I’d hoped to have a pretty substantial post today. I fell short of having time to edit the beast into shape. I apologize but hope to have that soon.
I also hope to soon have an announcement about a Mathematics A-to-Z for this year. But until then, here’s this.
Several years ago in an A-to-Z I tried to explain cohomologies. I wasn’t satisfied with it, as, in part, I couldn’t think of a good example. You know, something you could imagine demonstrating with specific physical objects. I can reel off definitions, once I look up the definitions, but there’s only so many people who can understand something from that.
Quanta Magazine recently ran an article about homologies. It’s a great piece, if we get past the introduction of topology with that doughnut-and-coffee-cup joke. (Not that it’s wrong, just that it’s tired.) It’s got pictures, too, which is great.
This I came to notice because Refurio Anachro on Mathstodon wrote a bit about it. This in a thread of toots talking about homologies and cohomologies. The thread at this link is more for mathematicians than the lay audience, unlike the Quanta Magazine article. If you’re comfortable reading about simplexes and linear operators and multifunctions you’re good. Otherwise … well, I imagine you trust that cohomologies can take care of themselves. But I feel better-informed for reading the thread. And it includes a link to a downloadable textbook in algebraic topology, useful for people who want to give that a try on their own.
nebusresearch 
How would a Mathematician approach John Lennon’s query in “A Day in the Life” and finding out how many holes would fill up the Albert Hall?
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Oh, first, by asking how big a hole is. Like, is it three-dimensional? Is it a fixed volume? Second, by asking whether holes can overlap or not.
If holes can’t overlap, then the holes-filling-Albert-Hall problem is a packing problem: how to fit the greatest number of these uniform things inside this volume? That’s a difficult problem because it’s quite easy to get a good answer and quite hard to show there’s no better answer. For uniform spheres and a rectangular space, you mostly end up with … well, the stacking you get stacking oranges out on a farmer’s table. But that’s also very regular, like, you create these rows and columns of spheres that can stretch forever in each direction. I’m not sure that it’s proven that there aren’t better approaches from including slight irregularities. And that’s interesting, not just because of the prospects for stacking oranges. It turns out to match naturally some questions about how to form crystals, and a lot of stuff wants to be crystals.
If holes can overlap, then, yeah, you could just have
infinity'' holes. But that's boring, since that same answer applies to everything that fits at least one hole. So the mathematician would look at what's the absolute smallest number of holes needed to cover the space, that is, to make sure every speck inside Albert Hall is inside at least one hole. That smallest number is thecover set”, and it’s a core idea in measure theory, the study of how big volumes are. This in turn sneaks into probability questions and calculus and a bunch of fractal questions too.LikeLike
Great pointers, thanks! It’s always a pleasure to return to your blog after a while! Right now, I am learning a bit of topology – in a few days I might be interested in “de Rham cohomology” ;-)
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Thank you so! And thanks for the kind words; I’m gratified to host a blog people enjoy returning to.
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