Let $M$ be a real $C^k$-manifold, $\mathscr{H}_x$ be the $\mathbb{R}$-algebra of germs of $C^k$-differentiable functions at $x \in M$. It is easy to show that $T_x M$ can be embedded into the space of derivations of the form $v: \mathscr{H}_x \to \mathbb{R}$. In his book on Lie groups and Lie alg...
I'd rather say geometric insights. It is a rather remarkable fact that there is (essentially) a one-to-one correspondence between hyperbolic metrics, complex structures and conjugacy classes of (good) representations of the fundamental group to PSL(2,R)
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each of those gives a different way of looking at the same things with its own technical advantages and they can be combined to give much sharper tools
It often happens that you get families of Riemann surfaces, which then correspond to paths in the moduli space, so it is of interest to understand the geometry of that moduli space, which is quite intricate
There's a french astérisque "Travaux de Thurston a propos des surfaces" or something, which I liked pretty much, but I seem to remember that you don't read french.
@Alexei: then look at the work by the master himself: Thurston, Three-Dimensional Geometry and Topology, that is easily found on the web...
"a propos" aargh, "sur"
Travaux de Thurston sur les surfaces, Astérisque, 66-67, Soc. Math. France, Paris, 1979
@Alexei: Thurston: First, though, we must be more precise about what we mean by two hyperbolic structures being the same. There are in fact two important notions of equivalence, giving rise to two spaces: moduli space and Teichmuller space. ...
... Informally, in Teichmuller space, we pay attention not just to what metric a surface is wearing, but also to how it is worn. In moduli space, all surfaces wearing the same metric are equivalent. The importance of the distinction will be clear to anybody who, after putting a pajama suit on an infant, has found one leg to be twisted.
I have to wonder if that withdrawal was for real or they're just saying what others want to hear. That sort of business is too lucrative to just let go of.
You take a lot of pomegranate arils, add a shitload of sugar while heating and stirring, the arils let out their syrup.
This one is not a try, I did that a year ago. With vastly less pomegranates too!
Last year I had like one small one, now I used two big ones. I actually drowned the chicken in the syrup. If I had a sterilized container I might have saved a half for future cooking.
Am I the only one who has the impression that the rotational symmetry of the Gravatar algorithm has the unfortunate side-effect of producing things reminiscent of the swastika and other ugly stuff?
The motif is always to have a 90° axis. I would suspect that the ones that cannot have reflection symmetry will invariably look like a swastika of some sort...
@robjohn No, I doubt I need an actual proof reading for this sort of questions. I do it myself usually. Just wondered if you ruined my statement about long answers that no one is gonna read :-)
but as far as I see, POM isn't but about "what numbers are" i.e. it doesn't help in finding new stuff in mathematics, like new fields in mathematics. Am I right?
Seeing how I think that I am missing a badge, and I don't think it is a good idea to open a thread for every time someone thinks he is missing a badge,
I hereby post a thread so people could post answers whenever they think they are missing a badge. Answers can be given in comments, since they a...
@quantumelixir Hi there. I am sorry. I hadn't seen yours before starting in on my answer. I am reading yours now.
I will comment when I am finished.
Oops, he is gone.
he thinks his answer is similar to mine. It looks so much more complicated, but I think it is simply the same idea of breaking things up in the sum of positive and negative numbers. He seemed upset that I didn't acknowledge his answer in mine.
"the discriminant is a symmetric function in the roots which reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots."
@Jack: The point of the thread is not to delete the answer when the badge problem is fixed, but rather to add the solution into the answer so future people will be able to use this knowledge too.
I am trying to read the last bit of quantumelixir's proof, but he has written |x| as max(x,-x), which while correct, because the expressions for x are non-trivial, it makes for complicated reading.
@AsafKaragila I can undelete if you want, but basically when I looked at the site again, I had the badge. No message from a human or anything, and I certainly didn't do anything.
@tb I have checked some of your papers and they suggest to me that you're not really an analyst. Or at least, it is analysis which I know nothing about. What field is that?
@Jack: Interesting. I think you should undelete it for at least a little bit, if no one comes forward as the one fixing the problem then perhaps you should delete it indeed.
@Jonas: I don't know. I'm somewhere in the middle of abstract harmonic analysis, homological algebra and geometry. Use a lot of functional analysis (mainly for understanding group actions that arise from a geometric setting).
But also a lot of abstract nonsense (category theory)
@Jonas: My main work is on so-called bounded cohomology, a gadget introduced by Gromov for topological spaces that has many surprising applications in what is called rigidity theory. (Discrete subgroups of Lie groups, symmetric spaces, hyperbolic geometry, etc.)
I like functional analysis a lot, but I mainly use the soft, abstract stuff. A bit of structure theory of some special spaces, but no hard estimates.
Okay. Nice. I'm doing less abstract things. For the next four years I will try to adapt/find a theory for harmonic analysis with a non-doubling measure.
What's your take on Stanford offering simplified versions of their courses on the internet for free? Could something like that be done for advanced mathematics?
@Asaf: maybe this is very silly. So there is a model of ZF in which there is a non-separable set of reals. Can you say anything about its structure? In particular, can you sort of prescribe some of its properties (closed, or somewhere in the Borel hierarchy?) (I don't know much about the actual forcing argument, it's just idle curiosity).
I'm thinking of ai-class.com, but there are db-class.com and ml.class.com too. Those are simplified in the sense that less material is covered, and grading is totally internet-based
@tb This model is the basic Cohen model, the original proof that ~AC is consistent relatively to ZF. The set itself is Dedekind finite, it has no countable subsets.
If one exists, then one exists which is unbounded.
Simply by taking a homeomorphism of (inf S,sup S) with R (given, of course, that S is the Dedekind finite set)
Note that if S is such set then every convergent sequence is converging within S, due to the fact that it is eventually constant.
Interestingly enough, the basic Cohen model is a model of the selection principle, thus the Prime Ideal Theorem and its equivalents, such as the ultrafilter lemma.
Silly question 2: so open are defined as usual, using intervals. Closed sets are complements of open and these closed sets are certainly sequence-closed. The converse is wrong, probably?
