Summer of Math Exposition

Very cool idea combining a story with an explanation about a math topic. I think it was well executed both the story and the math parts. Just for the attention of the listener, it would have been great to have the whole dialog spoken. With 1h, it’s quite long and hard to focus. Maybe a bit of music would have been nice. Overall great job and a good voice. Keep it going.

Cool format! I wasn’t expect a math video to have this kind of dramatic flair.

Congratulations on the video! And thank you for being inspired to make a video with such an important and meaningful topic.

Critical feedback: The film-imperfections flickering set the tone nicely, but quickly became distracting, especially over text and equations. It’s a little hard to keep track both of the story elements and the math elements at the same time.

I’m not sure the terminology of the “smoothness” of the graph is exactly right. The graph itself typically just refers to the edge/node structure, and f is a function defined on the vertices. I also don’t think people really refer to f which result in small values of fLf as “smooth”. Maybe “flat” is better. Typically people refer to fLf as some kind of “(spring) energy” or “variance” of f.

I also think skipping the derivation of fLf = sum squared differences across edges is a mistake. Writing L as BB^T where B is the incidence matrix is quite insightful IMO.

I found the analogy you made between eigenvectors and xrays confusing. I agree that the entries of Lf record the distance between the value of f at a vertex and its neighbors. But when you multiply that by f (rather than, say, take the L1 or L2 norm of Lf), you are already taking an unevenly weighted sum of those entries, so the idea that Lf isn’t disproportionately weighted in any direction when f is an eigenvector seems to be to be a red herring.

In the example you draw, you seem to suggest that the action L [1, 2, 3]^T = [4, 5, 6] is an eigenvector equation.

I fail to see how Lf is a “shadow” of the graph. Perhaps the idea you mean is this: the coordinates of the first k eigenvectors of L supply nice embeddings of the graph into R^k?

The numbering of the eigenvalues of L starting from 0 is nonstandard.

Minimizing fLf subject to the constraints f != 1 and f is an eigenvector I don’t think is as intuitive as the usual constraint of f \perp 1.

e^L is not equal to L + L^2 + L^3 + … I see the intuition you’re going for here, but I think this is really much too handwavy to be helpful. I also think its odd you use this matrix exponential, but then switch to the formula involving the eigendecomposition right away.

I like how he presented math without boring us in old school style

I thought the explanations were good, and I liked the use of very concrete examples.

The tone felt weird using this heavy topic of dementia as the flavortext of this analog/lo-fi-styled mystery, in a way that I thought was more distracting than helpful.

Adding in a story to the video was unique, and the application of spectral graph theory for diagnosing dementia was clever. However, the pacing was kind of slow, and the ending is a little corny. The analogies were also a bit lost on me. They would probably make sense on a second watch through, but at that point, I feel like the story would then hold the video back. Either way, the slow pacing would provide some time for the viewer to digest the information, and even though my experience in spectral graph theory is limited, someone else might find this video perfect. The idea behind the flashlights and bells made sense, and they would provide a good visual intuition for someone who is struggling with the notation. Good entry, but I think I just was not the target audience for the video