one point to note: If you argue that the result of the integral should only depend on $r=\sqrt{x^2+y^2}$, then without loss of generality one can assume $(x,y)=(r,0)$ with $r>0$
Suppose $\omega_1,...,\omega_d$ are the basis for the left invariant 1-forms on Lie group G. Let $\alpha$ be a path in G, what does the notion $\int_{\alpha(0)}^{\alpha(t)}\textrm{d}\omega$ mean?
@Semiclassic @Andrii: Perhaps the crazy Russian professor intends you to do $(x,y) = (0,0)$ in very easy fashion and then differentiate with respect to $x$ and $y$ under the integral sign? Have you learned that? I haven't tried.
As weird as this topology is, it is still Hausdorff for $\Bbb Z$ and $(p)$. Is the $I$-adic topology on a ring Hausdorff in general? I think there are issues if $I$ is very big
@TedShifrin this 2 weeks will be review weeks, so ill start from the begining and skip those i feel confident with , ill focus on what you said now :D projections onto subspaces and diff equations
@quallenjäger: Each one is just a path integral as a function of the endpoint, @quallenjäger. At some point we can talk about left-invariant $1$-forms on a Lie group if you want. It's generalizations of $d\theta$ on the circle, for example.
@quallenjäger: In local coordinates, each one-form looks like $\sum f_i(x)\,dx_i$. You then put in the parametrization of the path and do a usual line integral.
@quallenjäger: You can only apply Stokes if the two paths form a path that bounds. The topology of the space now becomes relevant unless you're just staying local.
You need exact forms for that to hold, @quallenjäger. If they're invariant forms they can't be exact (because they'd have to be derivatives of invariant functions, hence derivative of a constant).
Anyway I have to go now, thanks for your exampes! I'll read more about the p-adic numbers and I'll probably have more questions later (or in the following days)
Anyone happens to be able to explain the significance of field extensions like from $\Bbb Q_p$ to $\Bbb Q_p(p^{1/p^{n}})$? I have an idea about field extensions e.g. $\Bbb Q(\sqrt{2})$ and so on and I have an idea about p-adic numbers $\Bbb Q_p$.
Have been cycling with an old friend of mine yesterday whom I didn't meet for 10 years. but we were training together for more than 12 years like 5 times a week ...
I'm gonna decline the offer to study in the wintersemester and reapply for the sommersemester instead so that I can work for a few months to save some money, and meanwhile I'm gonna apply for as many scholarships as I can so that I can finance the final three semesters of the master
I have (I think) a really good application for some scholarships from the DAAD so I'm going to apply for those and I'm going to do the Goethe-Institut Großes Deutsches Sprachdiplom
Yeahhh, I applied to study at the BMS in Berlin in like.. January last year, but the only grades I had to show them were those from my second year of studies, which were poor (due to some significant mental health issues that I had back then)
Can arguments be measured starting in the imaginary axis? I think it's necessary to start an argument in the negative part of the y-axis to find the Riemann-surface of $f(z)=\sqrt{z^2+1}$
