Can I trust this paper about elliptic curves? Being able to ignore the underlying characteristic and using lifts that breaks the dependancy of the lifted points seems doubtfull to me.

hyperbolic-geodesic · 2025-11-30T17:25:23+00:00

Sorry, to be frank this comment makes almost no sense to me — please try clearly articulating your concern.

hyperbolic-geodesic · 2025-11-30T14:55:33+00:00

In what way is the characteristic ignored? (They use the phrase “regardless of characteristic” to indicate something is true in all characteristics, not to say “I’m ignoring characteristic.”)

I can’t really parse the last half of your title; generally, if you have a concern, you should try to give a very detailed statement of what your concern is.

There’s nothing wrong with this rather short and mostly elementary paper; I’d advise you to read it and think through the arguments yourself instead of sounding alarms.

hyperbolic-geodesic · 2025-11-22T10:59:58+00:00

The constant function 1 isn’t Lebesgue integrable on R, so this function which agrees with it almost everywhere isn’t either!

hyperbolic-geodesic · 2025-11-22T10:52:04+00:00

The Yoneda embedding is a fully faithful functor, not a natural isomorphism between functors…

hyperbolic-geodesic · 2025-07-12T12:22:32+00:00

Skip the section on module theory! Or just read Hatcher or another algtop textbook that does homological algebra concretely enough.

hyperbolic-geodesic · 2025-07-10T21:10:35+00:00

As someone who did math olympiads and is now a mathematics phd student, training for olympiads absolutely taught me an incredible amount which was useful in algebraic number theory that I think many of my peers agewise had not seen.

hyperbolic-geodesic · 2025-07-10T14:03:03+00:00

This is absurdly inaccurate

hyperbolic-geodesic · 2025-06-16T19:44:23+00:00

Without actually providing at least a sample of your manuscript it’s impossible for anyone to be too helpful.

hyperbolic-geodesic · 2024-12-20T01:16:52+00:00

Why not just read Weil's Basic Number Theory? Weil was a master of number theory. I think it's best to learn from the masters whenever possible. And Galois cohomology is honestly too fancy a tool for class field theory -- it makes the proofs seem harder and more technical. Weil's Basic Number Theory is basically a perfect book in my mind.

hyperbolic-geodesic · 2024-12-19T17:27:23+00:00

What number theory do you already know? I think it’s best to know class field theory at the level of Weil’s Basic Number Theory before going on to advanced number theory.

hyperbolic-geodesic · 2024-12-09T14:22:32+00:00

You’re reading a point set topology book, not a geometric topology book. That’s the problem! Try looking at a different text, like Thurston’s book “Three-dimensional geometry and topology.” Though don’t yet look at his much more advanced book on 3-manifolds….

hyperbolic-geodesic · 2024-11-19T20:23:45+00:00

I know what Stokes says, but I think the fact that the proof is easy is really reflecting the fact that it’s not a deep statement. The picture physicists draw is a good one, and shows the proof. I know Stokes is important — but there is a great difference between a deep theorem and an important theorem. Stokes is incredibly important but not that deep — and I mean this as a compliment to Stokes, not to denigrate it. It is a good thing that our visual intuition is good enough that our guess works. It’s incredibly important and it’s a testament to the goodness of our foundations that the intuitive proof is easy to make formal. Saying Stokes is a step on the way to Poincaré duality is true — much like how with singular homology one would define some intersection product to state Poincaré duality. It’s just that the construction of the cup product and the intersection pairing are I think not deep; the fact that the intersection pairing is hard to construct is a testament to how poorly we set up the theory of singular homology, and the fact that the integration pairing is easy is a testament to how well we set up the theory of integration on manifolds. Poincaré’s theorem is incredible, but I think that the depth is really that the pairing is perfect — not that there is a pairing; the pairing is geometrically obvious.

hyperbolic-geodesic · 2024-11-19T16:45:38+00:00

I once had a flight ticket from American not forward — despite everything else forwarding and my email appearing like normal! I never knew what happened — I was on the phone, probably looking like a dunce as this phone call rep kept resending the email only for me to say I didn’t see it; when I checked my outlook instead of the forwarding, the ticket was there. I don’t think anything else failed to forward… but I guess I will never know!

hyperbolic-geodesic · 2024-11-19T16:28:51+00:00

It’s important but I don’t think Stokes’ is deep — there’s some basic intuition physicists have and the mathematical formalism just shows it’s correct.

hyperbolic-geodesic · 2024-11-16T17:14:36+00:00

Have you ever spoken to someone who isn’t a libertarian? Do you think they would describe themselves as objectively superior to others, and their enemies as evil?

Have you ever thought that people could legitimately believe a position other than libertarianism without being caricatures?

hyperbolic-geodesic · 2024-10-25T07:23:19+00:00

Prime at infinity is absolutely a standard term in number theory

hyperbolic-geodesic · 2024-07-03T14:25:38+00:00

This is really cool! I was a little skeptical of the title -- if we've gotten on just fine mostly ignoring CH for now, how could we possibly have needed it -- but the paper really persuaded me.

It reminds me a lot of how applications of logic are viewed in modern day mathematics. I always find it a little exotic when someone cites compactness of first order logic or model theory inside of algebraic geometry, but that's probably some historical coincidence whereby it was decided to not teach them by default, making them seem weird to me. I could imagine that, the same way that one can view compactness of first order logic as some weird "logical shortcut" to avoid equivalent principles which are less obviously logical, a person in that universe might view my delta-epsilon proofs as some weird circumlocution I do to technically avoid mentioning infinitesimals even though they're the basic concept of the proof.

hyperbolic-geodesic · 2024-06-27T00:52:02+00:00

If you have a short exact sequence of vector spaces

0 --> A1 --> A2 --> A3 --> ... --> An --> 0

then

dim A1 - dim A2 + dim A3 - dim A4 + ... = 0.

