduce using the machinery of the cell they infect. Viruses have a protein coat. Viroids are just bare RNA - it doesn't even code for any proteins!
But all known viroids only infect plants. The first one found causes a disease in potatoes; another causes a disease in avocados, and so on. This raised the puzzle: why aren't there viroids that infect bacteria, or animals?
Now perhaps we've found them! But not quite: while obelisks may work in a similar way, they seem genetically unrelated. Also, their RNA seems to code for two proteins.
Given how little we know about this stuff, I think some caution is in order. Still, this is really cool. Do any of you biologists out there know any research going on now to learn more?
The original paper is free to read on the bioRxiv:
• Viroid-like colonists of human microbiomes, https://www.biorxiv.org/content/10.1101/2024.01.20.576352v1.full
I see just one other paper, about an automated system for detecting obelisks:
• Tormentor: An obelisk prediction and annotation pipeline, https://www.biorxiv.org/content/10.1101/2024.05.30.596730v1.full
There's also a budding Wikipedia article on obelisks:
https://en.wikipedia.org/wiki/Obelisk_(biology)
Hat-tip to @metaweta.
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What I want to know is: what's the nearest star with a planet with life? Red dwarfs are common, but their habitable zone is close to the star and many of these stars put out powerful flares, like Proxima Centauri. The combination is not promising.
For more on the nearest stars, try this:
https://en.wikipedia.org/wiki/List_of_nearest_stars
You can see a nice rotating 3d map of them here:
https://upload.wikimedia.org/wikipedia/commons/c/c9/Stars-within-11-light-years2.webm
The map below shows 33 stars within 12.5 light years of us, made by Richard Powel.
(2/2)
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We may think of the stars in the sky as fixed, but if we wait the closest stars will eventually drift away - and other stars will come closer. This graph shows how that works. The vertical blue line is today.
Today the nearest star is Proxima Centauri, just 4¼ light years away. It's a small red dwarf with 2 or 3 planets. Unfortunately it shoots out big X-ray flares. It orbits two other stars: one, called Rigil Kentaurus, is a bit bigger than the Sun, while the other, Toliman, is a bit smaller. They orbit each other every 79 years, while Proxima Centauri orbits both of them every 500,000 years.
But if we wait, various other stars will drift by and temporarily become closer!
Barnard's Star will swing by and tie Proxima Centauri for a short time 10,000 years from now. It's just a bit bigger than Proxima Centauri, and it has one planet.
Ross 248 will be the closest star for about nine millennia starting 30,000 years from now. It's another red dwarf, with huge starspots due to its powerful magnetic field.
Then Gliese 445, yet another red dwarf, will become the closest star for a while. During this time the Voyager 1 probe will pass within 1.6 light years of this star.
And so it goes. I wonder when a star will get really close to the Sun, like 1 or 2 light years? This could shake up the Oort cloud, the cloud of icy bodies in our solar system that stretches out for 1.5 years. Then we'd get lots of comets!
(1/2)
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2024-01-22
https://www.youtube.com/watch?v=V8gOqDDnPN4
Galois representations and motives: trying to understand the Langlands program, or at least the modularity theorem. Galois representations from torsion points on elliptic curves.
Consider the 5-torsion points of the Gaussian elliptic curve. These 25 points form a discrete subvariety of the elliptic curve, which is a projective variety, but we can treat this subvariety as an affine variety, and since it's discrete it is the spectrum of a special commutative Frobenius algebra. Since the 5-torsion points form a group, this algebra also becomes a bicommutative Hopf algebra. But these 25 points are also naturally a 2-dimensional vector space over the finite field 𝔽₅, and because the Gaussian elliptic curve is defined over ℚ, this vector space gives a representation of the absolute Galois group of ℚ.
Working with other elliptic curves and other primes ℓ, we get a map from the moduli stack of elliptic curves to the moduli stack of representations of the absolute Galois group of ℚ on 2d vector spaces over 𝔽_ℓ . This map of moduli stacks really arises, in a contravariant way, from a map of theories in the doctrine of 2-rigs: namely, a map from the theory of 2d vector spaces over 𝔽_ℓ to the theory of elliptic curves! (A "theory in the doctrine of 2-rigs" is just a 2-rig.)
We would like to describe the theory of elliptic curves as the theory of a formal group together with extra relations which force our formal group to be an elliptic curve.
For more details see:
Nigel Boston, The proof of Fermat's last theorem, https://www.researchgate.net/profile/Thong-Nguyen-Quang-Do/post/Are-there-other-pieces-of-information-about-Victory-Road-to-FLT/attachment/5b0ab419b53d2f63c3ce5da8/AS%3A630893421006850%401527428121602/download/FLT+Boston.pdf
We would like to understand Chapter 4, and especially Theorem 4.28.
(5/n)
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A conversation with James Dolan and Chris Grossack:
2023-12-15
https://www.youtube.com/watch?v=TbRrYwMgkEY
Categorifying various attitudes to rings, or rigs, to get corresponding attitudes to 2-rigs. A commutative algebraist studies commutative rings while an algebraic geometer might work in the opposite category and think of them as affine schemes. The algebraic side is more 'syntactical' while the geometric side is more 'semantic'.
