@johncarlosbaez@mathstodon.xyz
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johncarlosbaez

@johncarlosbaez@mathstodon.xyz

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

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johncarlosbaez, to random
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I want to read this book: A Darwinian Survival Guide. Sounds like a realistic view of what we need to do now. You can read an interview with one author, the biologist Daniel Brooks. A quote:

...

Daniel Brooks: What can we begin doing now that will increase the chances that those elements of technologically-dependent humanity will survive a general collapse, if that happens as a result of our unwillingness to begin to do anything effective with respect to climate change and human existence?

Peter Watts: So to be clear, you’re not talking about forestalling the collapse —

Daniel Brooks: No.

Peter Watts: — you’re talking about passing through that bottleneck and coming out the other side with some semblance of what we value intact.

Daniel Brooks: Yeah, that’s right. It is conceivable that if all of humanity suddenly decided to change its behavior, right now, we would emerge after 2050 with most everything intact, and we would be “OK.” We don’t think that’s realistic. It is a possibility, but we don’t think that’s a realistic possibility. We think that, in fact, most of humanity is committed to business as usual, and that’s what we’re really talking about: What can we begin doing now to try to shorten the period of time after the collapse, before we “recover”? In other words — and this is in analogy with Asimov’s Foundation trilogy — if we do nothing, there’s going to be a collapse and it’ll take 30,000 years for the galaxy to recover. But if we start doing things now, then it maybe only takes 1,000 years to recover. So using that analogy, what can some human beings start to do now that would shorten the period of time necessary to recover?

https://thereader.mitpress.mit.edu/the-collapse-is-coming-will-humanity-adapt/

johncarlosbaez,
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@michael_w_busch - does Daniel Brooks suggest setting up small towns without access to surgery? I don't see that. But I want to read his book, to get more detail.

The interview comes to close to discussing these issues:

...
So you live in circumstances where people cannot identify the sociopaths before they’ve taken control. And that’s the subtext in the idea that one of the ways that we should deal with the fact that more than 50 percent of human beings now live in large cities in climate-insecure places, is for those people to redistribute themselves away from climate-insecure areas, into population centers of lower density, and cooperating networks of low-density populations, rather than big, condensed cities.

Peter Watts: Let’s follow this move back to the rural environment a bit, because it’s fundamental. I mean, you brought it up, and it is fundamental to the modular post-apocalyptic society you’re talking about.

Daniel Brooks: Sure. Not post-apocalyptic: post-collapse.
....

But they get distracted, and Watts never gets back to asking what this move to a rural environment would look like.

dpiponi, to random
@dpiponi@mathstodon.xyz avatar

Despite Wildberger being a bit off the usual conventional paths in mathematics, he's influenced me to the point where every time I write a line of code using an angle I ask myself if I could use an alternative "rational" representation.

https://research.unsw.edu.au/people/professor-norman-j-wildberger

johncarlosbaez,
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@BartoszMilewski @dpiponi - "He's such a dyed in the wool Platonist, I can't stand it. According to him rational numbers "exist" but infinities don't."

What I find most annoying is that people who say some numbers exist and others don't never go into much detail about what it means for a number to "exist". I could be perfectly happy saying numbers exist, with one definition of "exist", or saying they don't, with some other definition. I can even imagine saying some numbers exist and others don't, with some more contorted definition of "exist". But most people who say some numbers exist and others don't seem to use a secret personal smell test for existence. Numbers that smell bad to them don't really exist - and they're shocked that I lack their sense of smell.

johncarlosbaez, (edited )
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@codyroux @BartoszMilewski @dpiponi - if you're trying to declare that some things "exist" and others don't, and you don't want to define what you mean by this term, it's up to smell. Even if you carefully define what it means, in a way that allows you to rigorously prove some things exist and others don't, the definition is likely just a reification of your sense of smell. But that's okay with me as long as you don't think your definition of "exist" is the only acceptable one. I don't think there's any "true" definition of existence of mathematical objects.

(Personally I don't care at all about which mathematical objects "exist": it's a non-issue to me, and an utter waste of time. But I try to remain polite.)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@sprout @BartoszMilewski @dpiponi - I enjoyed Quine's book 𝑂𝑛𝑡𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑖𝑡𝑦.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pozorvlak @BartoszMilewski @dpiponi - that's hilarious! Reminds me of the guy who tried to dial 870247i and got the message "The number you have tried to dial is imaginary. Please rotate by 90 degrees and dial again".

johncarlosbaez, (edited ) to random
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When global warming gets bad enough to create millions of refugees, just imagine how nasty the EU and US will become. They may just send people out to the desert to die.

