Summer of Math Exposition

Around minute six with the simple example of a discont. function, I think the pace was a little high. When introducing touching, a few drawings would have been in order. Some dashed lines, some green blobs, you know the drill.

I get that it’s good content to mention your indignation of separable/computable, but honestly separable makes a lot of sense. Or, I’m sure it made a lot of sense when it was coined, and how are you gonna blame the people naming the term for how society looks now, much later?

Other than that, really fun and interesting video. I found it novel and memorable, in that it was not about coffee mugs/donuts. Neither was it about graph theory, Koenigsburg’s bridges, topological insulators or that esoteric math stuff.

It was about how touching, distances, calculus and computers are related! That was new to me, and it was explained in a good way.

Couldn’t agree with you more about renaming separability. I love how you explained the different core concepts of topology at the start of the video, and while some of the humor was jarring at first I found the jealousy/high school friends metaphor for useful. My favorite part was the last section where you explained why these concepts were useful, and I love how you used the common knowledge of the mug donut thing to introduce actual topology usefully.

As far as the criteria for the contest, this was a memorable video with a strong reason to care. It was about as clear as can reasonably be expected given the subject matter, but wasn’t particularly original.

I found that I learned a surprising amount about topology given the length of the video and it was just the right difficulty for a undergrad or newer grad level.

The closure axioms are a nice and under-presented idea, so I’m glad you discussed closures, and I really liked your discussion of the duality between open and closed sets with approximations and touching.

I’m not convinced that “qualitative shape” is worth totally abandoning as a heuristic for what a space is, and the tone felt needlessly combative, but I appreciate that you took a memorable stance.

Too much of textbook presentation

I think this is a great attempt at a video, with many great parts to it, but I think all together it ends up being worse than what the individual parts do. I think the problem at its core is that this video doesn’t seem to have a set target audience; it’s nominally about introducing what topology is and why its important, but the jokes only seem to work if you already have an understanding about the topology. So if you go into this video expecting to learn, you see a bunch of jokes fly over your head; which doesn’t seem particularly conducive to learning about it.

Basically, I think this video has some really, really good potential; the creator definitely has some passion, and talent with this. However, I think this video should have either made a better attempt to teach an audience, or just go all in on the mathematical humour; both of which I think the creator is more than capable of. As is it sort of struggles at both.

Disclaimer: the furthest I got with formal math was calculus and statistics, so maybe I’d get something more out of this video if I had taken an analysis course. But for me, I couldn’t really parse a lot of the analogies made, despite pausing and going back a few times. As an example, I didn’t quite understand how the stack of papers and accuracies related to the axioms of open sets. Also, as I was completely unfamiliar with the 5 concepts of topology, the tier list at the beginning went completely over my head. I also wondered about the difference between different “definitions” of topology or how to define a “space”. I assume all topologies must follow the same axioms, so where do the different “flavors” come from?

That being said, I was able to follow everything in the “computable” section and onward, and I thought the explanations were pretty good. The example of the step function not preserving approximations was particularly helpful. I think maybe by rearranging some stuff in the script, or maybe by providing an outline which “foreshadows” later sections, it would be easier to digest - like for example, the first 2 minutes of your other topology video were very enlightening!

I believe there is a need for videos that approach topology in this way and you laid out the reasons very well. However, this video does not yet fulfil this need. First of all, ai found that the video stayed too close to a static slide set format and did not leverage the potential of the medium. I found this particularly apparent when you discussed in the definitions at the start, where simply showing [0,1] and friends on a number line would have made it a lot more accessible. You mentioned at the end you already have a video discussing this in detail, but since each contribution should stand on its own, I think your definitions in the video should do so, too. I liked the analogy to morality. It’s a large jump, but it works. I think much of the pacing was too quick, e.g. I did not get the transition to accuracies. I think that would have needed a clean definition to mark that this term is the central point of the remainder of the video and what I need to understand by it. Then all remaining examples would have had a higher chance to land. I did not get how the you tried to tie it all together with engineers, graphics etc. Again, visualisations would have helped. In the end, I understood that topology is not about shapes, but I don’t think I got enough info to piece together your other messages. I believe that you are sufficiently ambitious in your goals, but the execution needs tuning to your audience.

I don’t know whether to call it unique or not, but your certainly have a different style compared to other videos I have seen so far. It was very enjoyable and I think I grasped some important ideas about why we need topology and what it really is. Nevertheless, at times I felt a little lost, probably because I am not very familiar with topology. Since you are aiming for an undergraduate audience, you could consider adding more visuals to make the explanations clearer.

I really enjoyed this take on topology from start to finish. I have taken a course in topology, hence the terms were familiar, but I think your analogies and explanations make the concepts clear to unfamiliar people too. And even for me, the closure axioms were new. And the way you drew connections between topology and other concepts was excellent.

Big plus for the message, that topology is more than just donuts.

Also, love the visuals (well, deer, viewer).

In the following are my nitpicks:

At 10:03 there is a typo, the intersection should run over $i$ up to $n$, not the indexing set $\mathcal{I}$.

A comment on calling countable listable: Your objection still applies. You can’t list them all, but you can assign a whole number to them (number them, hence enumerable, hence countable).

When talking about accuracies, especially mentioning the weather model, I was missing a footnote on chaotic systems. That is, even if they are continuous, the level of accuracy you would need at the beginning can render the whole approximation unfeasible.

A refreshingly original take on this! I don’t have a formal background in topology but thoroughly enjoyed all the novel connections to real analysis, calculus, etc. The real-world connections were great too, especially pertaining to the ubiquity of approximation in all of modern computing and science.

I also appreciate that you still used the familiar donut/mug example to help ground the notion of tearing, even if you state that this overused example vastly glosses over the true essence of topology.

A few definitions and statements went by a little fast, and there could be more showing than telling for some topics (ex. showing the neighborhoods for epsilon-delta as applied to a graphed function) but really great stuff and casually/conversationally presented!