johncarlosbaez

johncarlosbaez, 2 hours ago

@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.

johncarlosbaez, 17 hours ago

@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, 9 hours ago (edited 7 hours ago)

@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, 9 hours ago

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

johncarlosbaez, 7 hours ago

@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, 17 hours ago

@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, 9 hours ago (edited 7 hours ago)

@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!

johncarlosbaez, 19 hours ago

@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, 19 hours ago

@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, 9 hours ago

@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.