Oh wait, you don't have to imagine! It's happening already:

"A year-long joint investigation by The Washington Post, Lighthouse Reports and a consortium of international media outlets shows how the European Union and individual European nations are supporting and financing aggressive operations by governments in North Africa to detain tens of thousands of migrants each year and dump them in remote areas, often barren deserts.

European funds have been used to train personnel and buy equipment for units implicated in desert dumps and human rights abuses, records and interviews show. Migrants have been pushed back into the most inhospitable parts of North Africa, exposing them to abandonment with no food or water, kidnapping, extortion, sale as human chattel, torture, sexual violence and, in the worst instances, death."

The full story is here: https://archive.is/1tEgl

(1/2)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

"“There is Algeria, follow the light,” the Tunisian official barked at the Black migrants. “If you’re seen here, you’ll be shot.”

François, a 38-year-old Cameroonian, obeyed, jumping off the bed of a pickup truck near the desolate Algerian frontier. A day earlier, the rickety boat attempting to carry him and other hopeful sub-Saharans to Europe — including his wife and 6-year-old stepson — had been interdicted by the Tunisian coast guard in the cobalt blue waters off the coast. Still wet and cold, the group of 30 migrants, including two pregnant women, now walked toward their punishment: the desert.

Their ordeal — an odyssey of at least 345 miles from sea to sand, recounted by François and verified by matching GPS tracking on his phone with images and videos he captured during nine days of wandering — illustrates one example of the draconian practices being deployed in at least three North African nations to dissuade sub-Saharan migrants from risky crossings to Europe."

"Witness accounts and visuals reviewed by The Post place the Tunisian National Guard at the center of desert dump operations. Between 2015 and 2023, the German federal police deployed 449 staff members and spent more than 1 million euros to train nearly 4,000 Tunisian national guards. As the dumps were ongoing in November 2023, a 9 million euro border-management training center opened in Tunisia, funded by Austria, Denmark and the Netherlands."

(2/2)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@GreenFire - hear hear! So please, all USians, vote for Biden.

https://www.nytimes.com/2024/05/09/climate/trump-oil-gas-mar-a-lago.html

johncarlosbaez,
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@darabos - among other things, we need to shame the governments who are doing these things.

johncarlosbaez,
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@decapitae @darabos - the Washington Post one of the most mainstream of mainsteam media. This news is also in Le Monde, the most famous French newspaper. The story is still somewhat new, and you'll be seeing it in other places soon.

https://www.lemonde.fr/afrique/article/2024/05/21/comment-l-argent-de-l-union-europeenne-permet-aux-pays-du-maghreb-de-refouler-des-migrants-dans-le-desert_6234489_3212.html

julesh, to random
@julesh@mathstodon.xyz avatar

It's not just imagination, Englandandwales really is a conservative religious country compared to Scotland

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@julesh - someone should do a study to see if people with "no religion" are nicer.

johncarlosbaez,
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@bhaugen @julesh - unfortunately the British are still effectively in charge of visas.

johncarlosbaez, (edited ) to random
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A while back @pschwahn raised an interesting puzzle here:

You can define a well-behaved cross product of vectors only in 3 and 7 dimensions. The 7d cross product is weird because it's not preserved by all rotations of 7d space. But very smart people have told us there's a way to get the group of 𝟯𝗱 rotations to act on 7d space while preserving the 7d cross product. In fact, you can do it while also getting this group to act 'irreducibly', meaning the only subspaces of 7d space preserved by this action are {0} and the whole space!

The puzzle is: 𝗰𝗮𝗻 𝘆𝗼𝘂 𝘄𝗿𝗶𝘁𝗲 𝗱𝗼𝘄𝗻 𝘁𝗵𝗲 𝗳𝗼𝗿𝗺𝘂𝗹𝗮𝘀 𝗳𝗼𝗿 𝗵𝗼𝘄 𝘁𝗵𝗶𝘀 𝘄𝗼𝗿𝗸𝘀?

I got stuck on this so I asked some of my friends, and now Layra Idarani has outlined a nice way to do it:

https://golem.ph.utexas.edu/category/2024/05/3d_rotations_and_the_cross_pro.html#c063185

Interestingly he doesn't actually give the formulas; he just tells you how to get them. So I will need to do some work to check his answer! If you want to help out, that would be great.

Layra said "The devil of the details is in the eating". I thought the proof was in the pudding. Now I'm hungry for devil's food cake.

johncarlosbaez, (edited )
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@Dyoung @pschwahn - Whoops, thanks! Fixed.

johncarlosbaez,
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@Dyoung @pschwahn - Layra has now revealed all the details of his solution of @pschwan's problem:

https://golem.ph.utexas.edu/category/2024/05/3d_rotations_and_the_cross_pro.html#c063197

One probably has to read a bunch of earlier comments to understand what he's doing.

