A higher homotopy group for digital images

[@ChrisStaecker]

This talk was given for subject-matter experts. For a more widely accessible version of this talk, see here: kzbin.info/www/bejne/fJDEfJ-sgZeKpZI The paper is: "A Second Homotopy Group for Digital Images" by Gregory Lupton, Oleg Musin, Nicholas A. Scoville, P. Christopher Staecker, Jonathan Treviño-Marroquín, at arxiv here: arxiv.org/abs/2310.08706 My remark about "seats in the front" was a joke. The room was huge, and mostly empty.

[@MohammadAlmeqdadi]

It's amazing to watch a whole lecture of what essentially to me sounds like the Plumbus "How it's made" section from Rick and Morty. I appreciate people who spend their lives to study this kind of stuff. I think outside of your world, this truly sounds like an alien language theory, interspersed with English prepositions.

[@ChrisStaecker]

That's how we feel about each other too- if I walk into a random math research talk about a subject that I haven't already studied, then it'll all sound like gibberish to me. We're used to it.

[@MadsGDS]

That was very interesting, i don't really think i understood any of it, but now i know it exists

[@UltraMaXAtAXX]

As soon as you said "spider-like moves," the picture went from Strong Bad to Spiderman for me.

[@aaron62688]

I liked that! Even managed to follow most of it. Naïve Question: Does the sequence of colors for the final fill operations matter?

[@ChrisStaecker]

The order doesn’t really matter. The last one needs to be dark red, and the other two can be any two different (non-opposite) colors.

[@Rhino-Flea]

Fascinating

[@j.aaronkambeitz9848]

Am I crazy, or did you forget to fill with black before you fill with red?

[@ChrisStaecker]

Thanks for watching! It's not necessary to fill with black- the first two fills suffice to isolate the pink points, and then this means when we fill with red it will just be those little blocks around single pink points. If you did an extra fill with black or dark blue in there it wouldn't hurt anything, but it's not necessary. In the original version of the argument in the paper we did iterated fills by every color, one after another. But then we realized you only need 3.

[@j.aaronkambeitz9848]

After filling with red, all that should be left are those little 5x5 clockwise and counterclockwise pinwheels, right? But in the slideshow, when you fill with red, extra black cells appear that weren't there before. @@ChrisStaecker

[@j.aaronkambeitz9848]

For example, at 18:37, the lower leftmost pink cell's neighborhood has two whites, two blues, /one/ black and /three/ light blues. At 18:42, after filling with red, that same pink cell has two neighbors of each color, as it should for the pinwheel configuration. So one of the light blue neighbors got swapped for a black neighbor during the red fill operation. ?

[@ChrisStaecker]

@@j.aaronkambeitz9848 OK yes! You looked very VERY closely! Like I said, in earlier versions of the paper we were doing fills by all the colors. Later we realized that only 3 are necessary. But when I was making this talk I grabbed an old version of the pictures- so the one you see at 18:42 is actually the result of doing all 5 fills. So my presentation is slightly dishonest on this point. If you really do what I claimed I was doing, you still get the little 5x5 pinwheel configurations, but some of them may not actually use 2 of each color. This is part of what I said at 14:52 - there are slight variations on T which can themselves be rotated and tweaked to become either T or T^-1. In this example, at the spot you mentioned, you'll get a pinwheel with only 1 black and 3 light blues. This is to be expected. If you like, you can do one more step at the end to change it so that each pinwheel matches T or T^-1 exactly. Thanks for watching carefully! (There is one other detail which I omitted from the talk entirely but nobody notices. You can see in the paper that there is one additional "adjustment" that you need to do before the fills. This is why at 18:19 you can see that two of the black regions have a little chunk taken out from them. This is necessary in order to make the 3 fills work properly.) (And also I realized after I made all the pictures that the original picture was illegal- towards the top-left there is a dark blue diagonally adjacent to a light blue. I didn't want to remake all the pictures so I just left it in.)

[@j.aaronkambeitz9848]

Ah! That makes sense. I watch /all/ your videos very closely. I really enjoyed Curtsmas btw!

[@AleksyGrabovski]

First!