Maths Uni Chat

oh cool. Yeah, there's no easy way in general. If you're finding volume, surely you should be integrating over 3 dimensions, or are you just trivialising one in some way, and therefore not counting it?

umm need to count the volume! thanks for the advice, I'll discuss my work with my teacher once we get back to school :smile:

Reply 6961

MrShifty

9

Original post by Jake22

...all of the stable homotopical information is realisably cohomologically.

When you say all of the information can be realized, does that include explicit constructions of the members of those classes, i.e. can you say, via cohomological methods what the form of a given map will be.

For a slightly different example, say you have two modules and you're looking at the hom space between them. I can see how you might say things about its dimension and the space as a whole, but in principle would it be possible to have a statement such as 'all of the homomorphisms in this space are of such and such form'?

Sorry if this is a really obvious question or if I'm not explaining myself very well. My knowledge of either cohomology or algebraic geometry is a bit spotty at best!

Reply 6962

Zuzuzu

12

Original post by kerily

I avoid this by cunningly not doing mechanics :smug:

But I know what you mean! Methods is basically FP4 all over again with a couple more proofs in there, and while I actually quite like Algebra, I'm really not a fan of matrices in general.

Lucky for you with the mechanics... I was genuinely considering swapping to maths and physics, but then I realised I hated physics even more than applied maths.

And aye, I actually think I could enjoy Algebra if I wanted to. It's just the notation, though... You wade through about 5 pages of i,j,k, mxn, E, e, etc. then it's just like 'oh, is that it?'... Just seems like a load of grabarse to me.

At least we have Analysis :sexface:

Which is exactly like I hoped 'university maths' would be - problem sheets that take you a while and things you actually have to think about.

It's really bothering me that they just blithely hand out the answers to the homework whenever people ask. We get the Analysis problem sheets on Monday, and I have the tutorial for it on Wednesday, in which he pretty much told us how to do the homework, thus ruining the second half of it for me, because I hadn't got around to doing it yet :frown:

Aye, really enjoying analysis, me. That's probably a bit down to C/W, he's a quality lecturer. The problem sheets are always decent as well (well, I enjoyed the first two, anyway). My tutor makes sure not to give the answers out to the starred questions, he just shows us tricks all the time.

The induction on question 1 :sexface:

Or taking lots of time out of the lecture to insult people who asked stupid questions. :sigh:

I'm surprised your lecturer does that, tbh... After all, it's not an English conversation class.

Oh, and were you knocking round the union last night? Thought I saw you when I went for a smoke but couldnt be sure.

(edited 12 years ago)

Reply 6963

around

13

Original post by Zuzuzu

Aye, really enjoying analysis, me. That's probably a bit down to C/W, he's a quality lecturer.

Do you have a C/W complex?

Reply 6964

anonanon123123

14

Original post by around

Do you have a C/W complex?

._.

Reply 6965

Meteorshower

14

Ok, maybe when there's 200 people in a lecture but when it's more like 20 I don't think questions are all that inappropriate...

Reply 6966

shamika

16

Original post by Jake22

You could do a quite gentle, basic course and mostly just treat it as a work out in linear algebra.

For instance, you could talk about the classification of finite dimensional irreps. of SL(2,C) or so.

It is true that a lot of basic results are much easier with heavier machinery. For example, on my Lie Algebra course, we did a fairly longish proof of Weyl's theorem which would have been easier with the machinery of cohomology.

I was thinking about this...

1) This was a Cambridge course, and I'm fairly sure Cambridge doesn't do things by half-measures :smile:

2) More seriously (and I ask this not having ever studied Lie Algebras), what are Lie Algebras good for? Are they interesting enough to be studied in their own right, or are they used only to study Lie Groups? If its for Lie Groups then that application is sophisticated to teach in the second year (in exam term of all times!)

Even if it was a gentle introduction, surely there would have to be some application of the theory, otherwise you're just learning a bunch of axioms without understanding their significance...

Reply 6967

Oh I Really Don't Care

17

Original post by shamika

I was thinking about this...

