Summer of Math Exposition

Great video! It’s nice how you break down the mathematical thought process for approaching a problem. Especially for people who aren’t trained as mathematicians (like myself; I’m a physicist), it gives a nice comparison. I had previously encountered the Euler characteristic in my differential geometry class but only in terms of the Gauss-Bonnet theorem. Although you hinted at it in the video, I think it would be cool to highlight even more intersections and show how seemingly different fields play into each other (topology, differential geometry, and graph theory in this case). You also breifly mentioned some tools from electric circuit theory being applied! That especially interests me and I would like deeper exploration into that.

As for some more pedantic stuff, I would try to make your audio recording one go if possible to avoid sudden audio changes in the video. Not to say that it’s bad (in fact in my submission due to an error I had to rerecord some sections), but it gives it more professionality. Geometry is also a really fun math topic because you can actually see a lot of what goes on. I would like to see animations better linking the surface mappings to graphs, because it can be confusing and not clear how a graph corresponds to that map, especially in a faster paced video form.

The topic has always fascinated me. I was lucky enough to study it in a graph theory course. Here, you’ve also made some interesting generalizations. I hope your video can inspire more people to take an interest in the subject. I’d add a bit of background music where appropriate. Great work!

I find this an excellent video, so it’s hard to give notes. I’m not experienced in this type of problem, my brain didn’t quite catch up with replacing the map with graph. Even though I understood it on a surface level right away, when you show graphs with edges and polygon-like structures, my brain wanted to jump back to the map and its borders, even though the graph is kind of an inversion of it. So a little more visualization, recalls to the original map problem could have helped. But not sure, this is very subjective.

The topic is quite attractive, and here is a few details that I really like.

1. at the end of each chapter, rather than showing the next title, you show the that chapter first before transform it, that really remind me what I just learn and what is the next step.
2. Solving the map problem -> Map of problem solving is so clever.
3. the map of problem solving remind me how we get the solution, I really like it

I think this topic is hard to understand first, but with your video, I can understand most of it because of your clear explanation.

• Very clear explanations and animations! I’ve heard a lot about Euler’s formula and graph coloring problems before, but I learned a lot about the relationship between the two, and I was able to follow along no problem.
• Very minor comment: after you prove that $\delta < H_S$ implies that the graph is the graph is $H_S$-colorable, I think it’s worth reiterating to the viewer that this proof is only true when $\delta < H_S$ - a less mathematically inclined viewer might forget that that was the hypothesis, and you haven’t proven any general colorability yet

Overall a very solid entry, and fairly easily digestible too, for me at least. A really strong first entry too, do keep it up.

Very well structured!

Got kinda lost at near the end after the heuristic part. It seemed at the beginning of the video we were trying to quantify how many colors for a specific map, so I was confused when I saw we were only deriving the upper bound for that number given a surface. Perhaps the video could be framed better by introducing a “table of contents” at the beginning of the video where you say “we’re going to show this using this this and this”. Overall interesting video!

Overall well made, but mostly standard. A few inconsistencies and minor errors that might through people new to the topic off. E.g.

• you say K_4 is planar on a sphere but it is obviously planar on a plane, too, as you showed.
• You use the tours unfolding way earlier than you introduce it.
• I think you didn’t explain that the “outer” face is also a face in Euler’s formula. Stuff like that should be ironed out.

I wished there was a section purely showing different surfaces explicitly and their colouring numbers. I know a formula was provided but it’s always nice to see with your own eyes the result.

Other than that, I think it was a pretty decent video. I would’ve explained more about the “no intersections” rule as I was wondering about non-orientable surfaces the entire time.

I liked the animations, I can tell a lot of work has been put into this so well done :)

Now, how about the “map colour theorem but where the British Empire is all coloured the same”? :)

The pacing of the video was a bit off because it felt like more time was spent on giving examples for the Euler identity than working with the coloring problem

I don’t think the solution presented in this video (the loose bound) was explored enough - seemed like just algebraic manipulation rather than a “discovery”

history at the end was interesting but it was hard to see how the exploration presented in this video could be expanded to think about the harder problems

I was amused that you used the double-torus as the thumbnail, but that it violated the ground rule requiring a single component. Some of the text coincided with the Closed Captions box. Simple example given of induction. The intended academic level was respected. I liked the discussion about the Map of Problem Solving. The suggestion to become curious at the reasoning for Euler’s Formula when using pseudoedges on a torus was a nice touch. Multiple speakers broke up any potential monotony.

The structure and big-picture view in this video are excellent - it really makes the problem-solving process feel accessible. Your animations are very clear and help get your points about this across. It would be nice to see some more detail/clarity in each section - e.g. many viewers will have questions about why the Euler characteristic is invariant, and some more visuals to go along with the proof steps would be helpful. For a general audience, the problem might also feel like it lacks motivation. If your aim is to showcase the general map of problem-solving (which this video definitely does well), then opening with this could tie your presentation together. Overall, though I enjoyed seeing this intersection of topology and graph theory explained in this way!

This is a really good video. It’s an interesting (and somewhat familiar) problem that I would have had no idea how to solve, and I thought the solution was reasonably accessible overall.

The one part that really tripped me up was the idea of capping the minimum degree. Maybe it’s just a naturally confusing idea, but I think it would help to slow down a bit there and explain exactly what you’re trying to show. Even now, if I try to rehash it in my head, I have to stop and think about it: for a given surface, you’re trying to show that you can always guarantee there’s a node of degree at most δ. I think I finally have that right. Also, FWIW, I still don’t really understand what H_S is.

I watched the video three times and eventually got it, and overall I think it’s very well done. The animations are great. You should just really slow down at the tricky parts and perhaps re-explain certain things just to make sure they click with viewers. I’m realizing that I have this issue with my own videos.

One minor production thing I noticed: the audio is a little uneven, and in a few places I could tell that different sections had probably been recorded at different times or under different conditions. Not a huge deal, but I’ve found it easiest to get the whole video set up first and then record all the audio in one sitting. You can always cut out places where you stumble and start over, so it doesn’t really have to be a true one-take recording.