Summer of Math Exposition

Rather dry and technical. Very nice if you are currently in a math degree and want to get an overview/intro for this topic. But otherwise not enough motivation for a more general audience. Also, a few editing mistakes like at 03:40 where the assumptions vanish too early.

Good academic video. Like the calculus topic. The sound wasn’t very good.

I am sorry for giving such a low grade. I explain below my reasoning but wanted to start with a positive point: I do like the point that this video is making and the general storyline, extending a fairly well-known method to explain the sum formulation of the exponential, before then applying it to compute power of $e$. However, I believe the video could be improved quite substantially with minor changes.

The first improvement I would recommend is the sound quality. I mostly see two possible avenues of improvement.

• Avoiding the current level of distortion in the voice (especially at the beginning); I assume this can be changed by only distancing one’s voice from the microphone, avoiding directly blowing into it, and/or removing some enhancer effects added in post-production.
• Removing some of the extra noise, such as the tongue clicks and the sounds of movement (at the end of the video for example); this can usually be obtained by being more cautious with the recording setting, by cropping the sound clips appropriately, and/or by adding some low-level and generic background music or sounds. Naturally, there are always more costly options to improve sound quality that might also be considered here given the frequency of videos created by the author, but my judgement of the sound quality tried to ignore such expectations as they should not be placed on every creator, especially during such open-ended and international competitions.

The second improvement I can see relates to the general flow of the video. I have the following comments on that regard.

• I personally find the timing between the voice and the text on the screen a bit off; I usually prefer the subtitles (or in general here, the written parts) to appear with or slightly after the voice, so that the sound leads the dynamic of the images. I understand that this might be a personal preference, but nonetheless, certain dialogues were actually said right before changing “slide” and thus had me go back to re-watch the correspondence between what was said and what was written.
• The general visual setup of the “slides” is not homogeneous: I can see that some text was directly implemented into the video, while some text likely came from some pdf rendering of a tex document. I would personally consider this normal if the displayed equations were LaTeX-styled and the rest was more “standard” text, but here sentences are both standard and LaTeX. Moreover, the LaTeX parts were clearly a lower resolution and thus further broke the illusion.
• There were some green frames that are kind of unpleasant to see and easy to remove. Following on my previous comment, I want to provide two free ways to possibly improve the previous points. First, to include LaTeX equations from a pdf with the proper resolution, the software GIMP (GNU Image Manipulation Program) is free and can transform pdfs into images with arbitrary resolutions (I assume access to pdf versions of the equations, but otherwise invite the author to use Overleaf to create their own displays, also free in its basic version). Second, to remove green frames, and in general for video editing, the software Shotcut is free and provides a wide variety of useful features which I also use for my own videos.

The third and final comment I have relates to the math, more precisely the assumptions on $g$ within the different results. First, it is worth mentioning that $\left|\int|g|\right|=\int|g|$ and that the assumption $\left|\int g\right|=\int|g|$ is actually equivalent to $g$ being either positive or negative almost-everywhere on the considered set (assuming it is a real function). Second, the assumption that $\int|g|>0$ is equivalent to $g\neq0$ almost-everywhere on a neighbourhood of $0$. In conclusion, and given I assume the viewers of this video are not assumed to know about measure theory, I would simply replace the assumptions with ”$g$ continuous and strictly positive except possibly at $c$”. While the functions $x\mapsto(x-c)^k$ do not exactly satisfy these assumptions (when $k$ is odd and $x<c$ it is negative), this can be ignored either by adding absolute values, or by remaining within the case $x\geq c$. Also on the topic of the math, saying the polynomial works “fine” (7:32) is a bit underwhelming, as it is your main example of application… I would either recommend being more enthusiastic, or giving a more exciting example to explain why this one is only “fine”.

To conclude, I am sorry for being so harsh. I actually like the potential of this video and of this general type of content. I just believe there are several accessible fixes that would immediately greatly improve the overall quality and tried as much as possible to give clear and accessible instructions.