Summer of Math Exposition

That was a nice presentation of the obstacle problem! A bit dry and technical when it came to the actual formulas, though.

Great video overall! a wrap up in the end would be nice and make the solution more memorable!

Nice animations, appropriate level specification (this is legitimately undergrad or grad level). Brings two disparate concepts together (ice versus putting something under an elastic sheet). I was a little unclear on the restrictions of what type of obstacle we could use (does it have to be rounded or can it be any shape?) The example is always rounded but it seems like this applies to any kind of shape thrust up into the elastic sheet.

The glare is on the sheets is nice. How did you get that?

Pretty good but lacks that extra ‘wow’ factor and energy that would earn an 8 or 9. Probably a bit fast for non-math majors, but not everybody is going to understand every video, so that’s okay. But in my opinion it’s okay to slow down a bit to unpack these explanations a bit more or work through more concrete examples.

I really like the connection to put options.

The details of some of the math are hard to follow, depending on background, but this is probably a good companion to someone separately studying those details who wants to understand the problem more meaningfully.

The connection to melting ice could be developed more or maybe instead reframe the setup.

This was a good presentation of a cool problem I’ve never seen before. The animations were helpful. Unfortunately I didn’t have the background knowledge to fully understand it, but I can tell it’ll be great for people deeper into this area of math.

I’m watching this as I’m a bit tired, but I couldn’t quite follow this video. I couldn’t follow your notation, you could have talked a bit about it. I can kind of see how the membrane problem relates to the melting ice problem, but it is still a mystery how it relates to trading, and why the regularity of the boundary matters.

The opening was a very nice hook to open the video. Unfortunately I think this set a tone that was a bit confusing; as I was watching I became increasingly unclear who the target audience was. By the end it was clear that the target audience was basically me— someone with significant mathematical sophistication but knowing nothing about the problem.

I begin with my most concrete criticism: the proof is not properly placed in the video. I am glad to see that you return to and resolve the hook from the beginning, but the geometry of the singular set does not seem relevant for connection between the two problems at all. Because of that, I would rather see the connection and conclusion first, with the proof tacked on as an “appendix”. (FWIW, I did not find the proof particularly clear, but I tried not to hold that against you. Such proofs in videos are nearly impossible to make smooth; your effort was clear, and admirable.)

Throughout the video you show significant of restraint in the script and visuals: while the equations you are throwing around work in general dimension, the visuals frame the discussion as 2-dimensional, and until it is necessary in the proof to discuss the stratification (at which point I think it is safe to break this illusion) the narration does nothing to dissuade this interpretation. To be clear, this is a big positive, and actually one of the things I like most about your work here.

Less restraint is shown in the use of technical jargon: early in the video the narration casually mentions Sobolev spaces. This is a mechanical error: it’s not really needed for what else you discuss in the video, so I would have preferred to see this on screen and not mentioned in the voiceover, or at least with some kind of signpost that this is just a technicality to appease/orient experts. This is probably the most egregious example, fortunately, though I also feel like some preamble would be appreciated for Dirichlet energy, harmonic functions, and perhaps others.

That said, as someone who is familiar with harmonic functions: I do think your discussions of why they are relevant here, as well as why this constraint can be imposed on the obstacle, are excellent. I found them clear, intuitive, and memorable, and I felt the visuals complemented the script particularly well in these segments.