I dunno. Some stuff in algebra and number theory can be reasonably intense, not to mention differential equations stuff (which you're less likely to do, I realize).
yes, @Alessandro, definitely on the verge of starvation.
I've heard Grobner bases are a thing but know nothing of them. I would suspect based on extremely limited knowledge that modular forms are also useful. I guess also based on the fact that lmfdb.org is a thing
There are some words like baratro which is really read as bàratro while the standard for Italian would be baràtro, but you just have to know which words have an unusual accent if it's in the middle of the word
Is it possible to have two point $x,y\in X$, $X$ a scheme, with $x\in\bar{y}$ and $y\in\bar{x}$? It doesn't happen for affine schemes but I don't see it in general since not every pair of points is contained in an affine patch
I don't know if this is a good picture, but I like to think of schemes as being stratified by $x\leq y \iff x\in\bar{y}$, so that taking closure "goes downstairs". The clear example would be $\mathrm{Spec}\Bbb C[x,y]$ with $3$ levels, the generic point on top, $(f(x,y))$ in the middle and $(x-a,y-b)$ in the bottom. This also helps me think about generic points as being close to many points downstairs
@Alessandro in general in a topological space, the relation $x \mapsto y : \Leftrightarrow y \in \overline{\{x\}}$ (read as $x$ specializes to $y$) is reflexive and transitive and it is anti-symmetric iff the space is $T_0$.
Hopefully wherever I end up on the faculty is chill with people teaching classes outside their specialty. a good opportunity to learn things I otherwise wouldn't
@TedShifrin If you'd believe it, I really do just forget sometimes that I want to learn things. It happens with about every bit of math I'm interested in, about once a week.
@CaptainAmerica16 I like the book Combinatorics and Graph Theory by Harris, Hirst, and Mossinghoff. It covers combinatorics, graph theory, and the infinitary versions of these.
Demonark: I found it more frustrating teaching grad courses, because generally grad students just sit there like bumps on logs and don't participate. So I did a large variety of undergrad courses. I taught graduate complex several times, graduate diff geo a bunch of times, a few advanced grad courses, but didn't do as much variety there because I didn't teach that many grad courses.
@CaptainAmerica16 I think that as you know more math in future, you might grow to like some topics which you think you hate now. After all, many topics are related, and you will see the relations later on in the grand synthesis, the meaning of everything.
I did teach grad diff geo one year and after the first quarter all the students were undergrads. They were all very talented. Not a one finished a Ph.D. in math, though. One Ph.D. in econ, one lawyer, one theologist. Oh, I take it back. One guy did do a Ph.D. in geometric PDE. Very gifted.
@CaptainAmerica16 I sympathize in that HS math can be woefully boring. But be careful that you do not set a bad precedent for your future work---participation is often key, either for understanding or just for getting in the good graces of your instructor.
@TedShifrin I guess there are other users you may not miss because you didn't meet them, like Jonas Teuwen, Old John, and Theo Buehler. But when I was here long ago I talked a lot to them.
@CaptainAmerica16 I think if you can find the latest schedules of the Cambridge mathematical tripos, their undergrad syllabus, you will see the depth and breadth of the math covered there in the three year course. I like it because you can see all the subtopics there and how many weeks they plan to devote to teaching each subtopic. Very detailed syllabus.
@JasperLoy I was looking at syllabuses for a few colleges recently. So many of them just list remedial classes. At least in my area. Plus they were state colleges.
@CaptainAmerica16 I actually have no idea now why some teachers can write such detailed testimonials for their students. When I was a teacher for a while long ago, I just couldn't know that much about each individual student, because there were so many students in a class, and you don't spend very much personal time with each of them.
I participated in some classes more than others. I have some weird habits. Before I used to be way more impatient and would often see where a proof was going halfway through so I'd try to push things along. I started doing it less because if it was too fast, people would just ask questions and my efforts would all be for naught
My high school physics teacher hated me, but I don't think the students did. He couldn't understand or explain anything correctly, and so I spent a lot of time politely trying to explain things so that the rest of the class could learn something.
It was very sad, actually. He was a nice guy, but was dreadfully unqualified to teach the course.
One thing I do now a bit which hopefully helps is, in a class where I kinda have an idea what's going on, is asking a question about a confusion I had that didn't immediately come to mind the first time I thought about something
I have been watching youtube videos on British parliamentary sessions. Very entertaining, their sense of humour and the way they speak, especially the speaker John Bercow.
It seems that the fundamental group of any subset of $\mathbb{R}^2$ will not have an element of finite order.
Though the $3$-dimensional version is an open problem I couldn't immediately see why it is true in the $2$-dimensional case.
Please shed some light on this.
I know Hatcher has a question about the fundamental group of the immersion of the Klein bottle in $\Bbb R^3$. Does he have one about the Steiner surface in $\Bbb R^3$ (the immersion of $\Bbb RP^2$)?
My print books have barely been updated (through no lack of my trying). I finally got the multivariable book to get a second, mostly corrected printing. The linear algebra book got a second edition. ... The diff geo text is better, cuz I can update it whenever I want, and I have done so.
It seems that the fundamental group of any subset of $\mathbb{R}^2$ will not have an element of finite order.
Though the $3$-dimensional version is an open problem I couldn't immediately see why it is true in the $2$-dimensional case.
Please shed some light on this.
