@AkivaWeinberger hmm, I wonder if this will work: $\sin x = 0$ has solutions $x = n \pi$ for all $n \in \Bbb{Z}$ and then $n\pi$ is nonzero since $3< \pi < 4$ can be proved and $3n < n \pi < 4n$ for all nonzero $n$
I used to like stuff like $\varphi_* $ but if you try to do that with a group action and add the point it acts on it looks like $R_{g,*,x}$ which is trash
@TedShifrin I think I figured out the tangent bundle thing shortly after you left but then I woke up this morning thinking I did it wrong. If we have the coordinates $x_i$ on the base and $y_i$ on the fibre of $TM$, the tangent space $T_pZ$ of the zero section is the span of the partials of the $y_i$ or the $x_i$?
I thought it was $y_i$ initially
But then that doesn't make sense for the tangent space of the fibre
The random is I thought I found a very intuitive proof of irrationalilty of pi, only to forget that I need to show sin (p) is not an integer
and yup, Niven's proof is one of the examples of auxillary function proofs in transcendence theory
The function f is specially chosen so that it dies in just the right way to trigger the contradiction
it is somewhat related to the diophataine approximation proof strategies, but not quite the same
whereas diophatine approximation proofs rely on chopping out the tail of the difference between the number y and any rational, and show that difference dies much faster than if y is rational, triggering the contradiction
@s.harp on second though, I can't think of a really bad spot for $T_pf$ notation for the derivative of $f$, other than the fact it is in general more clunky than something which only uses a subscript, or only uses a letter out front.
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
== History ==
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
== Statement ==
If a and b are algebraic numbers with a ≠ 0, a ≠ 1, and b irrational, then any value of ab is a transcendental number.
=== Comments ===
The values of a and b are not restricted to real numbers; complex numbers are allowed (they are never rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
In general...
Abel prize winner Uhlenbeck 1st female winner + noted for contributions to mathematical physics eg gauge theory http://www.abelprize.no/c73996/seksjon/vis.html?tid=74011&strukt_tid=73996
@TedShifrin why don't you like calling the zeros of vector fields singularities? Just wondering, as I recalled you sighing about it a little while back.
They're a singularity in the sense that the one-dimensional line bundle given by the vector field has no fiber there. But to me a zero of a function is not a singularity.
One thing that bugs me is that hyperbolic tends to mean two different things.
Either that the singularity is generically stable (eigenvalues have non-zero real parts), or that its eigenvalues have different signs for the real parts
Wow I didn't realize my question about linearisation got a downvote. :(
Oh it was just yesterday, too.
@TedShifrin would you be fine with me answering it with the explanation yielded by your comment?
I wouldn't accept it, as I was going to wait to see if anyone else had anything to say.
Think about the $2\times 2$ real matrix corresponding to $a+bi$. In general, you'll get $2\times 2$ blocks with little $2\times 2$ identity matrices above the diagonal.
@user123 so if you think of a general real representation as a group homomorphism $G \to GL_n(\Bbb R)$, then an orthogonal representation is a group homomorphism $G \to O(n)$
well I don't like being directionless in self study so I'd rather start somewhere self contained. I'll check out other resources when having trouble obviously
@TedShifrin why would you say that? I haven't read bourbaki but some sort of compendium of knowledge in a field to look at seems great to me. I was reading cultural history of physics by simonyi and i enjoyed it but I wanted to learn the actual material instead
OH okay. Well I hope landau is good but i still like to have something to reference. Like the table of contents in each section would give me an idea of what I need to learn for that volume etc
so if you have a finite group $G$ and a representation $\rho:G \to GL_n(\Bbb R)$, then you choose an inner product $\langle -,- \rangle$ on $\Bbb R^n$ and then you just "average" it over $G$: you define $(v,w)= \sum_{g \in G} \langle gv,gw\rangle$ (check that this is an inner product)
this new inner product satisfies $(gv,gw)=(v,w)$ for all $g$, so this means that every $g$ acts as an orthogonal transformation
this means that when we choose an orthonormal basis wrt $(-,-)$, then you get that every element of $G$ acts via an orthogonal matrix
this is an important trick in representation theory
@MatheinBoulomenos i need to go, i'll be back in about an hour.. thanks for the help , i hope you will be here when i will be back to continue this question :P
I am currently going to start symplectic geometry hoping that I will understand something. Karen Uhlenbeck winning the Abel just gives me hope that I might be able to do something for some reason.
@Albas I know from a reliable source that being strong in algebraic topology can be a big help there, as there are a lot of ways to apply that and it is something that not too many people working in symplective geometry are that strong in.
Suppose that $G$ is the internal semidirect of $N \unlhd G$ and $Q \le G$. Is there a canonical of viewing $G$ as isomorphic to some external semidirect product of $N$ and $Q$ ($G/N$ by $Q$?)?
@Alessandro: Mike Artin told me years ago that he went into algebraic geometry because he found complex analysis, analysis, topology, etc., "too easy." :P
Like the development of set theory after the discovery of the antinomies, there was a stepping back from the precipice of Kunen’s inconsistency, a charting out of possibilities that remained, and with the passage of time, a growing confidence in the delimited edifice. However, unlike the emergence of the cumulative hierarchy and other guiding ideas that provided intuitive underpinnings for ZFC, it is doubtful that even heuristic arguments can be put forward for the optimality of Kunen’s result,
because of its basis in a specific mathematical contingency. The strongest hypotheses thus stand on much shakier ground, but their study has a natural appeal owing to the power and simplicity of the concepts involved as well as the possibility of some new apocalyptic inconsistency.
@Ultradark Basically just that. An animation of birds flying like that, where the movement came from the birds wanting to move forward and keep close to the birds in front
@AlessandroCodenotti I think something like 7 or 8 PhDs and postdocs from Italy, most of them at QGM
I passed my AG exam by the way, I should thank @loch @Mathei and @Ted once again for all the help! I don't think I'm going to do any more AG now though, I'm definitely not taking AG II
So if $f_{st}(x)=x^{st}$ is a family of functions with $t$ is a time parameter and the number of functions in the family is $|s|=n$, then is it safe to say that this family of functions is a family of functions?
Here's a problem I saw earlier in the AMM which I thought was appealing (though there's no accounting for taste)
First, some notation. Let $I$ be the usual n-by-n identity matrix. Let $J$ be 1 on its skew-diagonal and otherwise zero. Let $R=\text{diag}(1,2,\ldots,n)$.