Summer of Math Exposition

Really great entry that took me back to my complex analysis and algebraic topology classes! You picked an interesting topic that could effectively be explained in the scope of a blog post. I enjoyed your writing style and explanations, and your visualizations were very helpful.

Just a few minor things:

1. I struggled with depth perception in some of the 3D visualizations.
2. I think you spent a little too much time manipulating Taylor expansions.
3. It would have been helpful if you had written out the solution to the equation as a complex square root. That would have made it clearer to me why the sign flipped.
4. You never defined monodromy

What do we get if we don’t focus the vector bundle on just the unit circle? More Möbius strips? Can we get an example for another diffeq/topology pair, even if considered more briefly?

This was a surprisingly nice read! It hit all the points I look for in an article; it was concise and the useful, interactable visuals made it satisfying and understandable.

Couple points to note:

• The Taylor Series derivation was a bit sticky and it was up front, so it made me a little nervous ^_^;7 I think if you can manage to put some visuals in there it would be absolutely dope (I’ve encountered this flavor of Taylor shenanigans in numerical analysis before, so I assume for people who aren’t familiar, it would be even more difficult to follow)
• Vectors bundles could use some more support, and I think you might want to mention the part about how they’re meant to form manifolds (which is why it makes sense for us to sort of staple them together into the Mobius Strip)

Other than that, great work!

The article was clearly written and introduced a really nice connection between differential equations and topology. The only downside was that the pacing felt a little off. The introduction power series expansion was given a lot of room (and a recap) but the topology at the end (the new part for most people) was discussed very quickly. I though the visuals and interactives were well put together.

This is really great. It is clear and concise and does the important work of bridging ODEs and geometry.

The diagram with all the circles seems unnecessarily complicated. Later, you show a slideshow that explains very simply. In my opinion, you should lead with the slideshow.

You focused on one example, f. I would be interested to hear a few comments about other functions and shapes that can be made, any general important theorems, etc. But it’s good that you leave me interested to learn more.

Aha! This was a very nice demo that gave me some new intuition. Solid visuals, reasonable pace, decently well motivated. Well done.

I would argue that the audience for this might be a little different: I learned something important adjacent to my field (physics), and I am already past my PhD, so I would say this could be of use to graduate students and beyond (in fields where algebraic topology is not a required course). On the flip side, there are probably some high schoolers who would get it, but it might be tough.

I would love to see this extended a little. Ok, so we did the Mobius strip, the usual simplest case. Now how would we use this to solve something we can’t otherwise, or what other problem does this case help us with?

Well written and animated! Simplifies a complex topic very well.

This was such a cute article! I love the tone and scope: it is really important to have these kinds of low-stakes and playful explanations of advanced math. And I really love the general project of Hidden Phenomena! This feels like a slightly more mathematically mature version of Francis Su’s “Math Fun Facts”---reading this article felt illuminating and encouraging, without requiring a deep level of concentration to follow along.

Regarding the content of this article itself, the only thing I felt was lacking in the explanation was how we know that repeating the procedure of “pick a point near the boundary and solve the differential equation centered there” in a loop around 0 really flips the solution to the initial function that was centered at 1. If $f_1$ refers to the function we chose which satisfies the differential equation in the blue circle, and $f_7$ refers to the function we arrived at through this procedure which satisfies the differential equation in the red circle, and if $z_0$ is in the intersection of the blue and red circle, then how do we know that $f_1(z_0)$ and $f_7(z_0)$ has to have any kind of relationship with each other in the first place? An answer to this question, even if it has to be handwavy to keep the article widely accessible, would better convince me that the mobius strip has to do with $\frac{df}{dz}=\frac{1}{2z}f(z)$---which I really want to be convinced of! Other than this one point, I felt like this article was very good at balancing being thorough while not being so detailed that it fatigued the reader.

very nice and clear math. loved it

Great article, this was fun to read through.

A few minor points. The jump from talking about the Mobius strip to the equation is quite sudden and it feels disconnected. The second time I got confused was after the red blue circle mismatch. I think it’d be good to explicitly state that no solution to the problem exist on the whole C / {0} plane. By the end everything makes sense and it’s gratifying , but on the first read I felt lost in it for a lot of the time. For me it would help to just state at the beginning something like: solutions to this equation hide the Möbius strip. Then I know what I’m looking for in the math.

I like that this does show a good connection from topology to differential equations that does highlight a connection between two very different objects. Overall, good amount of technical background for an early undergrad/high school audience with good charts to help to understand the notion of how certain cases can fail, which does give the audience a good sense of the subject. (this really does help to convey why certain properties are interesting, by how they don’t hold).

Some of the animations visibility needs to be improved. It’s a bit small or lightly colored. Some of the terminology on using a Taylor series can be more exact. For high school, this would be a guess. (although, using complex analysis, we know such as solution would be analytical). I think it’d also make the connection more interesting if we could somehow visualize the study of Mobius strips are very different from the differential equation.