Summer of Math Exposition

The visuals are quite useful for what you are presenting; but I thought that your explanation of the tangent bundle was too rushed and could be potentially confusing. Maybe you should show in more details what the tangent bundle is, for instance for the circle since you can still picture it.

You speak slightly too fast in my opinion, but this is a minor point. Speaking more slowly also gives more time for your viewers to digest the content!

Video does a good job especially towards the beginning and provides one of the better visualizations and corresponding explanations on vector bundles as well as making them feel natural as concepts emerging from a intuitive question to ask. That being said, the video spends a bit too much time introducing novel jargon and is a tad too overambitious in scope. I do appreciate it touching on the state-of-the-art towards the end of the video, and the range of concepts introduced is admirable given the relative simplicity of the starting point, but ultimately it gets overburdened by the slew of novel vocab and concepts that ends up detracting rather than enriching the learning experience.

That was a pretty good video, I enjoyed it!

The Dark Souls 2 example is a really good analogy, it immediately made it click for me.

Visuals are really good, I appreciate the consistent color-coding and the 60fps frame rate really makes the animations buttery smooth.

Pacing is a little fast though, not enough time was given to let the given facts process in the viewer’s brain to be honest. I had to watch the video twice to understand it better.

Also, not enough information was given in the video itself to be able to understand everything, it’s not really a self-contained whole. I would’ve appreciated some more information about the notation used. For example, I had to look up that ”𝕊^n” generally means “n-dimensional sphere” in topology. Also, in the formula Tℝ^3|𝕊^2 = 𝕊^2 × ℝ^3 I don’t know what the vertical bar notation means in this context. I’ve seen it used before in Bayes’ theorem, but again, not sure how it would apply in this context. Finally I had to look up ”⊕” as well.

The explanation about trivial bundles was pretty good though, it reminds me a bit of the Minkowski sum.

Also, some background music would have been nice.

very convoluted and the ease went down as we got further into the video

Great video. I like the motivation, and it gives a great overview of differential geometry and topology. It also makes those fields of math seem very interesting and inspires people to learn more about them.

One criticism is that I think there were too many “black boxes”. A lot of theorems were just stated without a proof or even a proof sketch. If the proof is too advanced, a brief sketch of how it was proved would be nice.

I feel like this video had potential. Topology is an advanced topic that many people aren’t knowledgeable of, so it could’ve been really interesting to teach people about how many dimensions one needs to embed a manifold. But even after having watched the video twice, I still didn’t understand the lesson. There are a lot of words like “embedding”, “immersion”, “vector bundle”, etc. whose meanings must be very clear to you because you studied in this area. But for me, it felt like I’m being told statement after statement without the necessary background. So eventually I got lost in the explanation. The 3D visuals look great. I didn’t want to give an average score for this, but ultimately I can’t find myself giving a higher score to a lesson I didn’t understand.