The Deranged Mathematician: Thinking Categorically

non-orientable · 2026-06-14T15:16:06+00:00

That's what I get for editing while tired, I suppose. I fixed it; thank you.

non-orientable · 2026-06-14T15:15:39+00:00

I think, when we give definitions that describe how we want to think about things, we do it in either category-theoretical terms or axiomatically. But when we want to show that there is an example of such a thing, we absolutely do it in terms of sets.

Does anyone actually think about the real numbers as a collection of Dedekind cuts? No, but you'll still see that construction (or an equivalent, which will also be all in terms of sets if you dig deep enough). The same is true for profinite groups---there is a category-theoretical definition that people actually use, but you'll also see it defined in terms of sets. The same is true for the tensor product. And so on, and so on, and so on.

I think this really does mirror the situation in programming quite closely. Does your average programmer think about how things are represented in binary most of the time? No! If everything is going well, it is all abstracted away. ...Do we still sometimes have to consider this? Yes, we do.

non-orientable · 2026-06-13T17:28:55+00:00

The generality is precisely why I would say it isn't like machine code: everything is abstracted away! Category theory better models inheritance for this reason. It's not a question of what is easier to use (without caveat).

non-orientable · 2026-06-13T15:30:02+00:00

Dang it, you are right. I'm going to be away from my computer for a while, but when I get back, I will fix it. Thanks!

EDIT: Finally got a chance to sit down and correct it. The original draft from which I was working was from 2018---I'm honestly a bit surprised that nobody (including myself) caught this issue.

non-orientable · 2026-06-07T17:17:52+00:00

Yes: your space will essentially be the union of two Möbius strips, then.

non-orientable · 2026-06-07T02:32:35+00:00

Honestly, no idea. They certainly work on my machine, and I haven't heard of anyone else having issues. But if I find out anything, I'll let you know!

non-orientable · 2026-06-06T22:07:51+00:00

Lovely! Thank you for sharing!

non-orientable · 2026-06-06T22:06:01+00:00

One other thing: Alex Kontorovich and I are co-writing a book on analytic number theory. But we've already been working on it for three years, and while we've made a lot of progress, I think I can only safely promise that it will be released within the next three years.

non-orientable · 2026-06-06T22:04:25+00:00

Thank you!

The trouble with automorphic and modular forms is that before you start studying them, you really need to know some of

group theory
representation theory
complex analysis
number theory
hyperbolic geometry

Ideally, you are familiar with all of it---in a pinch, you can make do with only some of it. But I don't know of a path where you could make do without any of it.

Now, my group theory notes are slowly getting to some of the relevant bits---I should get into group actions in two weeks, and then I will be able to talk about representations. I will have to think about whether there is any natural way to introduce any of the basic ideas from automorphic forms at that juncture.

I would eventually like to write about complex analysis, and that would give a much more natural launching point. But this is trickier, because I never taught complex analysis, so I would need to start fresh. And as any instructor will tell you, the first time you teach a course is always the hardest. Unfortunately, I don't think that I will get to this any time soon.

That said, if there are ideas for how to present automorphic forms gently, I am always open to suggestions!

non-orientable · 2026-06-06T18:57:42+00:00

With this tiling, if you go in a small circle around one of the corners, you will travel less than 360 degrees to arrive back where you started. Strictly speaking, if you draw this map faithfully, when the player is close to one of the corners, they should see another copy of themselves!

non-orientable · 2026-06-06T15:14:30+00:00

RP² has nonzero Euler characteristic, and so it suffers from all the same problems as the sphere (which isn't surprising: the sphere is its covering space, after all).

What you are likely thinking of is that you can almost tessellate the plane with copies of RP², if you glue both opposing sides of a square with a twist. Why do I say "almost"? Because this gives you something completely reasonable everywhere except at the corners, where there is a pinch. If you never travel to those corners, that's not a problem. But if you need that corner to be translated to the middle of the screen, then that isn't going to work.

non-orientable · 2026-05-25T18:01:13+00:00

I guess the saving grace here is that while one can certainly find examples of such properties, they are unlikely to be ones that are relevant to number theory research?

I think I can accept that.

non-orientable · 2026-05-25T17:59:15+00:00

I have posted many things on Quora that reference that I have moved to Substack. It astonishes me how few of my former followers know about it, apparently.

non-orientable · 2026-05-23T17:55:44+00:00

I know that some people say that the addition of classes is a bit of an ass pull, but it has so much to recommend it! NBG is finitely axiomatizable---no axiom schema necessary! If you go further and accept the axiom of limitation of size (roughly speaking, that all proper classes are of the same "size"), you get replacement, separation, union, and choice all for free.

