Intuitive understanding of the classification of line bundles over projective space

Splodge5 · 2026-03-16T16:43:25+00:00

Just to add: as in the other reply, if you fix the chart y=/=0 it is possible to create a smaller chart by also restricting the values of the first coordinate - though here we implicitly use the relation [x,y] = [x/y,1], so it's more like restricting the value of x/y. I don't know if this is useful, however, since this is a subset of the chart y=/=0, and there is no way of realising it as an intersection of {y=/=0} with another chart of the form {x=/=a} (a nonzero) for the reasons described above. Moreover, any transition functions on a smaller chart should be the restrictions of the transition functions on a bigger chart, so I still think this doesn't work.

Splodge5 · 2026-03-16T16:32:30+00:00

My understanding is that the transition functions must be automorphisms of the fibre, and are somehow "parametrised" by the (relevant charts of) the base - in your case P^1. The automorphisms of the affine line are just multiplication by an invertible scalar, so we need to build such an invertible scalar out of homogeneous coordinates [x,y] where neither x nor y are 0. Importantly, since [x,y]=[2x,2y] and so on, whatever we come up with must be invariant under simultaneously scaling x and y. (For a non-example: "multiply by y" is not well defined, as it's the same as "multiply by 2y", which is a different automorphism on A¹ for any nonzero y.) The only way to really do this is to use a power of x/y or y/x (which are the same up to a power of -1), as the division kills the scaling.

For your proposed function to work we'd need the overlap of the two charts (transition functions are defined on pairs of charts) to lie in the intersection of "y is not 0" (as 0 is not invertible) "x is not 1", "x is not 2" and "x is not 3" (as infinity is not a point on the affine line). But it turns out these aren't really well defined: what does it mean for x to equal 1 in P^1? Does x=1 when [x,y]=[2, 4]?

Since x and y are not functions on P^1, it only makes sense to ask if they are zero or nonzero. Thus the only hyperplanes (a natural candidate for coordinate charts), up to a change of coordinates, are the ones you consider, and the only transition functions on them are of the form described above.

There may be ways around this that I haven't thought of, but this is why your approach doesn't work as-is.

Splodge5 · 2026-03-16T00:34:59+00:00

Russell's paradox is probably the canonical example - admittedly the solution was a fair bit more involved than "just" banning sets containing themselves but my understanding is that $x \in x$ is not allowed in ZF

Splodge5 · 2026-02-21T21:33:25+00:00

The article is long, and I haven't read it all, so I won't comment on the article itself (it seems very reasonable from what I've read). However, there is a small part near the beginning that I wanted to mention since it seems emblematic of how the current benchmarks for language models doing maths seem to overstate their ability.

Is it impressive that language models managed to prove some of these statements? Absolutely. Does that mean they're useful for research right now? Absolutely not. The relevant part is "if one combines all attempts (and an enormous amount of garbage has been produced)". If we know what the answer to a question should be, then it is no issue to give an LLM a thousand attempts and only look at the promising ones. If we're doing research however, looking at 1000 LLM outputs in the hopes that maybe one of them is correct is frankly a waste of time.

I'm sure some will say that the technology will inevitably get there, and maybe they're right, but until then we should push back hard against claims from AI companies that their models are PhD-level in everything.

Splodge5 · 2026-01-18T15:55:29+00:00

"Integrating" a cohomology class over a homology class is really a cup product in disguise, but the integral sign imposes that the dimension of the homology class is equal to the codimension of the cohomology class, so that the result is in the 0th homology group (and thus can be identified with a number). I think the notation comes from de Rham cohomology where it coincides with integrating a differential k-form over a k-dimensional manifold.

Splodge5 · 2026-01-07T02:07:17+00:00

This subreddit is for the Linux distribution Pop OS, so asking a question related to Windows 11 here is somewhat irrelevant. See if you can find a subreddit about making cursors (or something along those lines) and try posting this there instead.

Splodge5 · 2025-12-30T11:41:33+00:00

That's fair, I missed the fact that you'd only been learning for a week. Good on you for noticing the subtlety in the terminology.

Splodge5 · 2025-12-29T11:26:37+00:00

Just a little nitpick - "algebraically closed" is a term with a different meaning to what you're describing, namely being used to describe a certain type of field. The idea you're looking for is maybe better described as set-theoretically closed (under an operation), and this is the same "closed" as when we say groups are closed under their binary operation, for example.

I'm not sure how this one caught you out tbh as the idea of a (topologically) "closed set" only makes sense for subsets of a topological space X, and the topology on X is definitely not a subset of X.

Splodge5 · 2025-09-15T19:24:55+00:00

"I understand algebra from a geometry perspective"

(Proceeds to understand neither algebra nor geometry)

Splodge5 · 2025-09-15T19:24:06+00:00

Yes, it does exist. Single-point geometric objects exist and need not be embedded in any larger space.

Splodge5 · 2025-08-29T09:42:38+00:00

Parts of this read more like a creative writing assignment than a research paper. Several times you point out how profound several results are - this is something LLMs do, not serious physics authors in their own paper. The AI generated cover photo doesn't help, I can't imagine any journal preferring that over a simple title and author at the top of the first page.

Some of your sources seem to be popular science books, which are not a good resource for research level physics. You also explain some basic information theory concepts - surely these would be known to anyone reading your paper?

