So I had this blog a while ago, but didn’t really use it, so I started it up again, completely refreshed! I think I’m mostly going to use this to talk about my preparations for grad school. i.e. what I’m learning over the summer.
Before I can do any serious studying, though, I do have about 5 weeks of school left. I’m currently taking Topics in Geometry and Differential Manifolds. Topics in Geometry is sort of silly. We’re sort of following Artin’s Geometric Algebra. It’s kind of fun in the same way combinatorics is kind of fun, but I don’t think that it’s really a hot field right now. It’s very classical stuff–affine and projective geometry and geometric interpretations of matrix groups. Anyway, this class is full of much younger people who I don’t really know. This makes sense I guess, since I pretty much only know the math majors my year (and the ones who graduated last year, of course). So I took the midterm for this class last Friday after not really studying at all, and I hadn’t really been paying attention in class at all either. The questions were all *very* easy. Two of them I could’ve done before taking the class, and the other two were really easy applications of the stuff we’d learned. I didn’t really have enough time though, and I wrote basically nothing (I would say nothing, but somehow I got points for what I wrote…) for one of the problems, and was very hand-wavy with all the others. I was pretty sure I’d done really badly, at best a B-, but then in class yesterday, he said the average was 45/100, and it turned out that I got 75! I’m not really sure how this is possible, since I felt like I did the bare minimum of trivial things. I couldn’t even on the spot think of a normal subgroup of S_6. I just scribbled one down in the last second, and it wasn’t normal. How did the majority of people do so much worse than I did? There are even a few math majors my age in the class, who I’m assuming did just as well as I did, which means that the other people must have done even worse to get the average so low!
I also just took my midterm for my other class, manifolds. Manifolds is a much more important subject to know, apparently. I will need to know all this stuff if I want to do just about any kind of math (except maybe finite group theory or something). The thing is, this class is HARD. Nothing is intuitive for me. I get the “idea” of everything, but when it comes to actually writing proofs, I can’t do it. It’s so notation-heavy, and I feel like the notation of everything takes away all the elegance, whereas in algebra, once you figure out the idea of why something works, it’s really easy to write down. Anyway, the test was VERY easy. There were three problems and we had an hour and 20 minutes to do it. I finished second (there were 30 minutes left). However, in two of the problems I was maybe not rigorous enough. The first problem was something like: Is there an immersion from every closed n- without boundary into R^n? I gave the counterexample of S^1 into R: every nonvanishing vector field on S^1 must have a vector map to 0 under the differential, by continuity. The other problems were trivial, and kind of hard, respectively. The trivial one isn’t worth mentioning, but the other was: Is RP^2 x RP^3 orientable? I said “no” and tried to show that every atlas has a transition function that does not preserve orientation, but I’m not sure how correct all of the assumptions I made were. I probably should have done something using volume forms, but I’m a little afraid of them. The people in my class seem to know their stuff, though, so the curve might not be so much in my favor this time, but I guess I’ll see.
When I get to Michigan, my goal is to pass both the algebra and the topology quals. The algebra should be sort of easy to pass once I review over the summer, and the topology shouldn’t be too bad either. I will just have to learn more homology over the summer, since I think the manifolds class will have prepared me pretty well for that part of the class. So my plans for review start with Dummit and Foote (pretty much all of the stuff up through Galois Theory) and Hatcher. In Hatcher, I’ll try to get as far as I can, but definitely through homology. If I am able to pass both of those exams, then the classes I’ll take will be: Complex Analysis (then I won’t have to take the analysis qual), Algebraic Geometry, and ADVANCED algebraic topology!! I think the latter two classes will help me decide sort of what I want to do. I think, though, that algebraic geometry will be very very hard. In the course description, he says: “A solid background in commutative algebra is very useful; working out the exerises in the Atiyah-Macdonald book Commutative Algebra is a good summer project to prepare for the course.”
So I guess I’ll be going through Atiyah-Macdonald also! The algebraic topology class shouldn’t be too bad. According to the description, there are no exams, and about three to four problems assigned each week. I think that this class will be useful even if I wind up doing commutative algebra or something. I’ll learn lots of homological algebra!