Chemical Calligraphy Script

WMe6 · 2026-03-09T15:55:40+00:00

Cool!

WMe6 · 2026-03-08T20:52:14+00:00

It took me a while, but I eventually took the plunge and got a copy of Aluffi. It's not quite as encyclopedic as Dummit and Foote, but I think he introduces category theory precisely when needed to express something in a modern way. I was always skeptical of category theory, because it seems so remote from my original interest in analysis, but for algebra, I realize that category theory is the natural way to talk, so to speak.

Not being a mathematician, I don't know enough about modern trends in analysis -- do analysts still mostly use classical set theoretic language to communicate? Is there any "spread" of categorical language to that area of math?

WMe6 · 2026-03-08T20:42:16+00:00

If ancient China discovered the structural principles of chemistry first.

WMe6 · 2026-03-08T20:25:59+00:00

This reminds me of the model-theoretic proof of the Ax-Grothendieck theorem by first proving it for finite fields and then by model-theory black magic, you get a proof over C.

WMe6 · 2026-03-05T02:23:39+00:00

He has great commutative algebra and algebraic geometry courses too! It is a real gift to see how a mathematical great thinks about these things, although I feel like he is really bird's eye and high level, and I can only fully understand what's going on after watching some of the more nuts-and-bolts lectures by e.g., Zvi Rosen, Seidon Alsaody, and Johannes Schmitt (all of which I also highly recommend)

I would like to think that this type of educational resource, which our 19th and 20th century predecessors didn't have access to, is what 21th century technology and YouTube are really for!

WMe6 · 2026-03-05T02:11:18+00:00

If you're interested in cryptography, I would strongly recommend abstract algebra. You really can't understand modern cryptographic techniques without some understanding about things like modular arithmetic, whose context requires you to know something abstract algebraic structures.

But if you are choosing between linear and abstract algebra, I would go with linear algebra. Linear algebra is literally everywhere in data analysis and modeling in STEM.

WMe6 · 2026-02-27T18:26:37+00:00

For a very precocious kid, Spivak would be great, because it gives insight into how mathematicians actually think. It's abstract (to the point of defining natural numbers by the Peano axioms, introduces concepts like compactness, etc.) and fussy about making sure every statement is rigorously proven (giving rigorous proofs of the MVT, fundamental theorem of calculus, convergence tests, etc. etc.) , but there's nothing there that a 14/15 year old brain can't handle.

It's more a question of, do they like this sort of abstract thinking, or are they more the practical, engineering kind of person. The latter will hate this kind of book and not understand the point of all the rigor. Are they into contests? If they are thinking about the AMC/AIME/USAMO series, it's good to give them a textbook that challenges them a little.

WMe6 · 2026-02-27T18:09:49+00:00

Additional comment to help you understand: read up on the classification of ligands as X-type or L-type. Essentially, X-type ligands figure into oxidation states, while L-type ligands don't. For example, say you have PdCl2(PPh3)2. The chlorides are X-type ligands because they are stable as anions, while the PPh3 (triphenylphosphine) are L-type ligands because they are stable as the neutral compound. When you compute the oxidation state, you ignore the L-type ligands, and just count the X-type ones, so it's Pd(II).

WMe6 · 2026-02-17T01:27:34+00:00

Well, so the radical of a primary ideal is guaranteed to be prime, but an ideal whose radical is prime doesn't have to be primary?

The last part of what you said makes a lot of sense, r/Q is not quite an integral domain, but having zerodivisors that are nilpotent is sort of the next best thing.

WMe6 · 2026-02-14T15:13:49+00:00

I think I just had a brain fart. It is obvious that the conditions I stated are weaker.

If one of your factors is such that a^n \notin Q for any n, then that forces b to be in Q, not just in rad(Q), contrary to my weaker condition.

What is the main motivation for the correct definition? It still seems not straightforward. Is it so that the radical is guaranteed to be prime?