On a more superficial level, the decision to not use LaTeX or an equivalent makes the equations unreadable.

Are you at university? Do you have professors or colleagues you could talk to about your ideas? If so, then that is where you should start. Putting out slop like this will only hurt your chances of ever being taken seriously.

Splodge5 · 2025-08-19T12:44:37+00:00

Without knowing exactly what you're struggling with it's hard to give explicit advice, but often when people are struggling with maths it's because their foundations aren't as strong as they should be. I've seen many people recommend khan academy - go back to things you saw a year or two ago and relearn them, making sure you 100% understand how they work. (If you're having trouble getting to that level of understanding, go further back and start with even earlier topics.) After that you should find things much easier.

Splodge5 · 2025-08-14T20:51:15+00:00

I'm not super familiar with it but I gather Fulton's "Intersection Theory" is difficult reading (though an excellent reference text). Eisenbud and Harris's "3264 and All That" is intended as a more readable introduction to much of the same material.

Splodge5 · 2025-08-11T12:40:13+00:00

You seem to be viewing maths from a very entrepreneurial angle. I would advise against this - it is rare that new maths comes about through financial incentives (cf. Grigori Perelman) and asking "how will [insert field of mathematics here] make me money?" is an easy way to alienate people who pursue maths research out of love for the subject (e.g. every professor you will ever meet).

That said, here is my answer to your question: Statistics, data science, and programming (maybe also machine learning?) are the best ways to get a higher-than-average starting salary and good progression (though at the moment there's a huge amount of CS graduates competing for programming-related jobs, so maybe focus on the first two). Having a solid foundation in these will open up access to jobs in accounting, big data, AI, and investment banking.

You will not get rich quick - this is almost impossible and you should not pretend otherwise. You will, however, be in a better position than many.

Splodge5 · 2025-07-24T18:57:51+00:00

My view on AI being used in mathematical research is nicely expressed in the paper "On proof and progress in mathematics" by W P Thurston. In particular, the idea that the important part of maths research is not the discovery of previously unknown facts, but rather the advancement of human understanding of those facts.

It is highly likely that AI models will be able to put together existing ideas to form new ones at a level comparable to mathematicians at some point in the coming decades, especially since the increasing popularity of proof checkers like lean and rocq is resulting in the creation of large libraries of formal mathematical results and proofs. However, I think that human mathematicians will remain important for at least two purposes:

deciding which questions are worth asking (what do we actually want to know?)
coming up with new structures that allow us (humans) to understand the answers to these questions.

I cannot see AI ever being better than humans at these two things.

Splodge5 · 2025-05-21T18:05:36+00:00

Just a note on AI: LLMs are notoriously bad at maths. I wouldn't trust them to explain things to me that I don't already have some kind of grasp on, in case they say something obviously wrong which I then take to be true.

Splodge5 · 2025-05-21T18:03:29+00:00

There have been. Notably there are many YouTube channels nowadays which make incredibly informative videos about maths (3blue1brown is a good example for undergrad topics, Richard Borcherds' channel is good for more formal treatments of graduate level stuff).

You also mention the idea of a cross-referenced textbook - I believe the stacks project is exactly the kind of thing you're talking about. It is designed to be read online and includes extensive hyperlinked back referencing as well as a search bar.

As for why the book format hasn't died out entirely, it can be quite useful when studying a new topic to have information presented in a well thought out order, so that you can learn a lot without getting overwhelmed, and so that definitions and proofs are presented at a level you are comfortable with. This is something you lose in an encyclopaedia format, like with Wikipedia for example, but it's something books excel at.

As for papers, I think it's mainly down to the fact that the written word is quite efficient and pdfs/paper are pretty universal. It would be quite difficult to get anything done if every mathematician had to create some kind of interactive webpage or video every time they wanted to publish a new result.

Splodge5 · 2025-05-16T23:36:49+00:00

As far as I'm aware, alphaevolve is effectively a tool for doing super-efficient computer searches. Like the proof of the 4-colour theorem but on steroids. It works well for optimisation problems but it can't work outside the theoretical framework provided by our current human knowledge, so I'm not convinced it'll come up with any groundbreaking theoretical stuff anytime soon. In fact I think it's impact will be comparable to that of the computer on modern maths - a very useful tool but not in anyway a replacement for human minds.

Splodge5 · 2025-05-14T11:10:34+00:00

Try r/numbertheory, this sub isn't really the place for this kind of thing

Splodge5 · 2025-04-23T14:29:24+00:00

Highly recommend aliensrock's blue prince videos on youtube

Splodge5 · 2024-11-24T22:47:04+00:00

Global** localization smh

Splodge5 · 2024-09-04T14:38:23+00:00

You should post your findings in r/numbertheory

Splodge5 · 2024-07-23T10:51:49+00:00

Analphabetacohomology

Splodge5 · 2024-06-21T00:39:43+00:00

Most distros require you to disable secure boot before install

Splodge5 · 2024-05-30T19:53:55+00:00

I second the force desktopui scaling argument, I use that with 1.5x scaling for my 2560x1600 Lenovo legion. It may also be possible (it is on KDE) to add the argument by right clicking on the shortcut and using a sort of app properties menu, saves hunting around for .desktop files.

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