Part of the requirements for my Honours is that I record a short 4-7 minute digital talk, which is then distributed to all the other students and staff at my university’s mathematics department. The video doesn't have to be related to my thesis, and it should ideally be accessible to all the other Honours students. I have already chosen a topic (a Galois theoretic proof of the Fundamental Theorem of Algebra), but it got me interested in learning about other results that have a pithy/clever proof. It doesn't even need to be a proof, just a subject which would make for an excellent short mathematics video.
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23$\begingroup$ I dislike this trend of requiring students (or even researchers) to give 5-minute talks on a mathematical topic. Mathematics requires time and patience and should not be restricted to such a short time. Imposing such forma mentis to students can do everything but good. $\endgroup$– rtsssCommented Jun 17, 2021 at 10:24
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1$\begingroup$ @rtsss it depends on the goal. If you want to show the piece of beautiful mathematics for not mathematicians, five minutes may well be enough. But this is rather about, say, Pythagoras theorem or infinitude of primes than Galois theoretic proof of fundamental theorem of algebra. $\endgroup$– Fedor PetrovCommented Jun 17, 2021 at 12:17
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11$\begingroup$ @rtsss Most disciplines require time and patience. Learning how to condense information and make it accessible to a broad audience is a difficult and useful skill. And as a mathematician, I certainly get a lot of training from acquaintances asking "What is it that you do, exactly?", and I wish I had a better mastery of the skill. $\endgroup$– Najib IdrissiCommented Jun 17, 2021 at 12:26
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$\begingroup$ This is an interesting discussion topic, but far too broad for an MO question, I think, unless narrowed down to something more specific. There are many, many, many different kinds of thing that can make a good short talk — concisely showing proof in a known field, or surveying a (sub)field, or showing some nice example, or giving a bit of history… And all of these will depend heavily on the tastes and background of the audience and the presenter. $\endgroup$– Peter LeFanu LumsdaineCommented Jun 17, 2021 at 17:43
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9$\begingroup$ There is some positive purpose in practicing giving "elevator pitches", meaning communicating some_form of a serious thing in a ridiculously short amount of time. I am well aware that there is a tradition in math in which one says that this-and-that are impossible discuss without years of preparation... but by now I think this is both very-bad PR, and a bit arrogant. $\endgroup$– paul garrettCommented Jun 17, 2021 at 19:29
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5 Answers
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I understand from the question ("talks accessible to everyone") that the audience will be broad. For inspiration, you might look at Math talks that blow your mind.
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I would suggest the 3Blue1Brown video on The essence of calculus, from minute 2 to 7. It explains how to find the formula for the area of a disk, as an preliminary to integrals. Not research level anymore, though.
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I suggest fractals as a topic that can be understood relatively easily and leads to some incredibly beautiful mathematics. You can define the Mandelbrot set without using any more advanced topics than quadratic functions and complex sequence convergence and then there's any number of videos on the internet (of varying lengths) to show the complexity arising from the very simple problem setup.
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3$\begingroup$ I agree in theory, but I think the Mandelbrot set is one of those things which has started to become a bit over-represented in popular expositions of mathematics. $\endgroup$ Commented Jun 17, 2021 at 22:28
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1$\begingroup$ For 5 minutes the length of coastlines could be a better topic --- $\endgroup$ Commented Jun 18, 2021 at 0:50
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I made some fancy animated diagrams of platonic solids, showing the way they nest inside each other, duality relations, axes of symmetry and so on. I think that's a good basis for a short talk. I made some videos myself, but I'm sure it's possible to do better. If you wanted to modify the diagrams, the code is available (but not well-documented).
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$\begingroup$ Your animated diagrams are beautiful, but the 12 edge midpoints in a regular octahedron form the vertex set of a cuboctahedron, not a regular icosahedron as claimed in your nesting animation. You need to take points that are closer to one end than the other to get the icosahedron. $\endgroup$– IJLCommented Jun 18, 2021 at 9:25
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$\begingroup$ @IJL The vertices of the nested icosahedron are not the midpoints of the edges of the octahedron. A typical vertex of the icosahedron is $(-\tau^{-1},\tau^{-2},0)$, where $\tau=(\sqrt{5}+1)/2$. This has $-\tau^{-1},\tau^{-2}>0$ with $-\tau^{-1}+\tau^{-2}=1$, so it lies on an edge of the standard octahedron, but not at the midpoint. $\endgroup$ Commented Jun 18, 2021 at 9:39
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$\begingroup$ Exactly. But your nesting page does use the word `midpoint', although it does also give the formula for the point. $\endgroup$– IJLCommented Jun 18, 2021 at 9:41
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1$\begingroup$ You're right, the explanatory text is wrong. I will fix it. $\endgroup$ Commented Jun 18, 2021 at 9:43
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1$\begingroup$ @IJL Incidentally, I have a vague memory that I learned about the coordinates of the dodecahedron from you years ago, although I can't remember the context. So thanks for that! $\endgroup$ Commented Jun 18, 2021 at 9:53
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I have a fascination for 'simple' topics (which often turn out to be not so simple), so here a couple of ideas:
- What is a number (set theory or even category theory?)
- What is continuity (intuition of $\epsilon-\delta$)
- Cantor's diagonal argument
There is a huge amount to say about each of these, of course, but I think the essence of each can be boiled down to a short talk. Example - natural numbers:
You have a bag of apples and a bag of oranges, and for each apple you have an orange: they are in a sense the same 'size', if you will. That is the essence of a number: the property that all sets of the same size possess. This can then lead on to relations, functions, equivalence etc etc.