You might think the geometric interpretation of a 2-rig is typically some sort of 'categorified affine scheme', but that's not always true! For example, if you take the 2-rig of modules of a commutative ring R, its spectrum is the same as that of R.
However, most 2-rigs aren't module categories of rings. Take a quasiprojective variety X and look at the 2-rig of quasicoherent sheaves on it, QCoh(X). When X is an affine variety QCoh(X) is equivalent to the 2-rig of modules of a ring, namely the ring R with Spec(R) ≅ X. But when X is a projective variety this is not true.
The free commutative ring on one generator is ℤ[x]. If we think of this as a space it's the line, which is an affine scheme. Similarly, the 2-rig of modules of ℤ[x] is the 2-rig of quasicoherent sheaves on the line, which is an affine scheme.
On the other hand, the free 2-rig on a line object is the 2-rig of ℤ-graded vector spaces, which is equivalent to the 2-rig of algebraic representations of the affine group scheme GL(1), or comodules of ℤ[x]. If we think of this 2-rig as a kind of 'space' it's the algebraic stack BGL(1).
Next consider the free 2-rig on a line object equipped with a line object equipped with a monomorphism to I⊕I, where I is the tensor unit. The corresponding space is the projective line.
(4/n)
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2023-11-27
https://www.youtube.com/watch?v=AGqjsDBUzUw
Zeta functions and structure types. There is a kind of structure R_L we can put on finite sets, i.e. a species in Joyal's sense, such that an R_L-structure on a finite set is a way of making it into a finite field. We can extract from this a Dirichlet series:
https://ncatlab.org/johnbaez/show/Dirichlet+species+and+the+Hasse-Weil+zeta+function
The exponential of this is the Riemann zeta function. The slice category of species over R_L is a Grothedieck topos: an object here is a way of making a finite set into a field and putting some further structure on it. Similarly, there's a species R_{L,p} such that an R_{L,p}-structure on a finite set is a way of making it into a finite field of characteristic p. Again the slice category of species over R_{L,p} is a Grothendieck topos. Putting the double negation topology on this and forming the category of sheaves, we get a Boolean topos, which is the geometric theory of algebraic closures of 𝔽ₚ. This topos is also the category of G-sets where G is the absolute Galois group of 𝔽ₚ, namely the profinite completion of ℤ:
https://en.wikipedia.org/wiki/Profinite_integer
If you take a commutative ring A and hom it into the the algebraic closure of 𝔽ₚ, you get an object in this topos. From this you can perhaps get one Euler factor of the L-series of this commutative ring (or its corresponding affine scheme).
(3/n)
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2023-11-13
https://www.youtube.com/watch?v=eNZW_REeKY8
More on possible relations between tuning systems and Coxeter groups. Just intonation involves a group homomorphism from ℤ² to GL(1,ℝ), sending the first generator to 5/4 (a just major third) and the second to 6/5 (a just minor third). Similarly equal temperament involves a group homomorphism ℤ² to GL(1,ℝ) sending the first generator to 2^(1/3) (an equal-tempered major third) and the second to 2^(1/4) (an equal-tempered minor third).
Similar concepts led to the 'Fokker periodicity blocks':
https://en.wikipedia.org/wiki/Fokker_periodicity_block
which are related to the Tonnetz:
https://en.wikipedia.org/wiki/Tonnetz
A hexagon of triads containing a single note. How the PLR group is a quotient of the Coxeter group {∞,∞,∞} with three generators which acts as isometries on the hyperbolic plane preserving the infinite-order triangular tiling:
https://en.wikipedia.org/wiki/Infinite-order_triangular_tiling
This group maps onto the Coxeter group {3,∞} which is the symmetry group of the infinite-order triangular tiling.
(2/n)
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Hardcore math post
My friend James Dolan and I have been talking about math for decades, lately for 2 hours a week on Zoom. More recently @hallasurvivor has joined us. Here are some of those conversations. These are not supposed to be entertainment: just mathematicians explaining stuff to each other, and struggling to figure things out. But I'm glad to get these videos out there, and some people are actually watching them.
2023-11-06
https://www.youtube.com/watch?v=07YB-ENbys8
Here we're talking about the mathematics of tuning systems. I'm wondering about some possible relations between just intonation and Coxeter groups - see the series starting here:
https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/
How the PLR group acts on the Tonnetz (or 'tone net'):
https://en.wikipedia.org/wiki/Tonnetz
https://alpof.wordpress.com/2014/01/26/an-introduction-to-neo-riemannian-theory-9/
Triads as a torsor of the dihedral group D₁₂:
https://math.ucr.edu/home/baez/week234.html
https://arxiv.org/abs/0711.1873
Jim has some better ideas about this next week!
For more on this whole series of conversations, go here:
https://math.ucr.edu/home/baez/conversations.html
(1/n)
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From "The Hill":
Sheinbaum, in her daily morning press conference Wednesday, said the body of water shared by Cuba, Mexico and the United States is internationally recognized as the Gulf of Mexico, adding that North America was historically marked on maps as Mexican America.
