/sci/ - Science & Math

[Anonymous No.10381914]

Algebra is one of the major fields of pure mathematics, so is also considered one of the major courses in the undergraduate math degree. It should NOT be in the top three along with linear algebra, advanced calculus, and ODE. These are the most important topics in the undergraduate math major to do well with old applications in other fields or for more advanced work in mathematics.

What is just CRUCIAL from algebra is the real number system, and it is FAR better to see this system in action in linear algebra, advanced calculus (completeness, compactness, continuity), and ODE before going for Dedekind cuts, Cauchy sequences, or whatever to 'construct' the reals from the rationals. Besides, for the reals themselves, the really astounding properties are best treated as side topics in measure theory, e.g., from Oxtoby's Measure and Category and Gelbaum and Olmsted, Counterexamples in Analysis. The really good stuff, even the philosophical stuff of the continuum hypothesis, even the simple stuff such as the Cantor diagonal argument that the reals are uncountable, just is NOT a main part of 'abstract algebra'.

[Anonymous No.10381916]

For groups, rings, fields in general, Galois theory, etc., the world is still waiting for much in useful, put food on the table, applications beyond just the simplest parts of these.

I know, I know: Group theory gets used in quantum mechanics and elsewhere in physics, but the group theory itself that gets used is just the trivial parts. E.g., never hear of Sylow's theorem, the Jordan-Holder theorem, the Feit and Thompson work on simple groups, etc. For group representations, I did mention that as an application of linear algebra but, again, need to know very little about group theory to do group representations. Heck, I wrote a paper on multidimensional, distribution-free hypothesis tests based on groups of measure-preserving transformations, and, STILL, the group theory needed was trivial. Similarly for the use of group theory in ergodic theory, ordinary differential equations, and integer linear programming. Groups are nice, but really need to know only about 10 pages of the basics and can pick it up in an hour whenever need it. Or, you want group theory to attack Rubik's cube?

Yes, Hamming used some finite field theory in error correcting codes: Now that work and a dime won't cover a 10 cent cup of coffee. Instead, coding theory has moved on. Yes, yes, I know, from A. Wiles we finally have a proof of Fermat's last theorem; other than Wiles, who made any money with that?

[Anonymous No.10381919]

Algebraic geometry is building expensive houses on-spec that stand empty too long. In the real world, we didn't even do that in the real estate bubble. There's just no significant promise of return on investment there, or elsewhere in abstract algebra. E.g., the US NSA pushed hard on finite field theory for years before RSA showed that they had been wasting their time.

US mathematics has had a long, disastrous, self-destructive love affair with abstract algebra, algebraic geometry, algebraic topology, algebraic number theory and, thus, has been a major contributor to shrinking Federal research grants to mathematics, shrinking math departments, mathematicians who would swap their Ph.D. for an electrician's license, and the technology world putting mathematics on the back burner if not in the trash bin. Can cover nearly all of abstract algebra in one word: Useless, and that's a VERY serious problem for that field, mathematics, and our economy.

[Anonymous No.10382417]

>the absolute state of analysis

[Anonymous No.10382501]

>>10381914
>>10381916
>>10381919
You have no idea about what you're talking about, do you now mate?
>I know, I know: Group theory gets used in quantum mechanics and elsewhere in physics, but the group theory itself that gets used is just the trivial parts.

In retrospect, sure. But that's not how physics and mathematics work. You can't foretell what specific results/theorems will have applications. You just work on the field. Heck, groups when they were conceived weren't exactly like the polished groups we know and study today.

I do agree that Linear Algebra, Advanced Calculus, and ODE are absolutely necessary for a math major. I am sure they are required in all colleges. However, Abstract Algebra is also absolutely necessary for a standard degree in mathematics. Algebra, believe it or not, is one of the foundations of modern mathematics. And yes, both pure and applied. Giving math degree to kids who don't know Algebra is preposterous.

[Anonymous No.10383252]

>>10381914
> Measure and Category

> ergodic theory
> no algebra
> pick ond

[Anonymous No.10383319]

OP is clueless, likely a major who learnt some basic math and now thinks he knows it all.
The algebra required for understanding fundamental theoretical physics is not trivial and goes way beyond Lie groups. C*-algebra formulation of QM, representations of CCR, connections and cohomologies, etc. It all has algebraic roots. I definitely couldn't go through my course on geometric quantization back in uni without proper understanding of some advanced algebra.
There are plenty of books that explain how it all fits in (e. g., Choquet-Bruhat), but saying what OP is saying is basically to admit one's ignorance.

[Anonymous No.10383352]

>>10381919
>the US NSA pushed hard on finite field theory for years before RSA showed that they had been wasting their time.
Are you implying that all cryptography doesn't work? Or that cryptography is somehow a minor thing we could do without?