by $\overline{V}$, i mean the dual of $V$. and the $coeval$ and the $eval$ morphisms are the bordisms $\emptyset \to M \sqcup \bar{M}$ and $M \sqcup \overline{M} \to \emptyset$ (which, after $Z$-ing, gives the coevaluation maps and the evaluation maps resp)
i am not yet sure why the morphism $Z(\subset)$ is the evaluation map $Z(pt^+) \otimes \overline{Z(pt^+)} \to \Bbb C$, though i am probably missing the obvious here.
@DanielFischer I wanted to write the following. Given that $f$ is a complex function. If we have $\frac{\partial f}{\partial t} = iC\frac{\partial^{2}f}{\partial^{2} x}$ could you state that $\frac{\partial f^{\ast}}{\partial t} = -iC\frac{\partial^{2}f^{\ast}}{\partial^{2} x}$, where $C$ is a constant and $f^{\ast}$ is the complex conjugate?
@JohnDoe One markup remark first, * in maths screws up the rendering in chat, use \ast instead. Then, supposing that $t$ and $x$ are real variables, then yes, that follows, since $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial x}$ are real operators (which means they commute with complex conjugation).
@DanielFischer When you say that they are 'real operators' do you mean that the complex component is ignored when you take the partial derivatives? So in other words you consider $f$ and $f^{\ast}$ as real valued functions?
@JohnDoe No, an operator is real if it commutes with complex conjugation. $\operatorname{op}(arg^\ast) = (\operatorname{op} arg)^\ast$. You could say that the partial derivatives act on the real and imaginary parts of $f$ separately.
@Balarka: This is far more general than just what happens on the point, and is really just linear algebra. Look at Lurie's proposition 1.1.8. That the map $p: Z(M) \otimes Z(\overline{M}) \to k$ is a perfect pairing gives you a canonical isomorphism $Z(\overline{M}) \to Z(M)^*$, obtained by sending $b \in Z(\overline{M})$ to $a \mapsto p(a \otimes b)$.
It looks like you've got the right idea up there. I'm not reading carefully, though. If you think what you've done proves it for general $M$ then you're probably right.
@DanielFischer Is it true that if $f$ tends to $0$ as $x$ tends to $\infty$ then $f^{\ast}$ tends to $0$ and $\frac{\partial f}{\partial x}$ tends to $0$ as $x$ tends to $\infty$?
@Hippalectryon I also managed to calculate in a very easy way one of the toughest integrals ever posted on I&S and MSE $$\int_0^{\infty} \frac{\cos(x)}{x}\left(\int_0^x \frac{\sin(t)}{t} \ dt\right)^2 \ dx$$
Hey, does someone have a reasonably simple explanation for what the difference is, exactly, between varieties and manifolds? I see the words used interchangeably sometimes, but that's probably not something one can generally do, right?
@JohnDoe $f$ tends to $0$ if and only if its conjugate tends to $0$. For $\frac{\partial f}{\partial x}$, things are more complicated, that need not tend to $0$. Consider something like $\frac{\sin e^x}{x}$.
@Danu They're not at all the same thing. A variety is a sort of algebraic analogue of a manifold, if you'd like. A smooth real or complex variety has, as its underlying topological space (as a subset of $\Bbb P^n$ with the analytic topology), a topological manifold.
@Hippalectryon Some months ago, when I first tried that, I failed to do it in an easy way. I had to use computational systems in some points of the proof. Then a brilliant idea came to mind ... I think to make a poster with the proof and post it on one of my walls, really. I love that.
@MikeMiller Sadly, I haven't the slightest idea about varieties. Is this something a normal algebra book (I'm about 2/3 through Vinberg's book) would cover, or would one really have to study some algebraic geometry?
You can say some nontrivial things without learning about schemes or anything. Here's the wrong way to think about them, as an algebraic geometer: a $k$-variety is the solution set in $k^n$ to some set of polynomials.
@JohnDoe No. If $f$ tends to $0$ and $\frac{\partial f}{\partial x}$ tends to $c$, then $c = 0$. But $\frac{\partial f}{\partial x}$ doesn't in general converge to anything if $f$ tends to $0$.
@Danu: The right way is to think of them somehow as if they're an algebraic manifold, with "coordinate patches" and "algebraic" change-of-coordinate formulas
Varieties aren't really objects embedded in $k^n$, just like manifolds aren't really objects embedded in $\Bbb R^n$
@MikeMiller it's nice to know they're the same, it confirms the thought that what the p-adics do beyond localizing is happening away from the rationals. I'll think about your argument
My attempt. This is by no means closer to the answer, but I want to address several equivalent forms that might be helpful for future calculations.
First, from Landen's identity of the following form
$$ \mathrm{Li}_2(z) = -\mathrm{Li}_2\left(-\frac{z}{1-z}\right) - \frac{1}{2}\log^{2}(1-z), \qu...
@Hippalectryon I also have the full, neat solution to the one above. I did it the last days.