“I mean obviously ‘Gulf of Mexico,’ the name is recognized by the United Nations, an organism of the United Nations. But next, why don’t we call it ‘Mexican America’? It sounds nice, doesn’t it?” Sheinbaum said, waving to a historical map projected on a screen.
“Since 1607,” Sheinbaum added, in an apparent reference to the map. “the Constitution of Apatzingán was for Mexican America, so we’re going to call it ‘Mexican America,’ it sounds nice, doesn’t it? And Gulf of Mexico, well, since 1607 and it’s also recognized internationally.”
Sheinbaum’s constitutional reference pointed to the country’s first proposed founding document after declaring independence from Spain in 1810 — a constitution that was never in effect and mandated, among other things, Catholicism as the official state religion.
The Apatzingán Constitution of 1814 did refer to the budding country’s territory as “América mexicana” pending an “exact demarcation” of the territory that would eventually become Mexico.
And numerous historical maps during the colonial era referred to North America as either “Mexican America,” “Mexicana,” “Septentrional America” or a combination.
One 1544 map designated North America as “Baccalearum” in reference to the abundance of cod along its shores.
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Trump suggested renaming the Gulf of Mexico the "Gulf of America".
The president of Mexico made a counter-proposal. Why don't we call north America "America Mexicana"... like in the good old days!
😆
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It's obviously not working. Insanity is doing the same thing over and over again and expecting different results.
It's time to stop saying "climate change is largely responsible for this" and start admitting "we are largely responsible for this".
And to our so-called leaders, and the super-rich: "You are responsible for this. We will make you stop."
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Here are the equations of magnetohydrodynamics, or MHD for short!
The fields here are
• the velocity of our plasma, 𝐯
• the density field of our plasma, ρ
• the electric current, 𝐉
• the electric field, 𝐄
• the magnetic field, 𝐁
The quantities in boldface are vector fields while the density ρ is a scalar field. I'm assuming the plasma's pressure is some function 𝑓 of its density, so that's why you see 𝑓(ρ) in the equations. There are also three constants:
• the magnetic permeability of the vacuum, μ₀
• the electrical conductivity of the plasma, η
• the viscosity of the plasma, μ
But my blog article explains step by step how we get these equations, and a few basic things we can do with them. That's the fun part.
I do not explain how the magnetic field can get 'frozen in' to the plasma. Maybe I'll do that some other time. I'm not completely happy with the usual story about that, so I'd like to expand on it a bit.
(3/n, n = 3)
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Here's my little initial foray into the wonderful world of magnetohydrodynamics:
https://johncarlosbaez.wordpress.com/2025/01/01/magnetohydrodynamics/
I show how to derive the basic equations, and I use them to explain magnetic pressure and magnetic tension. All this stuff is standard. I just never learned it in school! I was too enamored with the charms of pure mathematics and 'fundamental' physics.
If you're scared to look at my article, I'll still show you the key equations. (Do my next posts need a content warning for the math averse?)
The animated gif here is from
• Philip Mocz, Create your own constrained transport magnetohydrodynamics simulation (with Python), https://levelup.gitconnected.com/create-your-own-constrained-transport-magnetohydrodynamics-simulation-with-python-276f787f537d
(2/n)
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When enough electrons get ripped off the molecules of a gas, it can become so electrically conductive that long-range electric and magnetic fields dominate its behavior. Then you've got PLASMA.
Plasma rules the world of astrophysics. It spans an enormous range of densities and temperatures, from interstellar space to the Sun's core.
For some reason I never seriously studied plasma until a few weeks ago, when the Parker Solar Probe penetrated the boundary separating the solar wind from the Sun's upper atmosphere - its corona.
Trying to understand this, I started reading about the equations of 'magnetohydrodynamics'. These are a combination of the equations for electromagnetism and the equations describing fluid flow. Not all plasmas are well described by the equations of magnetohydrodynamics - they're approximate - but many are. And these equations describe a bunch of weird things that plasmas do!
First of all, in these equations the magnetic field is generally more important than the electric field - as the name implies.
Second, when the electrical conductivity of the plasma is very high, the magnetic field tends to get 'frozen in' to the plasma. In other words, you can visualize the magnetic field as a bunch of 'field lines' that move along with the flow of the plasma.
But third, these magnetic field lines have pressure: parallel field lines tend to push each other away. And they have tension: curved field lines tend to straighten out!
And as the field lines do these things, they push the plasma around.
The math of this is pretty fascinating. The equations are terribly hard to solve, but beautiful to contemplate.
(1/n)
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If you're anywhere near the Pacific Palisades in Los Angeles, my thoughts are with you. Thousands are fleeing the fire, bulldozers are pushing abandoned cars off the road, and the winds are supposed to increase tonight.
Global warming means fire season is all year round, here in California.
Does anyone know if the Getty Villa Museum has caught fire? The SF Chronicle says it has, but other sources just say the grounds are on fire. I only care because it has 125,000 artifacts, many from ancient Greece and Rome.
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text/gemini