1) This was a Cambridge course, and I'm fairly sure Cambridge doesn't do things by half-measures :smile:

2) More seriously (and I ask this not having ever studied Lie Algebras), what are Lie Algebras good for? Are they interesting enough to be studied in their own right, or are they used only to study Lie Groups? If its for Lie Groups then that application is sophisticated to teach in the second year (in exam term of all times!)

Even if it was a gentle introduction, surely there would have to be some application of the theory, otherwise you're just learning a bunch of axioms without understanding their significance...

Most probably a pre-cursor to the following terms course.

Standard route to introduce things in a known realm then next time formalise and abstract afa undergrad. lectures are concerned.

They have a variety of uses but I think they are usually paired with a differentiable set when studied.

Reply 6968

assmaster

1

I love reading through the maths thread. Everyone seems to be hating their course just a little bit, which reassures me.

Plus this year, I seem to have turned into an actual douche - in lectures, I keep correcting people I'm sat near to when I hear them say "that shouldn't be that" and even as I'm saying it, I think "you're a douche you're a douche" but I just can't help myself saying, "actually, it follows from this bit". They look appreciative but I'm pretttty sure they're like "you're a douche". Does anyone else get this feeling?

Although I actually couldn't help going into douche overload when I heard this girl say "the integral from 0 to 10 of x is 50? That is definitely wrong" and her friend goes "yeah, that's a typo for sure". ????

Reply 6969

around

13

I quite like all my courses this year (although Topics in Analysis is a bit of a doss [we just proved the IVT]), but I seem to always be the douche who asks questions during lectures this year...

Reply 6970

Mr Dactyl

Original post by assmaster

I love reading through the maths thread. Everyone seems to be hating their course just a little bit, which reassures me.

Plus this year, I seem to have turned into an actual douche - in lectures, I keep correcting people I'm sat near to when I hear them say "that shouldn't be that" and even as I'm saying it, I think "you're a douche you're a douche" but I just can't help myself saying, "actually, it follows from this bit". They look appreciative but I'm pretttty sure they're like "you're a douche". Does anyone else get this feeling?

Although I actually couldn't help going into douche overload when I heard this girl say "the integral from 0 to 10 of x is 50? That is definitely wrong" and her friend goes "yeah, that's a typo for sure". ????

I also hate those little chats you hear, not sure why though. I've never bothered to intervene though, can't be bothered.

Reply 6971

jj193

14

I like this thread too.

Reply 6972

boromir9111

17

Monday tomorrow :sigh:

Reply 6973

Oh I Really Don't Care

17

Original post by Mr Dactyl

I also hate those little chats you hear, not sure why though. I've never bothered to intervene though, can't be bothered.

I get the urge to slam someones head on the desk when they correct somebody else

brb talking loudly about the point the lecturer just made and also using mathematical terms constantly

brb confusing what the lecturer just said and what they were saying

brb FFFFFFFFFFFFUUUUUUUUUUU

Truth be told I don't take notes in the lecture - with the internet I find it easier to just read them and miss the lecture.

Does anyone find lectures useful anymore? I understand why they were once used though they're definitely obsolete for maths at least imo - especially when combined with a tutor system.

Reply 6974

boromir9111

17

Original post by Oh I Really Don't Care

I get the urge to slam someones head on the desk when they correct somebody else

brb talking loudly about the point the lecturer just made and also using mathematical terms constantly

brb confusing what the lecturer just said and what they were saying

brb FFFFFFFFFFFFUUUUUUUUUUU

Truth be told I don't take notes in the lecture - with the internet I find it easier to just read them and miss the lecture.

Does anyone find lectures useful anymore? I understand why they were once used though they're definitely obsolete for maths at least imo - especially when combined with a tutor system.

Glad to see I'm not the only one :five:

Reply 6975

Hedgeman49

17

Original post by Oh I Really Don't Care

Does anyone find lectures useful anymore? I understand why they were once used though they're definitely obsolete for maths at least imo - especially when combined with a tutor system.

None of my lecturers this semester are providing online notes... so yes, lectures are pretty useful for me!

Reply 6976

Jake22

18

Original post by MrShifty

When you say all of the information can be realized, does that include explicit constructions of the members of those classes, i.e. can you say, via cohomological methods what the form of a given map will be.

