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bolbteppa

23:02

Mathematical gr seems to be like any math subject, yeah you can get results, but people can't even factor some basic 6'th degree polynomials, that's a research problem, not a field with low hanging fruit

0celo7

I have no idea what that means

Sir Cumference

Wait, what's the difference between a manifold and a variety

Balarka Sen

Varieties are subsets of $\Bbb C^n$ given by $f_1(z_1, \cdots, z_n) = \cdots = f_n(z_1, \cdots, z_n) = 0$

Where $f_i$ are polynomials in n variables

You can insist on $\Bbb R^n$ if you want

dmckee

Does anyone else sometimes wish they could up-vote a bounty?

0celo7

I would say that the difference is one is a variety and the other is useful

Sir Cumference

23:06

@dmckee I wish sometimes I could bounty an upvote

Would be nice to get some more upvotes for tough-written answers

bolbteppa

A: What is the difference between a variety and a manifold?

A variety is usually defined to be the zero set of some polynomials. I don't know much about this, but I think people usually call manifolds that can be realized as varieties "algebraic manifolds". Take for example the polynomial $x^2+y^2-1$, then the zero set of this is $x^2+y^2=1$, or the circl...

Balarka Sen

Not all varieties are manifolds.

But they are manifold almost everywhere

outside where they are manifolds they have singularities

Semiclassical

as an example, $(y^2+(x-1)^2-4)(y^2+(x+1)^2-4)=0$ for real $x,y$

zero set is two circles of radius 2 centered at (1,0) and (-1,0)

if you only had one of the circles or the other, then it'd be a manifold. But where they intersect at $(0,\pm \sqrt{3})$ it's not locally Euclidean and therefore not a manifold

Sir Cumference

Why does geometry/topology whatever need so many damn terms

bolbteppa

It's hard to find books you can really trust and know you're not learning something badly

I don't know one on algebraic geometry

Sir Cumference

23:18

@bolbteppa God this book is awful. Nothing it says is intuitive and I constantly have to look things up

Yet online is always 100x clearer

bolbteppa

What book

0celo7

oh my god kill me please

I've been using the wrong convention

Sir Cumference

@bolbteppa amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907

0celo7

why wouldn't you use something normal for algebra

bolbteppa

I have tried that book about 4 times and it's horrendous

Sir Cumference

23:22

@0celo7 That's what my uni is giving us, so I'd expect it covers what I need for the tests

But holy hell I'm spending 9+ hours on each chapter. Then I look it up and figure it out in 20 mins.

Why is the internet so much better than a university

0celo7

umm

there's something wrong with your university if that's the case

Sir Cumference

@0celo7 It's pretty well known, well not for physics but in general

0celo7

well known does not imply good teaching

Sir Cumference

But still, sometimes I feel like I'm paying for the piece of paper and not the actual education

0celo7

you're the example of that

Sir Cumference

23:25

@0celo7 Welp, I'm trying my hardest to understand this stuff.

0celo7

why are you even learning algebra

Sir Cumference

But this is hard. I love it, but it's taking a ton of hours

@0celo7 I dunno, it's required for a math major

bolbteppa

That's the point of uni, you don't go to get spoonfed you go to get thrown into a lions den and figure it out for yourself

Sir Cumference

Might as well go for that at this point

@bolbteppa Ik and I'm getting used to spending 13+ hours studying each day, but the time I'm spending studying math is time not spent on other courses

0celo7

fuckin hell 13 hours?

Sir Cumference

23:27

I've yet to find a way to study for two subjects at once

bolbteppa

(Only 13 :p)

Sir Cumference

@0celo7 On math alone. I often am pulling all-nighters to get the other homeworks out of the way.

Luckily math hw isn't graded which means I don't have to rush teaching myself too much, but focusing on all these things is driving me crazy

bolbteppa

There's a reason why math is considered one of the hardest subjects

(if not the?)

0celo7

what are you talking about, everyone thinks math is super easy

it's the normies who think it's hard

Sir Cumference

@bolbteppa True, but having a better intuition on pure math is making physics and other mathy-classes so much more enjoyable

0celo7

23:29

engineering is the hardest

bolbteppa

Yeah, it's worth it

0celo7

by faaaaar

Sir Cumference

It just sucks that once you get past multivariable calc/differential eq/linear algebra, Khan academy and a lot of great resources stop existing

And Wikipedia loves being very technical

bolbteppa

It's probably better to skim the Fraleigh sections, and spend like 3 hours trying to write the chapter yourself from online, then use Fraleigh directly

The last time I tried to read this book was the chapters on homology but it's so boring I put it down

Sir Cumference

@bolbteppa Oh speaking of which, I've got over 73 pages of notes I've written. It's basically a textbook at this point.

Well, typed in Latex actually

bolbteppa

23:30

1 page of Poincare's old book is better than everything in his book, even if you only get 1 basic thing from that page

Sir Cumference

@0celo7 Oh that reminds me, you mentioned I should take some engineering class. You have any recommendations?

@bolbteppa Tbh though I'm extremely worried about confusing myself when trying to figure this out on my own, or building an incorrect intuition that'll hurt me later. Then everything will come crashing down.

Math is really something that needs guidance and order when teaching. Each concept builds on the previous.

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