Summer of Math Exposition

Visually, some very stunning scenes, as well as sophisticated methods for displaying algebra and some animations.

Very strong motivation and historical context. Mostly clear about “what comes next” or putting things in simpler terms.

While the topic of abstract algebra isn’t novel or obscure, this presentation is a novel approach towards explaining group theory for me, including in its emphasis of Cayley’s Theorem and describing Galois’ work in its original algebraic context; along with making connections to physics.

Maybe the best “aha” moment was your example for Cayley’s Theorem, seeing Z_3 as a subgroup of S_3. Visually, it’s very clear.

I think my only room for improvement might be related to the scenes when you prove Cayley’s Theorem. Maybe it was just a lot going on at once to track visually, especially trying to read the paragraph at the bottom of the screen meanwhile. As beautiful as the background gradients were to shade in the sets, it was also somewhat distracting me from following your argument.

The production value of this video is high, but I struggled to grasp the message. I believe it would have benefitted from more often returning to concrete examples, whereas it turned very abstract towards the end. I‘d also suggest you trim the title since it does not match the video‘s tone.

Really great video, animations were really easy to follow + clear voice, it felt like watching a 3b1b video. The topic as a whole is definitely above me at the moment, but thank you for the introduction. Thanks for sharing!

Good visualizations of the Galois group and intro there. This is a bit weird, because usually before going into Galois theory, one would learn group theory before leading to this.

When we talk about an automorphism generated by g, that does already assume some implicit knowledge of group theory for a lot of sections of the videos, which is odd given that we’re using a casual definition of a group. The video can be organized to use consistent language throughout to tie it better.

I’m a big fan of seeing group theory visualized, and I recently learned Galois theory, so this was a nice refresher! There were some pretty nice animations in here, too.

I wasn’t a gigantic fan of some of the larger text blocks in the video. If you could pare them down or even reveal each little bit one at a time instead of all at once, I think that’d do the video much better. I think using you’ve used some the strengths of the medium of video to your advantage (especially with some of the visualizations), but there are certainly points where it felt like I could’ve just read what was on screen.

Overall pretty decent, though! I hope you make more stuff. Also, please include more zoo animals.

I really liked your explanation of Cayley’s theorem! The music was a little loud at times. I felt that the first part talking about automorphisms of field extensions felt a little bit detached from the rest of the video.

This is a solid work explaining Cayley’s Theorem, which almost feels trivial, once you understood, but very mysterious before that (much like the ‘bigger brother’ Yoneda Lemma).

What can be improved? The automorphism of Q was a bit fast, probably not many from the audience will get it. The Hamiltonian at the end is also very steep. Probably not a good idea to leave the audience with the feeling of having no idea. If it was just a hook for a next video, then it should be shorter. Perhaps the audio can be improved, I had to increase the volume to hear what is said, but then the music came out loud.

It is a nice way to remember a friend.