What is...representation theory?

Goal. Explaining basic concepts of representation theory in an intuitive way. This time. What is...representation theory? Or: Lets make life linear! Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. Representation theory is a vast field, and a representation could be of a group, an associative algebra, a Lie algebra etc., and these even in flavors. These subfields are similar, but then also different in nature. So to be precise, this video series is mostly about finite dimensional representations of finite groups and monoids. Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-rt-2022.html Representation theory. https://en.wikipedia.org/wiki/Representation_theory https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609/ https://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/representation+theory https://www.maths.usyd.edu.au/u/geordie/BMS.pdf Applications of representation theory. https://yetanothermathblog.com/2016/08/06/real-world-applications-of-representation-theory/ https://mathoverflow.net/questions/11784/fun-applications-of-representations-of-finite-groups https://math.stackexchange.com/questions/622/importance-of-representation-theory https://math.stackexchange.com/questions/564260/application-of-representation-theory https://math.stackexchange.com/questions/423195/applications-of-representation-theory-in-physics https://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html Pictures used. https://en.wikipedia.org/wiki/Tangent#/media/File:Image_Tangent-plane.svg Picture from https://www.youtube.com/watch?v=0qshwDxI8Y0 Picture from https://www.dtubbenhauer.com/Talk-ETH-2018.pdf Picture from https://www.gutenberg.org/files/40395/40395-pdf.pdf Picture from https://people.math.rochester.edu/faculty/doug/otherpapers/burnside1911.pdf https://chem.libretexts.org/Courses/University_of_California_Davis/UCD_Chem_110B%3A_Physical_Chemistry_II/Text/12%3A_Group_Theory_-_Exploiting_Symmetry/12.2%3A_The_Symmetry_of_Molecules Some books I am using (I sometimes steal some pictures from there). https://link.springer.com/book/10.1007/978-1-4614-0776-8 https://users.metu.edu.tr/sozkap/513-2013/Steinberg.pdf https://link.springer.com/book/10.1007/978-3-319-43932-7 https://math.mit.edu/~etingof/replect.pdf https://link.springer.com/book/10.1007/978-1-4612-0979-9 https://link.springer.com/book/10.1007/978-1-4684-9458-7 https://www.cambridge.org/core/books/representations-and-cohomology/9EE67E1280D8DB01AAC662530799AA8E https://link.springer.com/book/10.1007/978-1-4757-6804-6 Mathematica. https://arxiv.org/abs/1912.10969 (Mostly for representations of Lie groups and algebras) Magma. https://magma.maths.usyd.edu.au/magma/overview/2/17/11/ http://magma.maths.usyd.edu.au/calc/ SageMath. https://doc.sagemath.org/html/en/constructions/rep_theory.html https://sagecell.sagemath.org/ #representationtheory #algebra #mathematics