Math 202C: Lecture 0

As a student enrolled in Math 202C, you have necessarily completed Math 202B under the tutelage of Professor Brendon Rhoades, and as such are an expert on the representation theory of the symmetric group \mathrm{S}(d). You may however not realize that this theory is directly applicable to natural and compelling real world problems, some of which are sufficiently rich and interesting that they can be adopted as a posteriori motivations for the development of the subject.

In this course, we are going to take another look at the representation theory of the symmetric group through the lens of the following real world question: how many times do you have to shuffle a deck of cards in order to completely randomize it? We will try to answer this question completely and thoroughly, down to the last detail, and in doing so expand on your existing knowledge of the representation theory of the symmetric group, as well as place this theory into the context of the more general subject known as harmonic analysis on finite groups. This is a beautiful and very classical branch of algebra which enriches the representation theory of finite groups you learned in Math 202B with new perspectives gleaned from the theory of the Fourier transform in classical analysis. Representation theory and harmonic analysis can be completely united, and the resulting subject is the representation theory of topological groups, which we may briefly discuss towards the end of the course. However, we will primarily remain in the finite setting where everything is totally algebraic and easy to understand without any sophisticated machinery.

The mechanics of the course are as follows. I will provide you with a complete text on harmonic analysis on finite groups, in the form of blog posts which will appear here every Monday and Wednesday, in the approximate neighborhood of our regularly scheduled lecture time. Thus there will be two posts per week; reading through and understanding each of these will likely require somewhat more effort than absorbing a standard in-person fifty minute lecture. These two lecture posts form the asynchronous component of the course. That leaves our Friday lecture slot, which will be a two hour live stream session: the first hour will be a flipped classroom style experience, rather than a planned live lecture, in which we can collaboratively review, discuss, and expand upon the material from the lecture posts. The second hour will be my office hours, so open to any and all questions. In practice, it is likely that there won’t be much of a distinction between the first and second hour, but batching the lecture slot and office hours gives us a good chunk of time to work through things. This all happens on the Math 262C Discord server, which will be our main means of communication throughout the quarter: please sign up immediately using this link, https://discord.gg/SRKNEvNP. When signing up, please take your real name as your username.

The grading scheme for the course is as follows. The lecture posts will contain embedded problems for solution. You will be expected to solve these problems, typeset your solutions in LaTeX, and submit them for grading each week, prior to Sunday at midnight. The exact method for submission and return of your graded solutions will be decided this week, and announced on the Discord server. Furthermore, you will write a “guest lecture post” to be published on this blog on a topic related to the course material; I will prepare a list of possible topics, or you may choose your own topic and clear it with me. Your grade for the course will be an average of your graded problem solutions and your guest lecture post, and you may choose the weighting of this average however you wish, anywhere from 100% problem solutions to 100% guest lecture post. Thus, at the extremes, if you prefer to solve problems and have your solutions graded, you can skip the lecture post, and conversely if you are getting tired of solving problems and would rather throw yourself into a more long term project, you can put all your effort into writing a lecture post and forego submission of typed solutions to the problems.

At this point, I expect you have questions. Please sign up for the Discord server, and we can start talking and working out various logistics. The mathematical content of the course begins on Wednesday, with Lecture 1.

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