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@Dyoung @pschwahn - it's there: it's the group of rotations that rotates x, y, and z in the usual way. To understand this you need to start further back, e.g. with my failed attempt

https://golem.ph.utexas.edu/category/2024/05/3d_rotations_and_the_cross_pro.html#c063189

I laid out the strategy nicely but failed to guess a good choice of 7 cubic polynomials - I called them 𝑒₁,...𝑒₇ . They lacked the properties I wanted to check, but worse they weren't even orthonormal. Layra's polynomials P, Q, R, S, T, U, V have all the right properties - or so he claims.

By the way, it's super-easy to construct the 7d cross product on the imaginary octonions: just define

𝑎 × 𝑏 = (𝑎𝑏−𝑏𝑎)/2

All the work here is constructing an SO(3) group that acts irreducibly on the imaginary octonions while preserving this cross product!

ngons, to random
@ngons@mathstodon.xyz avatar

#TilingTuesday decagon dissection.

#Tiling #MathArt #MathsArt #Geometry

johncarlosbaez,
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@ngons - yes. Fun puzzle!

highergeometer, to random
@highergeometer@mathstodon.xyz avatar

Paiva Miranda De Siqueira, J. V. "Tripos models of Internal Set Theory" (2022)
https://doi.org/10.17863/CAM.78799

This thesis provides a framework to make sense of models of E. Nelson’s Internal Set Theory (and hence of nonstandard analysis) in elementary toposes by exploiting the technology of tripos theory and Lawvere’s hyperdoctrines. A new doctrinal account of nonstandard phenomena is described, which avoids a few key restrictions in Nelson’s approach: chiefly, the dependence on Set Theory (which is done by replacing a model of set theory with a topos as the starting point) and reliance on an internally defined notion of standard element. From the new perspective, validity of the schemes of Idealisation, Standardisation, and Transfer correspond to the existence of certain relationships between hyperdoctrines, leading to the new notion of a tripos model of IST. [... see more in the pdf at the link]

^^^ Thesis supervised by Peter Johnstone and Martin Hyland.

johncarlosbaez,
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@highergeometer - thanks! It's fun that we were just talking about nonstandard natural numbers and topos theory over on the Category Theory Community Server.

nancycomics, to random
@nancycomics@mastodon.social avatar

WELCOME EVERYONE!

johncarlosbaez,
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@nancycomics - thanks! They brighten my day.

johncarlosbaez, (edited ) to random
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Since May 1st, a small team of category theorists, computer scientists and epidemiologists have been meeting daily in James Clerk Maxwell’s childhood home in Edinburgh. We’re hard at work on our project called New Mathematics and Software for Agent-Based Models.

We're creating a general software framework for agent-based models in epidemiology. By now we've really entered the zone where all these ideas come together. We're equally likely to talk about details of opioid abuse models or using coproducts of representables to make our software more efficient. It's exciting!

First we came up with a general framework for 'stochastic C-set rewriting systems'. These are models where graphs or more complicated combinatorial structures change in a random way through local 'rewrite rules'. Each rewrite rule says that when a bit of your structure matches some pattern, you can 'rewrite' it to some specific new pattern. See the pictures below for a couple examples.

'Stochastic' is a fancy word for 'random'. In our models you specify the randomness in a carefully crafted way by associating to each rewrite rule a 'timer'. The timer says the probability with which the rule is applied - as a function of time. A timer starts whenever a new match to the rule appears.

Kris Brown has already created a program that lets you run these stochastic C-set rewriting systems in AlgebraicJulia. This is a Julia package for scientific computing with categories. But we're just getting started!

(1/2)

johncarlosbaez,
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@bks - Alas, Rosen's work has very little substance to it and has not influenced applied category theory very much. Very roughly speaking, successful applications of category theory to a wide variety of practical areas started with computer science, then spread to quantum physics, and continued growing, meriting its own conference series in 2018. You can hear my story of this here:

https://www.youtube.com/watch?v=DviWztxqMJk&t=5430s

starting at 1:30:30.

johncarlosbaez, (edited ) to random
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There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.

We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining

v⋅w= -½(vw + wv), v×w = ½(vw - wv)

We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.

The following stuff is pretty well-known: the group of linear transformations of ℝ³ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on ℝ³ because there's no linear subspace of ℝ³ that's mapped to itself by every transformation in SO(3).

Much to my surprise, it seems that SO(3) also has an irreducible representation on ℝ⁷ where every transformation preserves the dot product and cross product in 7 dimensions!

