johncarlosbaez

johncarlosbaez, 14 minutes ago to random

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/

dpiponi, 15 hours ago to random

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, 1 day ago (edited 1 day ago) to random

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, 15 days ago (edited 15 days ago) to random

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.