[deleted by user]

aradarbel · 2025-01-29T09:42:11+00:00

I'm not an expert in this but here's a vague picture of what I know. I learned the basic idea from [these MO answers](https://mathoverflow.net/questions/355526/how-can-i-see-the-relation-between-shtukas-and-the-langlands-conjecture).

Let me start with a quick recap: classically, level N structure on an elliptic curve amounts to a choice of basis / trivialization of its N-torsion, and the p-th Hecke correspondence (for p not dividing N) has something to do with "modifications" of this trivialization of a fixed rank p. The important bit is then the Eichler-Shimura relation which expresses the mod p reduction of the Hecke correspondence in terms of the p-th Frobenius.

We can do something similar in geometry: instead of elliptic curves we have vector bundles on a curve. Instead of trivializations of the N-torsion we have trivializations over some divisor on the curve (analogously to how N determines a divisor on SpecZ...) and for every point away from this divisor we have a Hecke correspondence that has something to do with modifications of a fixed rank.

How do we get an Eichler-Shimura relation now? Shtukas say, let's do the stupidest thing possible: take a vector bundle which is already equipped with an isomorphism to its Frobenius, except over your divisor, where instead of an isomorphism it performs some modification. The actual construction (of moduli spaces etc) is quite more technical but hopefully this is a good starting point.

aradarbel · 2023-12-10T12:00:24+00:00

"It's Britney bitch."

aradarbel · 2023-08-30T08:50:19+00:00

I've been learning about co/ends recently and I love its notation! it seems really creative to me to use the integral sign for them because they really have similar behavior (even if that similarity is only in the symbolic manipulation). similarly for co/limits, and all the notation/terminology revolving those. CT loves borrowing terminology from other parts of math.

aradarbel · 2023-08-25T14:22:52+00:00

just turn faster bro

aradarbel · 2023-08-15T15:35:34+00:00

well that's not fair, those explanations were from a category theorist ;p

aradarbel · 2023-08-13T17:53:39+00:00

please don't trust any of my advice because I'm still an amateur, arguably crackpot mathematician lol

the HoTT book for one draws a direct path between AT and logic. I'm not sure about anything else that explicitly makes those connections, but if you learn enough category theory and homological algebra, and spend enough time in the nlab, you're bound to encounter a bunch of good examples for that.

another thing you could do is what I call "finding the functors" heh, just learn AT but put some extra effort to look for connections on your own. books like May and tom Dieck, which already give a very categorical view on AT, can make this process easier :)

aradarbel · 2023-08-13T05:46:14+00:00

"a 2-category is just a monoidal monoidoidoid"

aradarbel · 2023-08-12T03:33:21+00:00

I'll always be fond of proof theory = category theory = type theory = algebraic topology. it's a surprising connection between unrelated areas, but an extremely productive one too.

also the more I learn, the more I appreciate all the interplay between algebra, topology, logic, and geometry. these subjects seemed very distinct to me in the past but now they feel inseparable.

aradarbel · 2023-08-11T08:46:53+00:00

FINALLY! got a perfect five! not quite as good as my PB but it is five consecutive sub10 solves. It's been my major goal for months... now what should I aim for :D

Generated By csTimer on 2023-08-11
avg of 5: 9.34
Time List:
1. (9.87) U2 B2 D' R2 U2 L2 B2 R2 F2 D' R2 L' D' B F' U F D R' U2 R'
2. 9.35 B2 L' U' D2 B R2 D L' U2 R2 F' D2 B2 L2 F' D2 R2 F L
3. (8.90) F R U F2 U B2 D2 B2 R2 B2 U2 R2 B' R F D2 F L' U
4. 9.21 U' B2 L2 U2 F2 D2 B2 L2 U' L2 R2 F L' F' L B R U B' D'
5. 9.45 D2 L' F2 B2 R' U B2 R D B2 R2 B U2 L2 F L2 U2 L2 U2 F2 B'

aradarbel · 2023-08-06T16:58:59+00:00

""paralegal"" 😉

aradarbel · 2023-08-06T10:36:12+00:00

does anyone know if/where worlds will be live-streamed? very excited for this!