Sometimes. In a simpler example, a theorem of Hopf tells you that each element of the nth dimensional singular cohomology of a finite dimensional CW complex is in the image of a natural map induced by a map from the n-skeleton into the n-sphere. Now, my knowledge of this theorem (and indeed all of algebraic topology) is now quite rusty but whereas this doesn't mean that you can always explicitly construct the map (outside of favourable circumstances), this is often already 'partially constructive' enough to be of much use. You haven't constructed the geometric map maybe but you know that in theory, it can be realised which is sometimes more useful for theoretical purposes anyway.

e.g. You have some natural action on the cohomology of spaces. You can calculate what it looks like explicitly on the nth cohomology of the n-sphere and you see that some element

\phi

acts as the identity. You then want to see what happens on an arbitrary space X so you use CW approximation to replace X with a CW complex K. You then consider the n-skeleton K^n. For a class

\sigma \in H^n(K^n)

, Hopf's theorem tells you that you can pull back to

\sigma' \in H^n(S^n)

via an algebraic map induced by a map

K^n \rightarrow S^n

so by the naturality of your action, the action of \phi on K^n is the identity. Then you consider the embedding

K^n \to K

which induces

H^n(K) \to H^n(K^n)

which again by naturality tells you that

\phi

acts as the identity on

H^n(K)

and thus by CW approximation on

H^n(X)

.

Original post by MrShifty

For a slightly different example, say you have two modules and you're looking at the hom space between them. I can see how you might say things about its dimension and the space as a whole, but in principle would it be possible to have a statement such as 'all of the homomorphisms in this space are of such and such form'?

Sure. In the simplest case, think about R-linear homs out of a free R-module of rank one. Then, you know by general nonsense what homs out of a finite rank free R-module look like. Then, finally, you notice that if you take a finitely generated projective R-module, you can use the projective basis (or general nonsense stuff about compact objects in abelian categories) and you then know exactly what homs out of fgp modules look like via the natural transformations:

Hom_R(P,-) \cong P^\ast \otimes_R -

Hom_{R^{\mathsf{op}}}(Q,-) \cong - \otimes_{R^{\mathsf{op}}} Q^\ast

etc. where

_R P

and

Q_R

are left (resp. right) finitely generated projectives.

Obviously, those are simpler examples but there are other situations where you can explicitly realise all of the maps in a hom space

(edited 12 years ago)

Reply 6977

Jake22

18

Original post by shamika

2) More seriously (and I ask this not having ever studied Lie Algebras), what are Lie Algebras good for? Are they interesting enough to be studied in their own right, or are they used only to study Lie Groups? If its for Lie Groups then that application is sophisticated to teach in the second year (in exam term of all times!)

They are interesting enough to also be studied in there own right. And in that setting, classification is a good enough motivation for machinery (and also, as I said, one could do a shorter, more basic course as an extended work out in linear algebra without having to develop too much extra machinery)

Think about group theory courses. These are often run without anything other than a small off the cuff passing reference to anything external yet in many first courses, one looks at classification of finite groups of certain orders.

One can do a basic Lie algebras course in an analagous fashion i.e. do some little bits of classification for an end unto itself.

Reply 6978

DFranklin

18

Original post by shamika

I was thinking about this...

1) This was a Cambridge course, and I'm fairly sure Cambridge doesn't do things by half-measures :smile:

Oh I think "half-measures" is entirely accurate in this particular case. One might even suspect they had 16 lectures, and 10 lectures of stuff they actually wanted to cover, and then thought "what can we use to fill the other 6 lectures?".

I'm not sure that's quite what really happened, but there were so few lectures for the last part (and as I said, this was also exam term, so the last few lectures were really when people were concentrating on exams) that I doubt there was much "lecturing with intent" going on...

Reply 6979

MrShifty

9

Original post by Jake22

Sometimes...

Thanks for that. I may dust off my long forgotten Homology, try and get proficient with all the stuff I've long ignored, and see if there's anything there which proves useful.

That said, the problem 'here' is that surprisingly little is known about the projectives.

That'ss exacerbated a bit by the fact that really I'm studying a family of algebras which is dependent on a couple of parameters, and finding concrete statements can very quickly be quite nightmarish - even classifying the simple modules involves dragging quantum algebras up, and even then the best we have is a recursive definition rather than a nice, closed expression.

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