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Curated by: Mathemaniac (96 videos)
Part 1 (previous video): https://youtu.be/6kfbotHL0fs Orbit-stabilizer theorem: https://youtu.be/BfgMdi0OkPU Burnside's lemma is an interesting result in group theory that helps us count things with symmetries considered, e.g. in some situations, we don't want to count things that can be transformed into one another by rotation different, like in this case of an isomer - rotating an isomer yields the same isomer; and in the case of music theory, we might not want to count chords obtained by transposition the same. This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. There is usually also something called Pólya Enumeration Theorem, which is a generalization of Burnside's lemma, and can be used to tackle a wider set of problems, but most problems that Pólya Enumeration Theorem can be applied to can also be tackled by Burnside's lemma, so this is usually more important. The theorem simply extracts the core of what we are doing (considering cycles for example) and put it in a nice generating function, which can be useful, but the notation and the computations required are very troublesome, and does not fit well too well with the theme of this channel. Non-mathematical applications like counting the number of isomers of an organic molecule (organic chemistry) and the number of trichords (music theory) are usually tackled by the theorem mentioned above, but this can really be tackled by Burnside's lemma, just with a bit more care. By the way, this lemma is not actually first discovered by Burnside, and the Pólya Enumeration theorem is also not first discovered by Pólya, but this phenomenon is also prevalent throughout mathematics and science, known as Stigler's law of eponymy. This is not a part of the "Essence of Group Theory" video series, because it is not "essence", but an application of the orbit-stabilizer theorem, which is in Chapter 2 of the video series: https://youtu.be/BfgMdi0OkPU ****RESOURCES FOR THE POLYA ENUMERATION THEOREM**** All the resources here illustrate the theorem using different examples, and the last one also presents the proof of the theorem. You will need to know cycle notation of permutations (a topic in symmetric groups that I previously said wouldn't be covered on this channel, because it is not very visual and that I couldn't give unique enough insight on YouTube) (1) https://www.whitman.edu/Documents/Academics/Mathematics/Huisinga.pdf (2) https://cp-algorithms.com/combinatorics/burnside.html (3) http://pi.math.cornell.edu/~apatotski/IHS2015/Lecture%2010.pdf (4) http://www.diva-portal.org/smash/get/diva2:324594/FULLTEXT01.pdf (5) https://www.math.cmu.edu/~af1p/Teaching/Combinatorics/Slides/Polya.pdf ****GOOGLE FORM**** Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! ****SOCIAL MEDIA**** Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time! #mathemaniac #grouptheory #abstractalgebra #burnside #mathematics