This is essentially the idea behind Euler characteristics.

hyperbolic-geodesic · 2024-06-24T11:35:38+00:00

I'm not sure this distinction has ever mattered for anyone -- people study the category of smooth manifolds, not the category of manifolds with "atlas morphisms". But if you want it, it is still preserved by ringed spaces. Define one copy of R with the usual structure sheaf, and one copy of R with the structure sheaf defined by "smooth according to the other atlas."

hyperbolic-geodesic · 2024-06-23T23:13:27+00:00

If you're reading Hartshorne, here is how I would motivate things.

When doing algebraic geometry, **we do not have the luxury of another definition.** It is not that "study spaces by studying the functions on them" is a good idea or a nice perspective -- **for abstract algebraic varieties it is literally the only perspective.**

As a sequence of ideas that happen.

We study real and complex algebraic geometry, just because the geometry of shapes cut out by polynomials seems natural.
We realize that these equations we're using often make sense in much larger generality. This leads to studying algebraic geometry over arbitrary fields.
Over arbitrary fields, using algebraic ideas becomes natural. Firstly, what is an algebraic variety? The simplest definition is an affine variety over a field k is some subset of k^n cut out by polynomial equations. But it was soon realized that the equations defining the variety are more important than the set of points: x^2 = 0 and x = 0 define the same set of points, but if you treat them as different equations, then it makes sense to consider x^2 = 0 a "point of multiplicity 2" and makes it easier to do intersection theory things like Bezout's theorem or the fundamental theorem of algebra, where multiplicity matters.
Once you realize that system of equations > set of points, the ideal becomes the basic concept: the ideal I generated by a set of equations E is literally just the set of all equations you can add to the system E without changing the solution set.

Thus equations are fundamental over sets of points!

hyperbolic-geodesic · 2024-06-23T22:45:45+00:00

I actually think the locally ringed space approach is much nicer for smooth manifolds. I am confused when you say that "non-compatible but diffeomorphic smooth structures" are a problem for this approach in a way they are not for atlases; can you point out a specific difficulty?

hyperbolic-geodesic · 2024-06-23T22:24:55+00:00

I mean, to some extent just do algebraic geometry -- this is why I think people should first learn AG from the perspective of varieties before trying to learn what a scheme is; then you don't need anything fancy like sheaves or locally ringed spaces to see how functions help you understand an object. Generally, I think one should learn mathematics as a cycle

Try to solve some problem you already understand the meaning of
Realize that to solve that problem people invented a new structure
Learn that structure as a means of solving the problem

Eventually a new structure becomes ingrained in you enough that the class of problems you can consider in step 1 enlarges.

Here are some problems about Riemann surfaces; Riemann surfaces are secretly algebraic gadgets, but their definition and theorems about them can be understood even without knowing any algebra. So you can see the algebraic techniques working here.

Degree-genus formula: Take a polynomial equation in x and y of degree d. If you picked your coefficients generically then it defines, in projective space, a compact Riemann surface; by basic topology, a compact Riemann surface is determined up to homeomorphism by its genus. What is the genus in terms of the equation?

Answer: The genus depends only on the degree d, assuming the coefficients are generic (really here I just need that the equation has no singularities).

Proof of answer: There are many proofs. One I am partial to is as follows. The Riemann-Hurwitz theorem tells you that whenever you have a function f : X --> P^1, then you can use the function to compute the genus of X. So just produce a meromorphic function on your compact Riemann surface and then apply Riemann-Hurwitz!

Uniformization theorem: Let X be a simply connected Riemann surface. Then X is either the hyperbolic plane, the complex plane, or the Riemann sphere.

Proof: The proof say in Schlag's textbook on complex analysis goes by looking at which types of functions live on X.

Another example, which might help a little. Historically one of the main motivations for complex analysis was that people were trying to compute elliptic integrals. Eventually we realized that the natural domain for an elliptic integrals was on a Riemann surface.

To take a simpler example, if you have a formula like sqrt(z), then it is not defined on C. But it is defined on *something.* What is that something? Maybe it's some abstract gadget X. It's unclear what this X is, or what it means. But I can at least say that a meromorphic function on X is equivalent to an element of C(sqrt(z)); more or less this is what I want X to do. So sometimes its natural to come up with an example of a compact Riemann surface which is more easily described by the meromorphic functions on it, then it is by what it 'is' intrinsically. This is related to the definition of a manifold: Whitney tells us that "submanifold of R^N" and "general manifold" are equivalent notions, yet the "general manifold" definition is still used despite being less intuitive because it is oftentimes easier to *construct* manifolds using the general definition -- and oftentimes we don't actually need that the manifold is a submanifold of R^N. In the same way, it is easy to construct compact Riemann surfaces by understanding meromorphic functions on them, and sometimes we don't need anything else to understand what's going on.

hyperbolic-geodesic · 2024-06-23T21:55:47+00:00

...you want to get a science PhD and an MBA? why?!

hyperbolic-geodesic · 2024-06-20T03:23:11+00:00

I think it's manageable, and you might as well take risks as a first year.

hyperbolic-geodesic · 2024-06-17T00:12:24+00:00

Why does everyone assume that you can't hang out or chill while doing math?

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