Present and Future Projects
Some updates on the state of the Deranged Mathematician
Let’s discuss what projects are currently in play and what I have planned.
As mentioned in my update yesterday, the group theory lectures are, so far, the longest-running series I have done, and it is going to go on for a little while yet. We just got to covering actions there, and I’m quite excited to talk about actions on trees, the ping-pong lemma, and some of the applications of this to number theory. After that, we’ll finally get into quotient groups, the classification of finitely-generated abelian groups, and we’ll wrap up with some more specialized topics. I definitely want to talk a little about the classification of finite simple groups. There are a few other options.
Braid groups are very visual and have a surprising application to pathing.
Modular forms in number theory are defined by the fact that they are holomorphic, slow-growing, and (almost) invariant under the action of the group SL(2, ℤ) (or a commensurable group). It would be impossible to go into any real depth without bringing a bunch of other theory, but we could give a very rough introduction to some of the basic constructions.
I have a number of analogous lecture series that I would like to do afterward—certainly on number theory and linear algebra, but I would eventually like to cover real analysis, topology, and others. Before any of that, though, I intend to put a series that I would like to call “How to Survive Your First Proof-Based Class.”
It is not meant as a guide to pass any particular class—it isn’t even primarily meant as an introduction to any proof techniques (although some will inevitably feature). What it is is the accumulation of my experience and knowledge about how to think about and approach proofs, with some notes about why they are important, and you are being asked to learn them. But I intend to illustrate all of the philosophy and general principles with concrete examples taken from a variety of courses that might serve as one’s entry into proofs: geometry, real analysis, discrete math, and so on.
This series will not be under any paywall, in any capacity, ever. I consider it to be too important not to share, and I recognize that those who are most likely to need this resource are also least likely to be able to pay for it. What I will ask is that if you know someone who might benefit from this, please consider sharing it.
There are some other projects in the works. Somewhat unexpectedly, my article on JRPGs like Chrono Trigger and how they take place on a donut is stretching out into a multiple-part series.
It was always the plan to do a follow-up on the Klein bottle. But there are a couple of threads that this has opened. The first is that I still need to do an introduction on the Euler characteristic mentioned in both of the above posts—I’m thinking of doing this together with an illustration of how it is useful in computational geometry. The second is that many commenters asked why the torus and the Klein bottle were the only surfaces with the properties I requested: surely, the projective plane would also work? It doesn’t, but for slightly subtle reasons, and I want to do an article about this. It would have to be pretty illustration/animation-heavy, though, and so I might not get to it for a while.
I know that there has been interest in some follow-up to the post on L-functions: a gentle introduction to modular forms, or automorphic forms, etc. As I mentioned above, I might be able to do a tiny amount of that in the group theory lectures. But, in general, it is difficult: this is material where you really need to know some blend of group theory, complex analysis, and number theory to begin to do anything. How to make this palatable to the beginner? I don’t entirely know. I will continue to see what I can add, and where.
An easier ask is to look for examples of similarly complicated material that can be made easier to grasp, but in other areas. As an example, I would like to write an introduction to Banach spaces and functional analysis, readable to a clever calculus student. Or, perhaps, I could write about other parts of analytic number theory, like the circle method. If you have any suggestions/requests, please feel free to ask!
I like the idea of the series on surviving your first intro to proofs (whatever course that happens to be). In particular, one thing I struggled with (and occasionally still do) is that we learn about different proof "tactics" (to let a little Lean terminology leak in), and it was really unclear when something would be easier to prove via the contrapositive, etc. I felt like there were examples of each tactic, and a lecturer would often say, "here we need to use contradiction", but examples of exactly what the experienced person SAW in the statement that triggered that judgment -- or perhaps some examples of what goes wrong or becomes harder when you choose the wrong approach could be helpful. I feel like after a solid year of real analysis and metric spaces, I developed some feeling for that, but that feeling is probably rooted in the specifics of the material itself -- in seeing the repeated patterns in various analysis problems -- and not easily transferable to other branches of mathematics. I think that's the key thing -- how you start to see and catalog those proof patterns.
I also really like the idea of an introduction to Banach spaces and functional analysis, but this one's selfish, I'm starting to study it now and I find nearly all texts completely devoid of *motivation*. I think you get a taste of "why this giant edifice of machinery matters" late in real analysis -- often a second semester of real analysis in the US, when you look at function spaces and Picard-Lindelhof, and maybe your course gets to Weierstrass Approximation and Stone-Weierstrass (mine did not!), so you "get" that virtually everything we can't "directly solve" like you did in early calculus classes is something we're going to *approximate* with sequences of functions instead...but would it kill somebody to actually SAY that, and give you a bit of big picture? So I'd love to see a post on that, even if it was a fairly short one -- show me the big picture and where all this machinery will take me!
I'd personally be interested in seeing some material on character theory. There are so many and such beautiful patterns therein, and I think there's a great deal of room for your posts to make it «easier to grasp»!
(I would write such materials myself—in fact I intend to do so in the future—but right now my expositions are on rather more basic topics…)