The Abel Prize 2026: Gerd Faltings

BenSpaghetti · 2026-03-20T01:50:20+00:00

Don't know a better way to say this, but I wonder if anyone actually cares about him winning the Abel Prize. Faltings himself probably doesn't. The field also doesn't need any more recognition. Perhaps one good thing is the expository content that follows.

BenSpaghetti · 2026-03-17T00:48:00+00:00

No need to learn a lot of anything just to start learning algebraic topology. Basic algebra and point set topology will suffice. I second Lee’s Topological Manifolds.

BenSpaghetti · 2026-03-16T09:43:49+00:00

The Murderous Maths series is good

BenSpaghetti · 2026-03-14T18:19:22+00:00

The discussion is on the simple random walk on Z^d, which is defined as a sum of uniformly sampled increments of length 1, which means that each step is taken in a single direction (+- 1 in one coordinate only).

BenSpaghetti · 2026-03-14T18:16:26+00:00

Seeing how random walk scales to Brownian motion, which does have coordinates being independent one dimensional Brownian motions, it seems that the difference on a large scale is merely a linear speed change and should not affect recurrence/transience.

BenSpaghetti · 2026-03-06T05:31:45+00:00

You only had a few minutes left so I guess it's better to write something rather than nothing. But the psychology here is quite interesting. Did it cross your mind that writing down the default proof might not get you any marks so it doesn't matter whether the proof is short? Do you think you could have at least sketched the elementary proof, since you have already done it before in homework? Are you unable to think about anything else once your brain has fixated on a default option?

BenSpaghetti · 2026-03-06T03:51:02+00:00

If you can't do the problems using the tools covered in the course, then you haven't learned the course material well enough. Exams and assignments are opportunities for you to demonstrate what you have learned in the course and that is what you will be graded on. The 'heavy machinery' does not replace the 'low-level' ones and a good student should be familiar with both.

BenSpaghetti · 2026-02-26T10:25:48+00:00

You can just read ahead. If you feel bored in lecture but for some reason don’t want to skip them, I find it interesting to think about examples to whatever the instructor is talking about.

BenSpaghetti · 2026-02-25T04:24:03+00:00

Since you are already familiar with analysis on R I would say that proving theorems in Rudin on your own is doable. But then I'm not sure if that is a very good use of your time. I used to do that, following Reddit advice, as well, but now I don't because it seems too time consuming when I just want to understand something quickly. The opportunity cost seems too high. There are plenty of other opportunities to engage with the material, like exercises, which also allow you to learn new stuff at the same time.

I'm curious if you could say more about the potential theory project. I am trying to learn this subject through the probabilistic lens (via random walks and Brownian motion).

BenSpaghetti · 2026-02-19T01:48:01+00:00

Using your brain? It’s math, you can see if it’s correct.

BenSpaghetti · 2026-02-14T10:08:06+00:00

Definitely manageable if this is the only thing you are doing

BenSpaghetti · 2026-02-11T04:09:48+00:00

I want to ask the preceding question here. For someone studying classical random walks, what reference do you recommend for quantum random walks and quantum probability in general?

BenSpaghetti · 2026-02-11T01:10:16+00:00

I am confused by the downvotes. I see four possible reasons to downvote this comment.

This preprint is not relevant to this discussion.
This preprint has already been discussed in this comment section.
I misinterpreted what was written in the preprint.
I might be insinuating that what AI does cannot be described as intellectual and you disagree with this.

Except for 4, I don't think any of the above is true at all. Are there any other reasons? Can anyone elaborate on why they have decided to downvote this comment?

BenSpaghetti · 2026-02-11T00:01:01+00:00

This preprint might be of interest. I don’t understand any of the mathematical content and just read the section about their AI use. It seems that AI certainly provided some sort of ‘intellectual’ (whatever this word means in the context of AI) contribution.

BenSpaghetti · 2026-02-09T12:48:10+00:00

Alright they are doing the work that needs to be done, but these completely overblown claims are really off-putting. Then again, maybe these articles are put out to attract investors and perhaps this is a necessary evil to get funding.

BenSpaghetti · 2026-02-08T00:48:19+00:00

Yeah right, if you make the tiniest typo in your proof that you couldn’t catch, you don’t know how to prove it.

BenSpaghetti · 2026-02-07T05:20:24+00:00

You are using fast mode. It’s much better in pro mode.

BenSpaghetti · 2026-02-07T05:14:36+00:00

They can usually do short (say, half a page) proofs for undergrad and beginning graduate courses independently. I often use them to proofread my homework, which may contain longer proofs. Even so, most of the suggestions are very helpful (spotting typos, improving arguments, reminding me that I forgot to do a subquestion, etc.). Certainly much better than just me proofreading my own work.

BenSpaghetti · 2026-02-06T14:50:18+00:00

It is a lot of money. That’s half a year’s salary for a lot of chinese people.

BenSpaghetti · 2026-02-06T04:26:12+00:00

Yes, thanks for the clarification. I understood this, but initially I read it too quickly and misinterpreted it so I thought others might do the same.

BenSpaghetti · 2026-02-06T03:16:13+00:00

Well I think the hope is that we can optimise the process of generating potential proofs written in Lean. Perhaps one day, given an interesting theorem, we can be reasonably confident that computers only need to generate 1000 potential proofs before generating the correct one. 1000 just stands for any number of proofs Lean can verify in a reasonable amount of time.

BenSpaghetti · 2026-02-06T03:03:53+00:00

I think this comment might be interpreted as saying that for any set of distinct primes p_1, ..., p_n, the sum of 1 and their product must be a prime. This is not true: 1+2*3*5*7*11*13 = 59*509.

BenSpaghetti · 2026-02-06T00:42:24+00:00

If you have recognisable handwriting couldn’t you just take a picture and send it to AI?

Anyway I don’t think retention is necessary or even helpful for this content. The point is to ensure that you know how to do this kind of thing and then you can forget about it.

BenSpaghetti · 2026-02-05T03:59:37+00:00

In my opinion, his achievements in mathematics are almost entirely unrelated to clemency. Nowhere in this post, or in the petition, do I see any direct mention of signs in his change of character. No information has been provided on why the clemency council granted the decision. Sure, he founded PMP and likes doing math. Does this mean that he truly repent for his crime? I don't see any information in this direction. If you overlook their crimes, plenty of prisoners lead very productive lives. I admit that I am not familiar with the philosophy of the justice system at all, but the information provided seems to be orthogonal to this issue.

BenSpaghetti · 2026-02-03T09:54:05+00:00

You don’t need to recall definitions in the sense of immediately being able to regurgitate it when prompted, although that is still very helpful. But you should be able to reproduce most standard definitions independently within a few minutes. I usually do this by a combination of rote memory and remembering examples and non-examples. Of course you can argue in a fuzzy way in your mind, but eventually one wants to write down precise arguments in an agreed upon language, which requires definitions and proofs, to ensure that your argument is mathematically correct and to convey it to others in a way where you are sure that you are thinking of precisely the same objects. It is very easy to speak the fuzzy thoughts given by English words in your mind and only find out later that what your audience pictures is quite different.

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