What happens to the rejected edit suggestions? Will they be presented to the user who made the suggestion (so that they can learn why the reason for rejection)?
@robjohn Doing that has at least two problems: 1. The post is unnecessarily touched by too many hands; 2. The suggester never learns or realises that there is scope for improvement.
[I guess an alternative is to leave a comment to that user, explaining what can be improved...]
@robjohn I guess that helps a lot actually. Once @tb shot down my TeXing skills in a comment =). Just kidding, but that made me I realise quite a few issues that weren't quite obvious.
My Algebraic Topology lecturer produced a counterexample to the obvious generalisation of this question via, of course, the homology a certain algebraic variety...
Let $R$ be a (commutative) ring, $\mathfrak{a}$ an ideal. Then there is a canonical map $\textrm{SL}_2(R) \to \textrm{SL}_2(R/\mathfrak{a})$. The question is, are there $R$ and $\mathfrak{a}$ so that this map is not surjective?
Speaking of SL's, is it true that every volume preserving continuous map $\mathbf R^n \to \mathbf R^n$ is linear? [$\operatorname{SL}(n, \mathbf R)$ are the volume-preserving linear maps.]
Not at all rude. Shoot your question :). // You may not get an answer right away, but if someone reads it and has something to say then they will ping you.
@Srivatsan No, this is very far from true. The group of volume-preserving maps of $\mathbb R^n$ is infinite-dimensional if $n \geq 2$. A standard method to get examples is using Hamiltonian flows and Liouville's theorem on flows
@robjohn Read that way, yes. Because, if $g$ is a generator and $h$ is any other generator, then surely $h$ can be written as a power of $g$ (since $g$ is a generator) =)
@robjohn I don't think so. :)
Unless you want me to make up some questions on the fly.
@Srivatsan I wasn't worried about that, after all acceptance points are a minor portion of the reputation points. I just wondered whether there was something I'd missed.
@CamMcLeman How much algebraic topology does one have to know to do publishable original work (even the most minor work)? I assume with the state of the art today that just Hatcher and Bott/Tu will not suffice.
@Potato: That's a tough question, but I think it's safe to say that if you were solid on everything in Hatcher and Bott/Tu, you would be able to begin working on publishable original work, even if you weren't yet at the very frontier of modern research in algebraic topology.
Oh wow, really? I guess it does reduce to having a good advisor who can choose good problems.
My conception of research is that you read all the advanced monographs and have everything down pat before beginning research, and I don't think it works this way in reality.
@robjohn Kid style pizza. You take a pita bread, cut it open and put some tomato paste, cheese and olives then into the toaster oven for a few minutes. Of course I topped it up a notch with pesto, three kinds of cheese and further additions.
@robjohn I guess the middle school idea is good in its own way because it makes the high school program, grades 9 - 12 into a 4 year program which is what the bachelor program is in university...
@robjohn I agree it does depend on the "maturity" of the student, but this way all the students get to try out a 4 year program before deciding on university, in my opinion.
In high school, they also choose their own classes. But they are watched and counseled more closely than in college.
@Skullpatrol I don't think the length of the program is much of an issue. For kids of high school age, I think 3 years may be the equivalent of 4 years at college age.
@Skullpatrol time is different as people age. For most people, it passes faster as they get older.
and I don't think that the length of the program is what needs to be tested. It is more the choosing of your own classes, and management of your own studies, that needs practice.
@Skullpatrol I don't think that most high school kids are capable of even conceiving of a four year plan. That is too long a time period for someone that old (14 or 15 at the inception).
and I think that the idea of high school is to master more basic skills than a four year plan.
Even some college students are not really ready to make a four year plan.
@robjohn As I said, I agree it does depend on the "maturity" of the student.
@robjohn One of the basic skills that needs to be mastered is PreAlgebra and of coarse Algebra, but I think this gets lost in the middle school to high school transition. And now having a 4 year high school program with students thrown into Calculus classes leaves me wondering if students are able to "mature" mathematically before moving on to new areas of study in math?
@Skullpatrol You seem to now be assuming that all students will be taking calculus. You have jumped from general maturation into mathematical education. These things do not generally correlate.
Is it true that all a non-algebraist needs to know about groups is contained in Lang's Algebra? The group theory section seems a bit light on exercises...
Also, this is just a general knowledge question, but I've had a very good mathematician tell me that a solid knowledge of linear algebra is absolutely necessary to be a good mathematician. But when I look at the qual courses I am supposed to take in grad school, none of them appear to use linear algebra that much, if at all.
Ok, I guess I will just have to work through it. My original meaning was that for 80 pages of text it seems like too few exercises, but if they are as meaningful as you say then it should be alright.
Do you know how sometimes you know that you're supposed to prove something, have no idea nor the inclination to find out, but you figure out that if you just ignore the fact that you need to prove it - it will be fine?
@robjohn General maturation and mathematical education do not generally correlate? Is it not generally true that the behaviorly more mature students are the ones that are willing to put in the time to do their homework and thus progress further and sometimes faster in subjects like math and the sciences?
I think that someday I'll teach a course in set theory, and the final project would be to write an extensive chapter about something, and slowly but surely I'll construct Lecture Notes written by the students and edited by myself. That would be awesome.
