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Sheafification – The optimal path to mathematical mastery: The fast track (2022)

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Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#41

Post by griffzhowl »

If you have a good grasp of multivariable calculus and want to learn differential geometry, I can't recommend highly enough a book that isn't on this list, Loring Tu's Introduction to Manifolds.

I'd tried a few well-regarded diff geo books (Isham's Modern Differential Geometry, Morita's Geometry of Forms, Schutz's Geometrical Methods) but always got to a stage where there was some lack of clarity, something seemed assumed that wasn't explicitly mentioned, or just inscrutable notation that didn't seem to have been explained previously. In contrast, Tu's book is smooth and pellucid in its clarity. Very enjoyable

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#42

Post by asplake »

seanhunter wrote: post #29

There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a fou…

> "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated

Now you make me feel old! I had the second edition just as it came out in 1984. A long time ago but I remember it as one of my favourites.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#43

Post by graycat »

asplake wrote: post #42
seanhunter wrote: post #29

There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a fou…

> "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated Now you make me feel old! I had the second edition just as it came out in 1984. A long time ago but I remember it as one of my favourites.

I heard from somewhere that Herstein was an Emil Artin student at Princeton.

Herstein went to the extra trouble to make his linear algebra also work for finite fields.

In the back he has group representation theory is a small nutshell.

Also in the back he does linear programming, but his treatment is obscure and for no good reason. Since then nearly every treatment, beginner or advanced is not obscure at all.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#44

Post by griffzhowl »

graycat wrote: post #43
asplake wrote: post #42

Earlier quoted context omitted.

> "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated Now you make me feel old! I had the second edition just as it came out in 1984. A long time ago but I remember it as one of my favourites.

I heard from somewhere that Herstein was an Emil Artin student at Princeton. Herstein went to the extra trouble to make his linear algebra also work for finite fields. In the back he has group representation theory is a small nutshell. Also in the back he does linear programming, but his treatment is obscure and for no good reason. Since then nearly every treatment, beginner or advanced is not obscure at all.

Math genealogy project says Herstein was a student of Max Zorn (the guy the lemma is named after), but Zorn was a student of Artin! Zorn doctorized under Artin in Hamburg in 1930, before Artin moved to Princeton.

https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6526

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#45

Post by bbor »

figassis wrote: post #10
bbor wrote: post #7

Earlier quoted context omitted.

Oof. Not usually one to comment on technical issues, but that's a rough one to get for a (presumably) completely-static site! Dr. Sheaf, please consider serving your site via a dedicated service like S3. This is a solved problem <3

He probably doesn’t know how. This message is em from Wordpress. There are many WP providers where you just signup and start writing. They don’t tell you that if you get to HN front page, your db will die. I don’t know any static blog providers. He’d have to roll out his own or use Medium/Substack.

Fascinating, thanks for sharing your expertise. I've never published a blog, obviously. This just seems like such an absurdly simple use case... If blogs aren't static, what is???

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#46

Post by easygenes »

stavros wrote: post #26
easygenes wrote: post #14

Earlier quoted context omitted.

> I don’t know any static blog providers. Bear Blog comes to mind (no connection): https://bearblog.dev

That doesn't look static, it's a Django app, no?

Ah yeah, I guess I just assumed it was static because every time I've been to a Bear blog it has been so light and snappy.

Just did a little looking. Micro.blog [0] is a true hosted static site offering; just hosted Hugo. There's also Publii [1], an open source cross-platform desktop static CMS that has one-click push to a variety of cloud CDNs (e.g. Netlify or Github Pages)

[0] https://micro.blog [1] https://getpublii.com

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#47

Post by gsf_emergency_2 »

seanhunter wrote: post #29

There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a fou…

Have an apoplectism

https://agorism.dev/book/math/ag/royal-road-algebraic-geomet...

Also iirc, various books titled Royal Road to Geometry, which might make more sense given that IMO geom was the first to get crushed by chatbots

Ymmv, but denying existence of royal road to math seems like a narcissism limited to the most elite layity

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#48

Post by atoav »

seanhunter wrote: post #29

There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a fou…

Of course there is no single best road to mathematics, mainly because the way humans are able to think isn't uniform.

There are people with more or less Aphantasia, so people who can't or have a hard time forming mental images. Then there are others who can rotate 3 bezier curves in their head and plot the intersections.

For students of the latter category relying on mental images is a great way to teach them, for the former it is catastrophical.

Anybody who has thought anything mathematical should be aware of the fact that different people prefer different ways of learning.

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#49

Post by stared »

mananaysiempre wrote: post #27
stared wrote: post #16

I am not sure if I agree with the list. I mean, one red flag is the frequent mention of Landau & Lifshitz. It is considered a "standard textbook", but I feel it stuck there by inertia. There are quite a few choices of both less boring and more insightful. (Back when I was reading such stuff, 20 years ago, the Feynman Lectures provided orders magnitude more insight. And fun.)

The Feynman Lectures are in a very different category from Landau & Lifshitz. You shouldn’t use L&L as your first university-level physics textbook (too difficult and divorced from the real world), but you probably shouldn’t use the FLP as your second, either, provided your first was good (too easy to be worth the time spent, though it’s still useful as pleasure reading that also fills in things you might have missed…

Sure, if one wants to move deeper then Feynman Lectures are a nice entry point - at least for some people, everyone has different background and taste.

For L&L - well, not sure which ones I read (or: try to read), but likely electromagnetism and classical mechanics.

For quantum mechanics, I used to suggest these: https://p.migdal.pl/blog/2016/08/quantum-mechanics-for-high-...

Re: Sheafification – The optimal path to mathematical mastery: The fast track (2022)

#50

Post by muhdeeb »

nathan_compton wrote: post #35
bwfan123 wrote: post #33

The author misunderstands knowledge and mastery. Mastery comes from problem solving and practice - not from reading books. So, I would advise students to limit what you read, but spend a lot of time in problem solving. Get the basics in place. Start with euclid's elements and master that first.

Weird! I think Euclid is fine as a historical document and interesting in a broad sense, but its kind of silly to start with a document from 300 BCE when you could start with, for example, "How to Solve It" (Polya). And even that text could use a rewrite to make it much more readable. I have completed substantial education in both mathematics and physics and I would say one of the weaknesses of the standard system of…

Well put!

You make a very good point. The historical trajectory of a field's development is an extremely haphazard presentation of the ideas. It belongs in a separate curriculum, and I'm grateful that the history of science exists as field for that purpose. It's nice to study the history after one has understood the material; that way we can see what ideas prospered and faltered, and what might be ready for reinterrogation.

Why should we study antiquated and easily falsifiable models before we get to our modern and less-easily falsifiable models?

Drawing on common intro chemistry: the plum-pudding model of the atom is cute in historical context, but a real distraction from what our best understanding of what atoms are, for which we have much better evidence than the helium-nucleus scattering experiment that first suggested a dense, charged nucleus in gold atoms. We really only need the old plum pudding as a counter example, yet fail to explain the experiment in enough detail to justify including it. What probably began as a fastidious attempt to provide full context to a landmark experiment has at this point completely degenerated into historical trivia about the structure of British desserts, and yet remains prominent in educational material.

Math is a bit better off in this respect.

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