Cambridge Year

Sunday, June 24, 2007

Exam Results

So my exam results are kind of funny. I got a distinction overall, which is the highest possible grade. On most of my exams I got above 80. (An alpha is anything above 70, so they were all good.) My worst grade (other than additive number theory) was on local fields, which really suprised me. I got a 71 on that, and the only thing that I can figure out that I lost more than small amounts of points on would be proving that there is one extension of the absolute value. From what other people in the class have told me, I was supposed to prove that there is only one norm on any finite dimensional vector space, but COME ON. That's basic linear algebra. And yes, in linear algebra that result is proved for R or C, but the proof given in the class wasn't different from that... it just replaced those with K. GRRRRR. Anyway... =)

I got a 16 on additive number theory. That's a gamma-, which is even worse than a gamma... which is already not good for much. =) What surprised me was my reaction to the grade: I thought I might be upset or something, but I think that it's hilarious. 16 percent! I've never done that badly on a test! (And yes, I didn't study for it or anything, but still! 16 percent!)

Tuesday, June 19, 2007

The May Ball

The period between the end of exams and the announcing of exam results is called May Week. Of course, it's not in May, and it's two weeks long, but who's counting? During this time there are a lot of concerts and people just hanging out and letting off steam. I mean, it's a huge university where most of the students have just finished exams that count for all of their grade for the year... it's pretty understandable that people want to party. So most colleges have a May Ball.

The May Ball is a formal party which lasts all night. Since Trinity is the largest college its May Ball is "the best." (I'm not sure exactly how this is measured, but everyone seemed to agree that it was the best. Even the people from other colleges. I haven't really talked to anyone from St. John's (Trinity's rival college) but I'm sure that they would disagree. =) ) So people buy tickets (in pairs, you can't get single tickets) for about 220 pounds, get dressed up in formal wear, and go to this party. The ball starts at 9, but people start lining up at 7, because "queueing is part of the experience"... because, of course, standing in line is so much fun. But, when in Rome... and I suppose that standing in line is a particularly British kind of fun. So we showed up at 7, and stood in line. The college does try to entertain you at least somewhat: they have jugglers and a brass band and some a cappella singers to entertain you. But most of it is just standing in line for two hours. (Actually, here's something funny. You stand in line on the paths in Great Court. Each path is some flat paving stones surrounded by cobbles. Next to the line they had the following signs: "Gentlemen, please stand on the cobbles and allow the ladies to stand on the paving." Because the girls are wearing high heels, which would really suck on the cobble stones. (No, I wasn't wearing high heels, I was wearing sandals. I wasn't going to go to a party that's supposed to last 9 hours (and two more of standing in line) in high heels.)

Anyway, after standing in line you get let in to the main part of the party. There were about 5 different places to get food, two more with just drinks, one with fruit. There was a place with rock music and one with folk music and one with jazz. They also had a small room where they were playing nice classical music (also all live) where you could sit down and relax. There was a comedy club (where Tom and I watched four of the five performers... and I laughed so hard I couldn't breathe some of the time. We're going to have to start going to comedy clubs in Boston, cause I haven't had that much fun in a long time). They also had some really fun things: bumper cars (which they call Dodgeems... because you're supposed to dodge the other people, not bump in to them) and very large swings and a basketball-type arcade game. And they had fresly made doughnuts (which were really good) and a tent with chocolate fountains and strawberries and marshmallows and bananas on skewers. (I went there several times.)

They also had a private fireworks show. Every May Ball has to have fireworks, but Trinity's are "the best." (Once again, I'm not sure what this means. Tom and I saw Robinson college's fireworks (which were just a standard small fireworks), and Trinity's were definitely better than these. We're also going to go watch St. John's fireworks today, and we'll see whether Trinity's are really "the best.") It was a really really nice show; I don't think I've ever seen a better one. It was nicely synchronized with music, and generally several things going on: nicer sparkly ones on the lower part of the sky and more explosive ones on the higher part. It was symmetric and nicely timed with music and completely impossible to reproduce for a larger audience. I was really really really really impressed.

The party is supposed to go on until 6am (at which point they apparently take a "survivor's photo"). We only lasted until 4, when the comedy club shut down and we couldn't find anything else to do. The main problem was that there wasn't a nice place to sit and just talk, since it was cold (apparently they usually have large patio heaters, but they didn't this year) and there were no nice quiet places to sit except for the classical music room, where you couldn't talk. So we decided we were pretty much done and went home. Overall, I had a really nice time, and I'm glad that I went. (Although hearing a punk rock/klezmer band was very loud and kind of scary.)