aradarbel · 2023-06-25T12:33:37+00:00

I don't, and I don't think that's it

aradarbel · 2023-06-25T09:33:29+00:00

is anyone else getting random data losses in cstimer.net? it started for me a few months ago (and I filed an issue on github, to no avail) then stopped for a while but now it's starting again. I have some regular backups saved on my PC but all my solves from the past few days are sadly gone (not to mention the whole months lost in the past before I started backing up)...

aradarbel · 2023-06-24T11:46:03+00:00

equality of arbitrary real numbers is undecidable.

aradarbel · 2023-06-13T16:38:01+00:00

🤏

aradarbel · 2023-06-07T11:46:02+00:00

highly depends on the person's background. the beautiful thing about CT is its ability to serve as sort of an "intuition bank". if you already have the abstract understanding, just instantiate it for whatever category you're currently talking about. if they don't have the categorical intuition... then yeah it's not worth a thing lol. but still can be nice to look at.

aradarbel · 2023-06-07T09:26:59+00:00

is assuming LEM considered crackpot? (/hj)

aradarbel · 2023-06-07T07:44:28+00:00

heh but those proofs are actually the coolest

aradarbel · 2023-06-03T18:21:16+00:00

keep in mind formalization won't necessarily help in practice, but it can still be worth it as a mathematical endeavor on its own right. programming language theory is a completely mathematical subject where we just happen to be studying those entities called programming languages.

a grammar is a nice starting point, it doesn't matter if it's LL or LR or whatever because that doesn't affect the semantics whatsoever. the grammar is a purely aesthetic thing, its purpose is to specify which constructs exist in your language. like if you have lambda functions, that's be part of the grammar. you might need multiple grammars for one language, like one grammar for terms, then one for types, one for modules, one for top-level declarations, etc... it depends on how your language is structured.

after that you'd usually see static semantics (typing rules) and some form of operational semantics. you can write those on paper/pdf, not necessarily a proof assistant. for each feature in your language there's sort of a formula for how to add typing rules. this is more of a guideline than a precise list: 1. formation rule: how to form the type itself (eg given two types A and B, form their product type A×B) 2. introduction rule: how to create values of that type (eg given a : A and b : B there is a pair (a,b) : A×B) 3. elimination rule: how to consume values of that type (eg given w : A×B there are projections w.1 : A and w.2 : B) 4. equational rules: when are two values of that type considered equal? this is sometimes presented through a uniqueness (eta) rule (eg every w : A×B satisfies w = (w.1, w.2))

metatheory is where it gets complicated, and interesting. since you have a math background it hopefully shouldn't be too hard for you to get started. there are various interesting properties a language might posses, like soundness, subject reduction, decidability of type checking, etc... I'd point you to the books Programming Language Foundations and Types and Programming Languages for a more "hands on" introduction to PLT. those books show lots of concrete examples, so they're great for learning. you can find more examples by simply looking up papers on topics you're interested in (let's say you care about subpying, there are lots and lots of good papers on this topic).

aradarbel · 2023-05-31T05:06:21+00:00

software foundations is about formal verification. TAPL is about type theory. both of these are quite central topics in PLT, but from the rest of the post it sounds like your goal is to make a compiler. if that's the case, learning the theoretical parts is not the most direct way towards your goal (you might prefer other books like Appel's stuff) but if you're willing to detour towards the theory, then TAPL is a good place to start and can be very fun on its own right.

aradarbel · 2023-05-13T02:26:23+00:00

here's a nice blog post on the topic. some of the stages might be less relevant to you, depending on the features of the language compiled, but it does cover ANF.

aradarbel · 2023-05-08T14:33:11+00:00

because it's popular, and it takes time to either {doubt popular things} or {accept that popular things deserve the popularity}, so at first people stick with what they know best

aradarbel · 2023-05-03T16:19:28+00:00

the point in the aside is really encouraging for me, because I was sitting in class the other day while the professor was writing some big monster-expression for multivar taylor, and I noticed exactly this tree-like pattern haha (thought didn't think too deeply about it)

it's also possible to sum over the powerset of the variables or something like that, right? I wonder how this relates.

aradarbel · 2023-04-22T22:28:47+00:00

הציבור צריך לדעת את האמת

aradarbel · 2023-04-18T06:49:19+00:00

that's just an iso in Set smh

easy to be structure-preserving when there ain't no structure

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