It's not news that SO(3) has an irreducible representation on ℝ⁷. In physics we call ℝ³ the spin-1 representation of SO(3), or at least a real form thereof, while ℝ⁷ is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on ℝ⁷ preserves the dot product. But I didn't know it also preserves the cross product on ℝ⁷, which is a much more exotic thing!

In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:

https://mathstodon.xyz/@pschwahn/112435119959135052

and I think I almost see a proof, which I outlined after a long conversation on other things.

The octonions keep surprising me.

https://en.wikipedia.org/wiki/Seven-dimensional_cross_product

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn - There is a space of all 𝔰𝔬(3) subalgebras of 𝔤₂, and I was guessing that those subalgebras that act irreducibly on Im(𝕆) are 𝑑𝑒𝑛𝑠𝑒 in this space. That's what "generic" means in this context.

Another thing I guess is that the conjugacy class of 𝔰𝔬(3) subalgebras acting irreducibly on Im(𝕆) has higher dimension than the other 3 conjugacy classes. This should be a lot easier to check.

johncarlosbaez, (edited ) to random
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Tolstoy: "Happy families are all alike; every unhappy family is unhappy in its own way."

Mathematics: "Real tori are all alike; every complex torus is complex in its own way."

To be precise, a 'n-dimensional real torus' is a real manifold of the form V/Λ where V is an n-dimensional real vector space and Λ ⊆ V is a lattice of rank n in this vector space. They are all isomorphic.

An 'n-dimensional complex torus' is a complex manifold of the form V/Λ where V is an n-dimensional complex vector space and Λ ⊆ V is a lattice of rank 2n in this vector space. These are not all isomorphic, because there are different ways the lattice can get along with multiplication by i. For example we might have iΛ = Λ or we might not.

And so, it's possible to write a whole book - and indeed a fascinating one - on complex tori. For example a 1-dimensional complex torus is an elliptic curve, and there are whole books just about those.

johncarlosbaez,
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@battaglia01 wrote: "In our situation, typically we're interested in lattices in ℝⁿ."

Who is "us"? I can certainly imagine people getting interested in lattices in ℝⁿ. But all the work I mentioned on Siegel modular forms is about lattices in ℂⁿ, and it's trying to generalize work on lattices in ℂ which takes full advantage of complex analysis (or in modern terms, algebraic geometry). You can think of Eiseinstein series as giving functions of lattices in ℝ², but most people who work with them and their generalizations treat them as functions of lattices in ℂ and take full advantage of that fact: this is a prerequisite for treating them as 'modular forms', which unlocks much of their power.

But you're going in a different direction, so everything I said in my last post is basically irrelevant!

You can define something like Eisenstein series for lattices in 4 dimensions in the way you suggest, treating ℝ⁴ as the quaternions ℍ. I don't know anything about that except that the first publication mentioning octonions was the appendix of a completely unrelated paper by Cayley called "On Jacobi's Elliptic Functions, in Reply to the Rev. B. Bronwin; and on Quaternions". This seems to be about Cayley's attempt to generalize elliptic functions from the complex numbers to the quaternions. Elliptic functions are closely connected to modular forms. So I wouldn't be shocked if Cayley had tried to study quaternionic analogues of Eisenstein series. Apparently everything in this particular paper was wrong except the appendix, so it was left out of his collected works. But Cayley wrote vast numbers of papers, and only god knows what's in all of them.

(1/2)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@battaglia01 - As you suggest, you can also write down a formula for Eisenstein series for lattices in 8 dimensions, treating ℝ⁸ as the octonions 𝕆. Maybe Cayley was even contemplating something like this! More likely he was trying to invent octonionic elliptic functions.

I've never seen anyone discuss quaternionic or octonionic Eisenstein series. I'll google those terms.

There are division algebras except in dimensions 1, 2, 4, and 8, but you don't need a division algebra to make sense of the usual formula for an Eisenstein series: you just need a power-associative algebra where every nonzero element has a two-sided inverse. (This is NOT the same thing as a power-associative division algebra: it's weaker!) There's an algebra of this sort for every dimension that's a power of 2:

reals, complex numbers, quaternions, octonions, sedenions, 32-ions, 64-ions,...

These algebras have zero divisors when we reach dimension 16, but still every nonzero element has a left and right inverse, and that's good enough to write down the formula for Eisenstein series!

For more see this:

https://math.ucr.edu/home/baez/octonions/node5.html

However, I think it's wise to see what (if anything) people have done with Eisenstein series in the quaternionic and octonionic cases, before tackling these still more exotic cases! I have never seen anyone do anything useful with sedenions.

(2/2)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@battaglia01 - actually every paper I've seen about "quaternionic Eisenstein series" is not talking about what you and I would mean by that term!

What you're interested in might be found by searching around under "moduli space of lattices in R^n" or asking about it on MathOverflow.

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