I only found out that Tom and I had tickets to the May Ball a few days ago. I'd screwed up buying the tickets, and the guy Tom emailed to try and get us tickets didn't reply, so we didn't think we were going. Then, at a Trinity party for math people (and yes, I went to a party, and I had fun) I was talking to Ben Green when this guy (Hugh Osborne) walked up and started talking to him. Ben introduced us, and mentioned that "Inna is very good friends with Tom Barnet-Lamb." At this point the guy asked when Tom was coming, and mentioned that he'd had an email from Tom begging for May Ball tickets, and that Tom "has this girlfriend, I don't remember from where, and..." and at that point Ben interrupted him saying that yes, he did, and in fact she's right here. And Osborne was pretty embarrassed and got us tickets. =)

WARNING: Dress details follow. Anyone who isn't interested should stop reading here.

Of course, this meant that I had almost no time to find a dress. (And Tom had to borrow a tux from his brother, which looked pretty funny.) Which sucked, since I didn't even really know where to go. So Lilian and I went in search of a dress for me. It turned out that 99% of the dresses here were strapless (and sold with a separate little shrug so that you actually had sleeves) and weren't big enough for me, so even if I were willing to wear a strapless dress (which I'm generally not) I couldn't wear any of these. After lots of looking I found a dress (which still wasn't really big enough for me, and I had LOTS of cleavage in this dress) but that at least would close and look OK), but it isn't actually my usual style. (I'll have a picture here later, but I don't actually have any pictures of my dress yet.) I also had to find a shawl to wear with it, which meant that I had a chance to shop in the market! That was kind of cool, since I hadn't done it before at all.

I'm glad that I went to this and saw it... although I'll likely never go to another party like this. I don't really understand why it had to be formal... since the formal wear seemed more in the way than useful for most of the party. (And all of the comedians were wearing jeans and T-shirts, which was amusing.) I think that if Harvard had a party like this it wouldn't be formal, but just a "fun day." (It has several of those, but none on this scale.) I'd be willing to go to that, if only for the comedy. =)

Friday, June 08, 2007

Seventh Exam: Elliptic Curves

I forgot this exam paper inside the exam room, so you'll only get summaries of problems. But still plenty of comments. =)

So I slept through all three of my alarms today. They go off at 8, 8:15, and 8:30 and I figured that at least one of them would get through to me. Apparently the stress and no sleep is getting to me, though, cause I woke up at 8:50 today. (And I'm really really out of shape... I was tired just sprinting to CMS.) My thought process went something like "Oooh, early... it's only 8:50... 8:50!!!! Crap crap crapcrapcrapcrap...." I wasn't late to the exam, though. Got there a minute ahead of time and wearing contacts and with my hair brushed. And wearing clean clothes. I was damn proud of myself... after I stopped panicking.

The exam was long. And by long I mean we should have had 4.5 hours instead of 3. The kind of long where you just want to turn in all of the class notes, and that wouldn't be too little writing. The kind where you're panicking the entire time because you won't be able to do all of the exam. The kind where everyone asks for more paper because the people setting up the room didn't prepare for that kind of exam.

1. Consider the elliptic curve X^3 + Y^3 + dZ^3 = 0 with identity (1:-1:0). Describe the group law on it geometrically. Find the points of inflexion. (I messed up on this part cause I was nervous. Not too badly, though.) Compute its Weierstrass equation. Prove that there are infinitely many rational x,y such that x^3 + y^3 = 7. And can I say "EEEWWWWWW"? EEEEEEEEWWWWWWW. Lots of computations, lots of icky worrying, and you have to find a point on the curve y^2 = x^3 - 432*7^2. 432??!! What kind of sadistic person gives a question with that kind of coefficient? On a test?! I'm surprised I made so few mistakes.

2. Let f be a power series in T with coefficients in R such that f(0) = 0 and f'(0) is a unit. Show that there exists a power series g in T such that f(g(T)) = T and g(f(T)) = 0. Outline where this fact is used in the proof of weak Mordell-Weil. OK, so first off, the computations in this proof are icky. Just cause there are a lot of coefficients to keep track of, and it was icky. On the other hand, it was pretty easy. But the second part was just brutal, cause this fact is SO far away from Mordell-Weil that it's unclear how much you're even supposed to write. How much can you assume? How precisely do they want us to point out where the fact is used? I just ended up writing a quick one-paragraph proof of weak Mordell-Weil and then pointing out where the fact is necessary... but maybe the question wanted us to prove that part in its entirety? What does "outline" mean? Does it mean "touch on the main points" or does it mean "write out in full excruciating detail"? Cause given the way things are structured in this place I wouldn't put it past them for it to mean the second. So I either did really well on this problem or really badly. I did well enough for me, anyway, so *pffffffft* to them if they don't think so.

3. EITHER write an essay on isogenies, including a proof that deg is a quadratic form, OR write an essay on heights and their place in the proof of Mordell-Weil. Almost everyone that I know did the second part because proving that deg is a quadratic form is just UGLY. And long. And did I mention ugly? And, once again, how much did they want in the essay? Heights are a pretty specialized construction, and there isn't really that much that they do... so is defining them, giving their properties and stating their use enough? Or do they actually want all of the gory details? I wrote two pages on this, and I'm not sure that it was enough.