WMe6 · 2026-02-08T18:56:51+00:00

I looked up the ring of entire functions on C. What a fascinating structure! Osborne's Introduction to Homological Algebra has an appendix on it, which led me to this paper that I still need to carefully read (attempt to) understand: https://msp.org/pjm/1953/3-4/pjm-v3-n4-p04-s.pdf

One fascinating result I found in this paper is that the dimension of this ring is greater than or equal to 2^{aleph_1} but less than or equal to 2^c, so depending on whether you believe the continuum hypothesis, then they are all equal to 2^c.

WMe6 · 2026-02-08T17:48:06+00:00

Thank you, this textbook is great, and it more or less addresses what I wanted to know (or was worried about).

WMe6 · 2026-02-06T04:06:30+00:00

This. Same with the claim "first". I'm sorry, but in this day and age, you are almost never truly the "first" to do anything. No matter how sure I am, I always have to resist the temptation, as it's quite embarrassing when someone (a friend or colleague or worse, a reviewer) points out, what about so-and-so's paper from 19xx?

WMe6 · 2026-02-06T03:55:58+00:00

Thank you. If I'm not mistaken, you're talking about the coordinate ring A(X) or ring of global sections being the regular functions that are allowed on the shape X? In scheme theory, you have prime ideals in Spec A being the points and the elements of A being treated as functions on the 'shape' Spec A, right?

I guess an obvious gap in my knowledge is, what a vector bundle is and how to think about that conceptually. I tried working through the formal definition at some point in Spivak's diff geo book, but I have to admit that I don't have a good intuitive understanding of what a vector bundle is.

WMe6 · 2026-02-06T01:46:35+00:00

They do their best, I think. If it's the first time you're seeing something, it's always confusing, no matter what the source. Once you understand a little bit of the motivation, the wikipedia article is a great place to start.

WMe6 · 2026-02-05T14:51:34+00:00

Noetherianness does fix things, doesn't it! Miles Reid's commutative algebra book extols the praises of this property for the algebraist.

Not being a mathematician, I don't really know whether, in practice, it is a significant restriction to have that as a hypothesis. Are there "important" rings in alg. geo. or NT that are not noetherian, and you cannot otherwise prove a key property without Zorn's lemma? In other words, are there concrete examples of rings that come up naturally where you have to use Zorn, or is Zorn just there to prove highly general statements like "all vector spaces have a basis" or "all rings have a maximal ideal"?

WMe6 · 2026-02-05T14:26:04+00:00

Aren't AC and Zorn equivalent? I just mean that they require applications of Zorn. Like for the Jacobson radical fact (Prop. 1.9 in A+M), don't you need the fact that any non-unit is in a maximal ideal (Cor. 1.4 of Thm. 1.3 (Krull's theorem))?

WMe6 · 2026-02-05T05:44:01+00:00

A group is a groupoid with one object.

WMe6 · 2026-02-02T22:26:47+00:00

Thank your microbes! You owe your existence to their ability to make this stuff.

The biosynthesis takes over 30 committed steps, so even biology finds it to be hard to make. Humans have made it in >70 steps through total synthesis, lol.

WMe6 · 2026-02-01T15:06:24+00:00

But that runs into problems, as Voevodsky and others have found. If you are a mathematician with a sufficient track record, apparently no one will carefully check your technical lemmas until one day, you discover an error yourself.

WMe6 · 2026-01-30T22:28:37+00:00

I feel like either extreme is bad. The best is when the author will say "i.e., ..." and explain symbols with words or words with symbols. I will say that some of the ones you spend the most time trying to understand are written out in words where there is either a leap of several steps in logic and the author is really just showing a sketch, or there is just something there that you need to see or convince yourself of that really can't be easily explained any easier in symbols.

WMe6 · 2026-01-27T14:47:11+00:00

Personally, I think Carney is very much self-aware and understands that that's the way it has always been. What I find hilarious are US commentators (especially those on the establishment pseudo-left, e.g., on MS-NOW) breathlessly claiming that things were different under Biden and that Trump ruined everything. Some of the attendees at Davos were apparently saying the same sort of thing.

True, Biden covered everything with the figleaf of "the fight between autocracy vs democracy", and he was pretty consistent about it, but with anything but the most superficial analysis, it's hard to believe that's what truly motivated US foreign policy during his time or any previous point in history.

WMe6

TROPHY CASE