Similarly one defines the exponential series to be $exp(a)=\sum_{n\geq 0}\frac{a^n}{n!}$ and the claim is similar that this is convergence because for each $k \geq 0$, the term of degree k of $\sum_{n=0}^N \frac{a^n}{n!}$ has a limit in $E^{\otimes k}$ as N tends to $\infty$.
@TedShifrin I thought the secret might lies in that the exponential series and log series are maps to the space $T(E)$ which can be thought in basis form $\sum_{n=0}^\infty \sum a_{i_1,...,in}e_{i_1}\otimes...e_{i_n}$
Thus, what tey mean is that each coefficent to the base $e_{i_1}\otimes...\otimes e_{i_n}$ should be finite.
I don't know him well. So I can't really pass judgment. I knew Rick Schoen pretty well, but he moved on ... and I've known Rafe Mazzeo since he was a sophomore in college. :P
I wrote an integrated linear algebra/multivariable calculus/analysis book. Mine was a bit more grounded than Leon's. We can't all be blown away like Stanford students.
I was wondering do you know if classifying holomorphic vector bundles is same as reals ? That is holomorphic one are one-to-one correspondence $[X,G_n]$ ?
yeah haha I mean in later part of my thesis I work with smooth projective varieties over C. I classified real vector bundles but it is not related to other parts of my thesis because that statement isn't true for holomorphic vector bundles @TedShifrin
Karim: Topological classification of complex vector bundles is fine. But holomorphic structure is much harder. Same thing with different complex structures on diffeomorphic complex manifolds.
And the Picard variety $H^1(X,\mathscr O)/H^1(X,\mathbb Z)$ gives you the torus that parametrizes different complex structures on the same complex line bundle.
sounds good @TedShifrin. I am just doing final revisions to my thesis. I will finish on Tuesday. I will go hang out with fellow graduate students on Thursday and drink wine and discuss philosophy. Starting from Friday I will start preparing for my PhD.
I am excited though I mean my knowledge in transcendental aspects and arithmetic aspects of algebraic geometry is good, but I still have long way to go.
haha. I was thinking maybe I should once a month give lectures to other graduate students about my work. I think that would probably help clarify certain things.
@EricSilva It was joke regarding your "cuz y'alls is 2 gud" comment, because it sounded African-American English in my ear (mainly because of the usage of the auxiliary "is" instead of "are", otherwise it may have been just Southern).
@TedShifrin I don't know how. To derive sinx=2sin(x/2)cos(x/2) I wrote A=x/2 in the equality sin2A=2sinA cosA. What value of A should I take to get $sin(x)=4sin(x/4) cos(x/4) cos(x/2)$ ?
@TedShifrin You know it is so weird that I have very close to photographic memory, but I tend to lose my keys and wallet all the time. I just lost my credit card yesterday.
Whatever input you think is worth giving, I would absolutely appreciate. I don't have a full list put together yet but I'll send you the places I've started looking at recently. Thank you!
I took MT from Brian White and we used Cohn's book. I'm pretty sure he used it just as a f* you lol. Anyways it turns out its much easier to do the exercises once you actually know something about measure theory.
How are you envisioning relating vector bundles to cycles? It's not quite like line bundles and divisors.
@Drew: Well, I had colleagues who told students (both undergrad and grad) not to take classes from me because I expected too much work/commitment. I think you've done just fine.
Karim: Assuming the title is appropriate, I would word it "Vector bundles and algebraic cycles"
After all, the results in this book were developed over the course of, I dunno, at least a century. And you're right. It wasn't meant to be personal. Just a demonstration that you can fit a lot in 350 pages.
LOL ... Yeah, unless the adviser reads each word carefully and fills the thesis with red ink (as I have done numerous times), the quality is generally poor. General statement.
Karim: You know that you have a tendency to be sloppy and say things that are wrong, so I worry that you haven't had this sufficiently criticized. But that's all I'll say.
@TedShifrin Everything else listed I've heard before, e.g. twistor spaces, harmonic maps, complex/conformal - quaternionic/octonionic geometry etc but not manifolds with exceptional holonomy etc. They sound amazing!
Well, for example, if you have a generic hermitian complex manifold, instead of the usual $SO(2n)$ you get $U(n)$.
It's not about once, actually, but "going around" means parallel translating around (not too large) closed loops and looking at the resulting linear transformation on the tangent vectors.
Am I missing something on this measure theory exercise?
Suppose (X, A, m) is a measure space, and f_n, f are measureable functions such that f_n -> f a.e. Show that there exists functions g_n such that g_n = f_n a.e. and g_n -> f everywhere.
Solution. Take g_n = 1_N * f + 1_N^c * f_n, where N = {f_n -/> f}
other things I have in mind is whether the powerset operation is a computable function, and whether there exists a hierarchy of definable functions that grow faster than the powerset operation
Ah, that will rule out sets of cardinality $\aleph_0$ being a valid input then, because then the output is necessary uncountable in ZF
then under this definition, $\mathscr{P}$ is computable since all its inputs are finite sets, and the members of their powerset can be enumerated in finite steps
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:
z
n
+
1
=
(
|
Re
(
z
n
)
|
+
i
|
Im
(
z
...
Gotta be glad that any nontrivial division by zero pretty much requires you to break commutativity and associativity, thus this article actually makes sense
You cannot really have a singular cardinal to be not reachable from below by definition, or that even in ZFC, there are infinite cardinals that are neither singular nor regular?
Let $p$ be a prime such that $p+2$ is Also a prime.
Define
$$ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $$
For $z \neq 1 $ and $re(z) > 1$ this $f(z)$ always converges.
If there are infinitely many prime twins , and prime twins grow like $O (n * ln(n)^2 ) $ as i...
Suppose we have a map $f$ of two variables that is $x$ and $y$, then is there any relation between the determinant of the Jacobian of $f$ and $f$ beng locally invertible?
Hi all! For a set $M$ of points in $\Bbb R^3$ with point group symmetry $G$, is the concept of "symmetry adapted" local coordinates something which is familiar to the mathematician?
I am asking because I'd like to understand how the distortion along such a coordinate effects symmetry. The gist of the SACs is that each one can be assigned to an irrep. Now the question is how distorting in that way "along" an irrep affects the symmetry.
If the product of $det(Df_{1})_{|(0,1;0)}$ and $det(Df_{2})_{|(0,1;0)}$ is positive, then $f$ is locally invertible
Well if the product of the determinant is positive then neither of the determinant is non-zero and hence locally invertible by Inverse function theorem
But I am thinking whether saying this would be wrong?
If the product of $det(Df1)_{|(0,1;0)}$ and $det(Df2)_{|(0,1;0)}$ is negative, then $f$ is locally invertible
Hello there! I wanted to find properties of parabola (vertex and focus) from its standard form of equation. How can I do it? Any answers from stack exchange? I can't find though.
Not to say that you can't do it with 1 at first, it's just not immediately clear what the starting of your "algorithm" does until some n_i further down the line where it reduces to doing what Rudin's does, roughly.
Let $p$ be a prime such that $p+2$ is Also a prime.
Define
$$ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $$
For $z \neq 1 $ and $re(z) > 1$ this $f(z)$ always converges.
If there are infinitely many prime twins , and prime twins grow like $O (n * ln(n)^2 ) $ as i...
