@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.

This profile is from a federated server and may be incomplete. Browse more on the original instance.

mjd, to random
@mjd@mathstodon.xyz avatar

This is the world's first use of the modern equals sign, from Robert Recorde's 1557 book The Whetstone of Witte.

(Screencap from Internet Archive's scan of the book: https://archive.org/details/TheWhetstoneOfWitte/page/n237/mode/2up)

(I also wrote a blog post a couple of years back explaining what it says if you are interested: https://blog.plover.com/math/recorde.html)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@mjd - that's a looooooong equals sign.

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

Joe Gore did graduate work on musicology and is an expert on medieval harmony theory. But when he played guitar on the Tom Waits song "Goin' Out West", he got handed a cassette tape of Tom Waits singing while pounding drums in a steel shed - and Waits had him play over that on an electric guitar tuned a quarter tone flat.

A great interview! He describes how Tom Waits gets great music from players by pushing them way out of their comfort zone. And near the end he gets into some serious harmony theory! I know just enough to know he's not faking it. I like the idea that in Lydian, the sharp fourth functions as a kind of second leading tone - a half-tone below the fifth, just like the usual leading tone is a half-tone down below tonic.

https://www.youtube.com/watch?v=0_wqAcj6EBU

julesh, to random
@julesh@mathstodon.xyz avatar

I just had an epiphany. The purpose of an review rebuttal is not to convince the reviewers to change their mind - which is obviously not possible except in vanishingly rare cases - it's to convince the programme committee that the reviewers don't know what they're talking about

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

@ProfKinyon @julesh "The purpose of an review rebuttal is not to convince the reviewers to change their mind - which is obviously not possible except in vanishingly rare cases - it's to convince the programme committee that the reviewers don't know what they're talking about."

Yes, and this is also the right attitude when arguing with jerks on the internet. You are really talking to the smarter people quietly looking on.

dmm, to math
@dmm@mathstodon.xyz avatar

Here's something I just learned: the lucky numbers of Euler.

Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.

Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.

Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).

The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.

h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.

References

[1] "Lucky numbers of Euler", https://en.wikipedia.org/wiki/Lucky_numbers_of_Euler

[2] "Heegner number", https://en.wikipedia.org/wiki/Heegner_number

[3] "Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1)", https://oeis.org/A003173

[4] "Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1", https://oeis.org/A003173

#luckynumbersofeuler #heegnernumber #euler #math #maths #mathematics

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dmm - you came tantalizingly close to telling your readers how to get the sequence

7, 11, 19, 43, 67, 163

from the sequence

2, 3, 5, 11, 17, 41

but it's probably best to leave that as an easy puzzle for them! Of course, 𝑤h𝑦 𝑖𝑡 𝑤𝑜𝑟𝑘𝑠 is a much harder puzzle.

dmm, to math
@dmm@mathstodon.xyz avatar

The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].

Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].

A few of my notes on this and related topics are here: https://davidmeyer.github.io/qc/galois_theory.pdf. As always, questions/comments/corrections/* greatly appreciated.

References

[1] "Heegner Number", https://mathworld.wolfram.com/HeegnerNumber.html

[2] "Splitting Field", https://mathworld.wolfram.com/SplittingField.html

[3] "Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).", https://oeis.org/A003173

[4] "Ramanujan Constant", https://mathworld.wolfram.com/RamanujanConstant.html

#galois #gauss #heegnernumber #ramanujan #math #maths #mathematics

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

Another fun thing: Euler's formula

𝑛² + 𝑛 + 41

which gives primes from 𝑛 = 1 to 𝑛 = 39 is actually related to the fact that 41 × 4 - 1 = 163 is a Heegner number. And this fact has (less impressive) relatives for the smaller Heegner numbers!

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dmm - it's discussed here:

https://en.wikipedia.org/wiki/Heegner_number#Euler's_prime-generating_polynomial

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

An epidemiologist having a category-theoretic revelation. My colleague Nathaniel Osgood, discovering how the process of converting stock and flow diagrams into causal loop diagrams can be captured by a left pushforward functor between presheaf categories. These two kinds of diagrams are both important in the modeling tradition called 'system dynamics', which is used in epidemiology as well as economics and other disciplines.

System dynamics:

https://en.wikipedia.org/wiki/System_dynamics

"Left pushforward" as a method of data migration:

David Spivak, Functorial data migration, https://arxiv.org/abs/1009.1166

highergeometer, to random
@highergeometer@mathstodon.xyz avatar

Two thematically related papers

Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn, "Projective Space in Synthetic Algebraic Geometry"

Abstract: Working in an abstract, homotopy type theory based axiomatization of the higher Zariski-topos called synthetic algebraic geometry, we show that the Picard group of projective n-space is the integers, the automorphism group of projective n-space is PGL(n+1) and morphisms between projective standard spaces are given by homogenous polynomials in the usual way.

https://arxiv.org/abs/2405.13916

=====

Matías Menni, "Bi-directional models of `Radically Synthetic' Differential Geometry", Theory and Applications of Categories, Vol. 40, 2024, No. 15, pp 413-429.

Abstract: The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model.
http://www.tac.mta.ca/tac/volumes/40/15/40-15abs.html

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@highergeometer - proving that the automorphism group of projective n-space is PGL(n+1) will sound a bit unimpressive to most algebraic geometers, since it's such a basic fact. I wonder if with almost equal ease the homotopy type theory approach can prove some much more impressive-sounding homotopical generalization of this fact.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@highergeometer wrote: "Yeah, but you have to see the definition of this synthetic projective space as a type to realise it's not immediate. And the automorphism group consists of endomorphisms of the synthetic projective space as a type, not via some explicit construction."

Showing you can still prove things in a formalism that makes them less obvious doesn't count as an advertisement for the formalism to me - it's mostly interesting to people already committed to the formalism. I was hoping that with some tweak one could consider the projective space of a "homotopy vector space", or something like that. (The main kind of "homotopy vector space" I know is a simplicial object in Vect, aka chain complex of vector spaces, but there should be more interesting ones that are spectra. I don't however know if the projective space still makes sense.)

sprout, to random
@sprout@mathstodon.xyz avatar

As a student, I once walked into a philosophy department's library and found a small gem discussing the origin and naturalness of different Boolean connectives. I remember several options for implication being discussed, a connective we nowadays learn about and whose properties are taken without question.

The paper below on 'why negation' reminds me of that experience some thirty years ago, though as research goes this paper is written with some quantitative arguments instead of only qualitative ones.

https://www.journals.uchicago.edu/doi/10.1086/715140

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@sprout - they could have titled the paper "Why not?"

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

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,
@johncarlosbaez@mathstodon.xyz avatar

@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,
@johncarlosbaez@mathstodon.xyz avatar

@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 )
@johncarlosbaez@mathstodon.xyz avatar

@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
@johncarlosbaez@mathstodon.xyz avatar

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

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

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@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

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

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 )
@johncarlosbaez@mathstodon.xyz avatar

@Dyoung @pschwahn - Whoops, thanks! Fixed.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@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

decagon dissection.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@ngons - yes. Fun puzzle!

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

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,
@johncarlosbaez@mathstodon.xyz avatar

@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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