johncarlosbaez

johncarlosbaez, 4 days ago

@dougmerritt - don't be sad that this doesn't fit into an infinite pattern. There are 'classical' algebraic structures, which fit into infinite patterns, and 'exceptional' algebraic structures, which don't. But most of the exceptional structures are related to each other! This means that there's a kind of pattern running through mathematics at right angles to our usual classifications. Nobody knows what this means, except for the obvious part:

"There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy."

johncarlosbaez, 4 days ago (edited 4 days ago)

@RefurioAnachro @dougmerritt -

A 'vector product algebra' is a finite-dimensional real vector space with an inner product I'll call the 'dot product' together with a bilinear operation I'll call the 'cross product', obeying three identities:

𝑢×𝑣 = −𝑣×𝑢

𝑢⋅(𝑣×𝑤) = 𝑣⋅(𝑤×𝑢)

(𝑢×𝑣)×𝑢 = (𝑢⋅𝑢)𝑣 − (𝑢⋅𝑣)𝑢

These imply a bunch more identities.

You can get a vector product algebra from a normed division algebra by taking the subspace of 'imaginary' elements, namely those orthogonal to 1. You can also reverse this process. Since there are only four normed division algebras, ℝ,ℂ,ℍ and 𝕆, there are only four vector product algebras! But you can also run this argument backwards, which is nice because there's a great string diagram proof that there are only four vector product algebras:

• Markus Rost, On vector product algebras, https://www.math.uni-bielefeld.de/~rost/data/vpg.pdf).

The four vector product algebras have dimensions 0, 1, 3, and 7. But only the last two are interesting, since in the first two the cross product is zero.

johncarlosbaez, 4 days ago

@dougmerritt - wow, I didn't know Rudvalis went into statistics. I don't know anything about him, except that there's a group called the Rudvalis group. I also don't know anything about the Rudvalis group! The sporadic finite simple groups are vastly stranger than the exceptional Lie groups (which fall out from the octonions in a simple yet mysterious way).

johncarlosbaez, 4 days ago

@dougmerritt - people are working on a simplified "second generation" classification, but it's easy to see why Rudvalis would want to try something new:

"The proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generation classification proof. This effort, called "revisionism", was originally led by Daniel Gorenstein."

"As of 2023, ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; & Capdeboscq, 2021, 2023). In 2012 Solomon estimated that the project would need another 5 volumes, but said that progress on them was slow. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from the second generation proof being written in a more relaxed style.) However, with the publication of volume 9 of the GLS series, and including the Aschbacher–Smith contribution, this estimate was already reached, with several more volumes still in preparation (the rest of what was originally intended for volume 9, plus projected volumes 10 and 11). Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof."

johncarlosbaez, 4 days ago (edited 4 days ago)

@SvenGeier - there may be elements of 𝐴 that aren't in the sequence 𝑎₁,𝑎₂,𝑎₃,.... In fact if 𝐴 is uncountable it 𝑚𝑢𝑠𝑡 have lots of elements that aren't in this sequence!

johncarlosbaez, 4 days ago

@SvenGeier - and even in the countable case, there's no reason every element of 𝐴 needs to be on your list 𝑎₁,𝑎₂,𝑎₃,.... So you need to deal with ones that aren't. Luckily it's easy: leave them alone!

johncarlosbaez, 6 days ago (edited 4 days ago)

@dougmerritt @mattmcirvin @idlestate @60sRefugee @pewnack -

Cool! To me the big mystery is not the chemistry of prometheum - since I'm not deep enough into lanthanide chemistry to be dying to collect the complete set of 14 and carefully compare them. It's the nuclear physics of prometheum! Why are all isotopes of this element radioactive?

I haven't checked lately, but it must be 'doubly odd' - an odd number of protons and an odd number of neutrons. That's a good rule of thumb but it leaves a lot of questions unanswered. (I'm sure there are people who know a lot more about these answers than I do.)

johncarlosbaez, 5 days ago

@mattmcirvin - makes sense; I should have guessed that since neutrons are a dime a dozen.

Maybe it's near some sort of 'anti-magic number'????

@dougmerritt @idlestate @60sRefugee @pewnack

johncarlosbaez, 6 days ago

@codrusofathens - A painter told us one of those now-closed windows was used to carry stuff into the building while it wa being built - the one that's now behind a 'press', a shallow closet common in these old buildings.

These windows face north, and keeping cool wasn't a big concern in Scotland - it tends to be chilly except for a short summer.