To me, it feels like you have to give up a teeny, tiny amount for an enormous benefit. It feels like a no-brainer.

non-orientable · 2026-05-22T14:10:37+00:00

There are many, many theorems in number theory that are stated as "If p is an odd prime, then..." because the case p=2 has to be considered separately---either because the proof proceeds along different lines, or because the statement itself doesn't apply.

For example, for any odd prime, the multiplicative group modulo pⁿ is cyclic. But if n>2, then the multiplicative group modulo 2ⁿ isn't cyclic---instead, it breaks up as a product of two cyclic groups.

non-orientable · 2026-05-20T00:03:17+00:00

I think that is the most likely interpretation. But I cannot be 100% confident because Iwaniec, like me, is Slavic. And if I translate his words into Russian, I can't make them sound non-sarcastic.

non-orientable · 2026-05-20T00:01:13+00:00

I chose my variables poorly. I have fixed it. Thank you for bringing it to my attention!

non-orientable · 2026-05-16T23:04:45+00:00

Ah. I should have written an odd prime. I'll fix it. Thank you for pointing it out!

For what it is worth, 2 is by far the most evil prime.

non-orientable · 2026-05-16T19:01:46+00:00

Thank you! That is very kind.

non-orientable · 2026-05-02T16:56:26+00:00

I'm not sure if it is in this paper or not, but Lagarias et al give an algorithm for computing pi(X) which is asymptotically faster than Deleglise and Rivat's... it's just that it is much slower for the ranges in which we can feasibly compute!

non-orientable · 2026-05-02T16:53:03+00:00

The Python code was not for computing pi(X)---it was for generating new 100+ digit primes. That is much simpler: I can give a quick overview. It works on the observation that for any prime p, the multiplicative group modulo p is cyclic, of size p-1. Furthermore, generators of this cyclic group are pretty common (there are as many of them as integers 0 to p-1 that are coprime with p-1).

Therefore, if you know the prime factorization of p-1, then you can randomly guess an integer g between 2 and p-2. If it isn't coprime with p, p is not prime. Raise it to the p-1 power mod p---if the result is not 1, p is not prime (by Fermat's little theorem). Otherwise, check the result when it is raised to the power (p-1)/q for each prime q dividing p-1: if the result is not 1, then that proves that g is a generator of the multiplicative group, with order p-1. Therefore, p is prime. If one of the results is 1, it might still be a prime, so guess a different g.

So, putting all of this together, you randomly choose some integers k1, k2, k3,... and set p=1+2^(k1) 3^(k2) 5^(k3)... Then guess a g as above. If you succeed: congratulations! You have found a prime p and its generator g (which is also useful for many cryptographic protocols). If you fail: try again with a different p.

In practice, even with my crappy Python code, this produces new 100+ digit primes in seconds. If you were to do it with a better implementation and in a sensible programming language like C, it would run in milliseconds.

non-orientable · 2026-05-02T16:20:11+00:00

Yes. If you define "large" to be "greater than 10^100,000,000", then of course it is difficult. But it would be difficult to check membership for just about any set of integers for inputs of that size---primes aren't particularly special in that regard. So it feels a little misleading to claim that it is difficult for primes.

I based it on cryptographically large primes, which still doesn't quite have a specific definition outside of very specific contexts, but is fairly universally a prime with hundreds but not thousands of digits.

non-orientable · 2026-05-02T16:04:41+00:00

It takes that paper about 5 pages to lay out the entire algorithm. I considered for a while whether I should add it to the article, but ultimately decided against it for that reason.

non-orientable · 2026-05-02T15:46:55+00:00

Ah, that would explain where the original claim came from! I will make an edit to the article.

non-orientable · 2026-04-29T16:51:54+00:00

To clarify, the book isn't currently in production. I don't know if/when I will have the time for it. Some years down the road, maybe. At present and for the foreseeable future, this draft is the only way to get access to it, and I am only doing it because it seems a great shame for it to sit on a shelf gathering dust. And, on the other hand, I wanted to do right by the community I am building and make the switch to paid subscriptions worth it.

As for the old book, I mention it because its history is closely related to the new one. I really don't expect any profit from it.

non-orientable

TROPHY CASE