@Hippalectryon All I say about mathematics (well, in terms of what I did) is always as I say it is. I admit that I might not say everything (I actually say nothing) about personal life since I don't enjoy such a topic. :-)
@Danu I meant Berger's Geometry. I'm not sure what category it falls into. Definitely affine, from what I've seen! There's definitely more to it though
Btw, when you guys hear of a [grad] student of physics trying to self-study some advanced math, do you get the same feeling I get when hearing about a high schooler trying to self-study string theory? ^^
@pjs36 Can you link it? I don't think I was able to find it through google
@Danu Score! And by the way his Geometry Revealed is pretty stunning, although much more expository in nature; not so much a course. Somehow a nearly-final draft seems to have leaked onto the web...
@Danu: Berger has a beautiful book on basic Euclidean/non-Euclidean geometry, but written for a sophisticated (French) college audience ... ultimately for people who will teach good high school students. Not part of the Riemannian geometry stuff ...
I never thought about Z_{(p)} very well. I could only think about it in terms of localization in general, where the idea isn't necessarily to forget about all the other primes (what about the primes below p in a general ring?), it's to forget about all the other maximals
I want to find the solution of the problem:
$$(t+u(x,t))u_x(x,t)+tu_t(x,t)=x-t, x \in \mathbb{R}, t>1 \\ u(x,1)=1+x, x\in \mathbb{R}$$
I have tried the following:
$$(x(0), t(0))=(x_0, 1)$$
We will find a curve $(x(s), t(s)), s \in \mathbb{R}$ such that $\sigma (s)=u(x(s),t(s))$
$\sigma'(...
@Kaa We established that $0<\sqrt2/2<1$. If $x$ and $y$ are rational numbers, stretch write down a linear function that sends $0$ to $x$ and sends $1$ to $y$. Where does $\sqrt2/2$ go?
hi, how would I calculate the probability that an undirected graph is connected when every node in the graph has at least one edge and there are as many edges as nodes?
It's pretty difficult to push into the about 20-30 minutes I'll have for it... especially considering I have no choice but to talk about handlebodies and the key terminology thereof...
@evinda i think you should also try asking your question on how to input the equation into wolfram alpha in the mathematica stack exchange so that you can check you answer in wolfram alpha.
@Mike, don't be at your age like one of my old colleagues ... who used to grade promptly and now takes weeks to get it done ... more proof that some of us should retire ...
Homework grade is a miniscule part of their overall grade (10%? might even be 5)
So I'd rather use it so that they learn from it rather than as an evaluative tool. Allowing them to redo their problems is a good way of forcing them to confront what they didn't learn.
@Ted: I talk about the general strategy one should do when solving a problem. I never give a full solution in class, before the homework or after. This means that if they want the points back, I'm fine if they use the idea, but they still need to think it through.
I've been telling my differential geometry students all semester that the "required" problems on each problem set (one each) would appear on tests and the final. How many of them have come to me to work on them ... either before or after?
I emailed a guy at Columbia today; he gave a talk recently but I can't find the work he talked about on the arXiv or elsewhere. (He appears to be a 5th year grad student.) I probably made his day by asking about his work, but I find sending emails to people I don't know just as terrifying as @Incurrence does talking to lecturers.
Do you happen to know how accessible "The Geometries of 3-manifolds" by Peter Scott is, and if it is a good source to learn about such things? Do you have a rough idea of what should be studied first? @MikeMiller
I answered that I am pretty sure, maybe incorrectly: If both are not normal, the first epimorphism is not a quotient map, so it isn't a short exact sequence, if one is not normal, then it splits, and you told me it doesn't split, and hence both L and Z(H) are normal
@Incurrence The symbol $\rtimes$ does not uniquely identify a group, although if you have a group you are working in there is an implied meaning (the corresponding automorphism is conjugation inside the group.)
>_<' The proof I just worked feels so dirty... full of case analysis. I'm sure I'm missing something obvious that should have yielded an analytical proof...
@DiscipleofBarney $G\cong N \rtimes H\iff$ $G=NH$, $N\cap H =\{e\}$ and $N$ is a normal subgroup of $G$ and $H$ is a subgroup, not necessarily normal $iff$ $G$ is a split extension of $H$ by $N$
My attempt. This is by no means closer to the answer, but I want to address several equivalent forms that might be helpful for future calculations.
First, from Landen's identity of the following form
$$ \mathrm{Li}_2(z) = -\mathrm{Li}_2\left(-\frac{z}{1-z}\right) - \frac{1}{2}\log^{2}(1-z), \qu...
I want to find the solution of the problem:
$$(t+u(x,t))u_x(x,t)+tu_t(x,t)=x-t, x \in \mathbb{R}, t>1 \\ u(x,1)=1+x, x\in \mathbb{R}$$
I have tried the following:
$$(x(0), t(0))=(x_0, 1)$$
We will find a curve $(x(s), t(s)), s \in \mathbb{R}$ such that $\sigma (s)=u(x(s),t(s))$
$\sigma'(...