@Skullpatrol You seem to be putting too much correlation between math and good students. There are lots of good students who will struggle with math no matter how or when it is presented to them. There are many who are good at math who are generally bad students and generally immature.
In abstract algebra you meet the same ideas, on a much looser environments (compared to finite dimensional vector spaces and fields, which are very strict structures)
Things like modules over rings, ideals and so on.
This then connects with algebraic geometry, algebraic topology, algebraic number theory, representation theory, abstract harmonic analysis and operator theory.
@robjohn I agree with you that I may be over generalising, and there will always be exceptions, thank you for sharing your views on this hypercomplex topic, and have a Happy New Year!!!
You're ambling along and then suddenly flag varieties pop up to solve your number-theoretic question. You're chaining together a bunch of dual space identifications and it's a mess.
Symplectic forms, orthogonal groups, whatever.
@AsafKaragila Spill it.
Matt E and I meet every week to talk about Shimura varieties, which are about as number theoretic as you can get. And most of the time understanding what's happening turns into some exercise in linear algebra.
@DylanMoreland I'm supposed to show that if $R$ is a nontrivial commutative ring, and $R^m\cong R^n$ then $n=m$. The hint says something about using the existence of a maximal ideal.
I'm not sure, but tell me if this is correct:
$R/m=F$ is a field, then $R^m$ and $R^n$ modded by $m$ are vector spaces and we know it's true there.
I have a simple question about differential forms. The book says that if a form is locally exact, it is closed, because of course $d(d\omega)=0$. But wouldn't this just show it is "locally closed"? How do we get that it is closed over the whole domain from it being closed over overlapping subsets, because the exactness depends on different functions.
Well I guess you know it's closed at each point so it's closed over the domain...
@ZeeshanMahmud I think the party line is that if the edit is appropriate, just go on and edit it. The bumps are a way to gather eyeballs for edits that turn out to be inappropriate, so they can't be disabled by the one who edits.
@Henning Ok. Also I was having problem with formatting. Listing was driving me nuts as well as unable to format the equivalent sign. Also I was curious how to make the stars 'dots'. Autoformatting partially made some of the stars dots...
@ZeeshanMahmud Also, if you don't indent the lines starting 1.2. and 3., then you get a proper HTML list if you make sure everything else in the list is indented with at least one space. Then you can use blank lines for paragraph breaks within the list items.
@HenningMakholm I will go ahead and submit the corrections and you can perhaps edit in the bullets? And I will see from there how to do the formatting...
@Potato Consider f(v)=(cos v, sin v) for 0<=v<=2pi. If h is a continuous angle function, then g(v) = h(f(v))-v maps (0,2pi) continuously into multiples of 2pi, which is a discrete set, and therefore g is constant. But that contradicts g(0) != g(2pi).
@Asaf I think you can just say: there's a surjective homomorphism $R^n \to (R/\mathfrak m)^n$, and the kernel is the submodule $\mathfrak m \times \cdots \times \mathfrak m$.
And then show that this last thing is $\mathfrak{m}R^n$. That's not so bad.
Dear all, I need help in figuring out how do I do the following: I use my gmail to access math.SE. Now, I want to delete trace of this ID and use only a new gmail account to access math.SE
@HenningMakholm The code you gave me does not create a perfect array. Last bullet in line 2 is in red letters called :\bul with the word 'let'; incidentally this is the same place where the code breaks up in 1 line
@ZeeshanMahmud Oh, sorry. You can only break lines where you'd be able to place a space; that is (simplified), before a non-netter, except after a backslash.
(Yes, that's a rather strange sense of "everywhere")
@Asaf So I think you could say, "I have a generating set of size $n$, so the dimension is $\leq n$. If $e_1, \ldots, e_n$ is the obvious thing (a $1$ in the $i$-th slot), then a relation $a_1e_1 + \cdots + a_ne_n = 0$ means $a_n \in \mathfrak{m}$, just looking at each coordinate.
@Dylan Reviewing the previous questions (some of which I prefer not to do, since I have quite the choice in this course - literally :-)) as well hints from A&M (the source) I think that tensor products should be fine.
If I want to find a singular 2-cube whose boundary is the circle with radius 1 minus the circle of radius 2 centered at the origin, does it suffice to map $I^2$ to the annulus bounded by those circles so that two of the sides overlap and cancel (so I would map the other two sides to $[1,2]$ and $[1,2]$ with the orientation reversed so they cancel)?
Then to use the fact that tensor product commutes with direct sums to have $\left(\bigoplus R\right)\otimes R/\mathfrak m \cong\bigoplus(R\otimes R/\mathfrak m)\cong (R/\mathfrak m)^n$.
@Potato To respond to a 3-hour old comment (sorry): Yes, this is not a non-uncommon misconception. But if you can get the basics of your field down ("basics" in an obviously understated sense), then you're ready to start research. In the process of that research, you'll undoubtedly have to start working through a slew of research-level papers, but it would be a mistake to try to pick those in advance.
@KannappanSampath I think the problem is not so much one of notation, but the fact that the OP uses no connecting prose between his formulas. Just a dozen English words at strategic places would have made his procedure totally clear.
