That's not horrible. Most of my students in the US didn't do too much better. Too much competition from the very top students at Harvard, Princeton, MIT, and China.
@user159870: Surfaces of revolution need to have symmetry about the axis of revolution. How do you see that?
That's different in different departments, so I can't answer that universally. But GRE subject test does matter, grades, and particularly letters of recommendation that say something specific. Plus it helps if there's someone in the department who's interested in having you as a student.
In choosing an advisor I went through the department one by one and asked if they needed me to draw commutative diagrams. Went for the first one who said no.
I had essentially a 4.0 (one B in a math course) and didn't get in everywhere I applied many years ago. And my GRE score was perfect, too. Just saying ... There are other ingredients.
My impression (though I hVent served on many admissions committees) is that one of the most important factors is your letters. There are grade and GRE cutoffs, but past that, what really matters is more subjective...
@TedShifrin I don't know. They're the best schools in the continent here (so that's top 20 world wide according to most rankings, though I realize those don't mean much...)
Yes. I should have told them to GTFO, because it's for lower division students, anyway, and also possibly to switch majors. Didnt think to until later.
I remember when our Physics 1 prof decided to do a mock-up exam and the average of over 300 students was roughly 4 out of 50+ points. That's when we realized shit just got real.
Ciprian imports IMO winners to study here, and beyond them there are some outstanding students I can probably count on one hand. Then there's a good number of good students; and then the remaining half or so make me cry.
The most exciting thing we did was the residue theorem (and the prerequisite Cauchy stuff that leads up to it). Keep in mind this is a 10 week course that is intended to get some engineers to come to it, too.
@AndrewThompson Sure, that's OK. But it's less important he did it alone or not, the beauty of that mathematics remains the same (at least to me). It's like visiting the heaven. Try some of his stuff and convince yourself.
That stuff is incredibly deep, profound, amazing, outstanding ...
Are there any fundamental prerequisites to learn measure theory? I've an ok grasp on some countable set operations, have developed the integral of regulated functions (as limits of uniformly converging step functions) and looked at continuous maps between normed spaces.
@user159870, it is often also helpful to look at slices perpendicular to the other axes, too. For example, to distinguish between $z=x^2+y^2$ and $z^2=x^2+y^2$ or $z^2=x^2+y^2\pm 1$.
All of those examples have circles when you slice perpendicular to the z-axis. How do you decide what the surfaces are? You have to look at other slices.
To see if it is a surface of revolution we always have to look if $z=\text{constant}$ is a circle. To say which surface it is, do we have to take for example $x$ to be constant? @TedShifrin
redo: [...] inclusion map is homotopic to one that factors through the covering.
The motivation for this comes from how we systematically remove singularities of disk in the proof of disk theorem (by considering coverings) + this is what happens in $\Bbb{RP}^2$:
take the generator $a$ of $H_2(\Bbb{RP}^2)$. Ignore that $2a = 0$ for a while. $2a$ is represented by the 2 fold cover $S^1 \to S^1$. We consider a little circle in the attached $D^2$ representing $2a$ (which is nullhomotopic, 'course, but ignore this), and the inclusion of that is homotopic to one which factors through the 2 fold cover.
We're trying to do something similar to $\Bbb{CP}^2$. Our 2-fold cover becomes branched at 2 points (because the degree 2 map S^2 --> S^2 is a double cover brached at 2 points). And the attaching map is a hell of a lot more messier.
Image of the degree 2 map $S^2 \to S^2 = \Bbb{CP}^1$ is not an embedded surface. But $S^2$ is, and I'm trying to get an $S^2$ in the attached $D^4$ in $\Bbb{CP}^2$ so that sliding that $S^2$ through the disk $D^4$ towards boundary and composing with the attaching map $S^3 \to \Bbb{CP}^1$ gives me a homotopy of the inclusion with the degree 2 map.
@MikeMiller Ok, nevermind, fair point. It's not as if I can get every branched double cover to work.
