@Huy, I checked out two of Lee's books; Smooth Manifolds and Topological Manifolds. I'm going to quickly browse through the second one to pick up thinks I've missed.

Question: In the first chapter, under the section algebra, he says "Since a matrix is invertible iff its determinant is nonzero, $\text{GL}(n, \mathbb R)$ is an open subset of $\mathbb R^{n^2}$". Why is this true? I don't see the connection between the first statement and the conclusion. And the conclusion doesn't seem at all obvious to me.

I browsed through the first four chapters and couldn't see this detail addressed at all.

Well, after looking in Loring Tu's book, I did see why the implication is true: it uses that the determinant function is continuous. But I don't see why that is true?

Now, I could probably work through the details of this, but as it's in the introductory section and it's not addressed in detail later, I think I may be over thinking it? Is there an obvious reason why it's true?

@RobertCardona: Polynomials are continuous.

@RobertCardona: The determinant is just a polynomial function of matrix entries.

Ahh yes, I see!

Thanks!

@RobertCardona: The important part is to realize how easy it can be to show that certain sets are open where you probably never thought of them as an open subset of $R^{n^2}$.

Just by exploiting continuity of a certain map.

There are very similar arguments for topological properties of other matrix groups.

Ok! I'll keep an eye out for that.

If you have time, think about other matrix groups you know, like SL, O, SO and complex versions of them and see if you can figure out anything about them (open, closed, compact, connected, etc) - given you already know a basic (but good basic) amount of topology.

I will!

This is not so much important for smooth manifolds per se, but some of them will be very frequent examples of such manifolds.

and there are very surprising connections between those matrix groups and seemingly different things, for example there is a very very close connection between $SO_3(R)$ and $S^3$.

@ChantryCargill: we're starting on 18 Dec, check out r/dgreading once every (other) day. there will probably not be a lot going on before that.

@Huy Okay perfect. Hopefully I can be relatively active. I'm going to be pretty busy travelling around Australia.

@ChantryCargill: I'd be worried about all the deadly animals. :P

They don't scare me too much. Although, the flies are absolutely horrid this time of year.

That's good for you.

I hear stories from friends about big spiders in the house.

I don't ever wanna go there.

Topological Manifolds | Smooth Manifolds | Riemannian Manifolds

So without any obvious research, is that Topological Manifolds -> Smooth Manifolds -> Riemannian Manifolds or do each stand alone independently?

@ChantryCargill: I think that's the order one would usually go, but you can do pretty much all simultaneously.

Maybe the first two first.

And then the third.

@Huy Okay, thanks a lot. I'm trying to formulate a reading list for the next couple of years to work on once I get settled back in Canada.

cool.

if you need any recommendations, you're always welcome to ask :)

Rereading Stewart Calculus, Started Tao's Analysis, was recommended Jacobson's Basic Algebra I and Basic Algebra II for Algebra. However, I still need (a/some) good differential equations book(s) including partials. Any recommendations/corrections?

@ChantryCargill: I think the general consensus is that Rudin/Apostol are the classic analysis books, though I never read them. I read Zorich (but not recommended for a first read), and Struwe's German notes. for Algebra, I also used a German book, though I think D&F is very good, and Artin is the classic.

I tried Rudin, but it was too terse for me. Tao's is much more readable imo.

@ChantryCargill: Ok.

I basically just need diff eq and complex analysis I think.

As for PDEs, I've never taken a course on them, only on functional analysis.

well in Complex Analysis the clear classic is Ahlfors

Ooo and it's online for free.

Which university would that be?

ETH Zürich.

oh, sorry, the notes are in German. only his notes on differential geometry and functional analysis are English. sorry.

Ahh, too bad. I'll make do with your other suggestion.

@ChantryCargill: be careful with the classics though. they're usually not made for a first course, in my experience.

and usually contain a lot more than one can cover in a typical one semester course.

but there are exceptions of course.

I've studied some complex analysis in the past, but I've forgotten most of it and the style was similar to a calculus course in rigour.

how can you do complex analysis without rigour?

the theorems are a lot lot less believable than in calculus

Present theorems and be more selective about the proofs.

I mean, don't get me wrong, there was an analsysis element.

I think that kind of teaching style is really bad for undergrads.

It didn't do me any good.

In graduate school sure, most students will read the proofs up and understand them.

My school left me feeling like I was missing out to be honest. That's why I'm planning on redoing my education from the basics onward. I want to feel like I can actually do graduate level mathematics.

@ChantryCargill: Sorry to hear that.

@Huy My fault really. I could have left at any time. It's up to me to correct it.

@ChantryCargill: Is the education system the same as in the US?

@Huy It's different. Ours is a lot better imo. I just happened to go to a weak school.

Why? Don't you think getting a degree in maths should require actual knowledge of it?

(about the "my fault really" part)

@Huy The problem is that the level of understanding can be debated. I think they should close down their program, but if people keep entering and graduating from it, then they have no reason to close it.

@ChantryCargill: Do you also have such horrifying tuition fees at university level?

About 6k per year in Canadian dollars.

For just the tuition.

Ok, that's not as bad as in the US at least.

@Huy Naa, and we have some really good programs in Canada too.

What I'd like to do though is study for 3-4 years and do grad school in Germany. I've heard there's more and more english programs being developed.

I feel like in the US that is one of the very problems of their education. Ted (retired professor on MSE) told me the other day that you can basically get a degree in maths nowadays by being able to compute a few limits. I think the problem is that universities in the US are private and not public and most of them are run as a business and not as an education institution. Hence for a business, the more the better.

@ChantryCargill: I can imagine. Though ETH is cooler. :P

Does ETH have english programs?

In graduate school, almost exclusively.

And yeah. Not only that, but students have multiple attempts at classes.

We have a lot of international students in graduate school.

I knew someone that took real analysis 3 times.

3 times is maybe the very most I can kind of accept to be honest

we're allowed to repeat classes once

I hear of other unis where you can basically study for as long as you want and repeat as often as you'd like ^^

Once I can see. I mean, sometimes it take a second time to truly get a topic. But if it's your second or third time and you're passing with a 50 or 60, that's not acceptable.

yeah, I agree

I actually asked my professor to repeat differential geometry

we have differential geometry 1 and 2, and I was really bad at 1 and my prof asked me if I needed to pass the course really badly because then he'd barely let me pass

and since I didn't I told him I'd retake it

and we were both very happy about how it turned out :)

You're there to learn after all.

exactly.

and it wasn't that I didn't learn because I didn't care, but because my time management went wrong.

you should definitely check the ETH as an option if you want to try Germany or so

(even though Zürich is not in Germany :D)

Yeah, my computer science specialization made it a challenge for me to time manage as well.

I wonder if I'd even get in to be honest lol.

(or more like pretty much all of them)

@ChantryCargill: Just as a last thing: What about topology and measure theory? Which books are you doing for that?

@Huy After beginning to read the text for this chat, I realize that I need to add that to the list. I believe Tao covers measure theory in his analysis 2, but I'll have to learn topology.

Ok. Try Munkres, but not all of it, just selected chapters. All of it takes ages. :P

I'm off to bed now, see you around. Check back in any time you want. :)

@Huy Thanks a lot. Night!