4. Outline a procedure that often works to compute the rank of an elliptic curve over Q. For p a prime show that the rank of the group of rational points on y^2 = x^3-p^2x is at most 2. Given an example where the rank is exactly one. All of the same questions as above apply to the meaning of the word "outline", of course... and the computations were ugly. And p=3 didn't work, so I had to do the work over again to try p=5. (Which did work, thank God.) And I kept worrying that I didn't write enough for the stupid "outline."

So all of these worries that I didn't write enough, and I still ran out of time. I actually mostly finished everything, but I wanted to add a couple more sentences to the end of 3. This test was evil. And stressful. And long.

But I'm done!!!!! Done done done done done done done done done done done odone done done done odne done dondod done ond eond oend onedone do edn eod done ond eond odn eondoe odned. Right. =) I can go back to my usually scheduled life now. You know what the first step is? Throwing all of this out:

(That's actually not all of it. I also have two notebooks filled with notes that I made to studly, and my desk is also covered with notes. I'll be so glad to get it all out.)

Thursday, June 07, 2007

Sixth exam: Additive Number Theory

I have now officially won some ice cream from Flip. He bet me that I would get an alpha on this exam, despite all of my protestations that I really don't know the material, that I'm not just saying that, and that my expectations are completely realistic. Well, out of the 3 questions that I needed to answer I answered about 80% of one and gave a three-sentence sketch of how to do another. So I can't have gotten more than 30% on the exam, which isn't an alpha, not even close.

It's kind of funny how I don't care, though... cause yes, I couldn't remember the formula for a Fourier transform of a function on R, and I couldn't remember how to prove the functional equation for the zeta-function (even though I've known proofs of that for a long time)... but really, why should I know them off the top of my head? Especially after 4 hours of sleep. And I know where I could look them up, and I know that I could figure it out eventually... so I just don't care. It's kind of liberating like that. =)

Wednesday, June 06, 2007

Exams four and five: 3-Manifolds and Local Fields

So this morning I had 3-manifolds. Can I say ugh? And maybe ugh again? I didn't have much time to prepare last night (I was really tired after my exam, and I took a nap) and so I studied for local fields (for which I hadn't even really looked at half of the material) instead of 3-manifolds (for which I mostly-sorta knew the material). So that was a mistake. Cause I ended up skipping some material that I could have done on the exam, and studying stuff that wasn't even referred to. Which is annoying. And was it such a bad assumption to make that a theorem that is never (a) proven and (b) used in the class wouldn't be on the exam? I didn't think so. But there was a problem that used a theorem that was mentioned in passing, which was really frustrating cause I remembered an approximate statement, but not enough to actually use it. Anyway... I'm not going to talk about that exam any more.

Then I had two hours to study for local fields. That turned out to be more than enough time... cause the class covered almost no material, and the exam pretty much just needed knowledge of Hensel's lemma, the definition of a local field, and some common sense. Gotta love exams like this one: plenty of time, easy problems, no knowledge needed. (Of course it'll turn out that I screwed something up completely, but when doesn't it?)

Anyway, now I'm going to rest a bit and then study for additive number theory. I ran in to PJ at the local fields exam, and he said that he was considering not showing up for additive number theory cause he doesn't know any of the material anyway. I kinda feel the same way, but I gotta try... I mean, a miracle could happen...

Tuesday, June 05, 2007

Third Exam: Ramsey Theory

Ugh. And can I say ugh again? I have to prepare for two more exams tomorrow, and one more the day after (that I'm woefully underprepared for... but everyone seems to be), and I have no energy. Hopefully reading blogs will help, but if it doesn't...

I've also been unable to sleep for the last couple of days. I keep starting awake, certain that I've missed a day and I've skipped some exams and I'm going to fail and MIT will reject me and badness will ensue. And yes, I know that I have nothing to worry about, that I've likely already done enough work to pass and that even if I don't get a distinction that won't be the end of the world (or even the end of anything, other than my year at Cambridge (and that will come anyway)), and that really I shouldn't be worried. Like any of that helps. Ha. I think that stress is contagious: because some of the students are really worried (as the exams MATTER to them, since Ph.D. spots are given based on exam results) everyone else picks it up. Not that thinking that helps, either.

Anyway, the exam today was for Ramsey theory. 3 of 4 questions to solve, 2 hours. I didn't expect it to be so stressful, because unlike my previous exams there was NOT enough time. The exam was maybe a little bit shorter than both algebraic topology and category theory, but a whole hour shorter. And I wrote more than I wrote for either of those exams. (One of my solutions was 3 pages long, another 2 and a half.) The questions were:

1. (i) Using Ramsey's theorem, show that whenever N is finitely colored there exist x_1<x_2<... such that the set {x_i + x_j | i != j} is finitely colored. This was easy; it was a 3-line solution only because I insisted on justifying everything.

(ii) Show that whenever N is finitely colored there exist x_1<x_2<... such tha the set {x_i + 2x_j | i < j} is monochromatic. This was even easier.

(iii) Show that it is not true that whenver N is finitely colored there exist x_1<x_2<... such that the set {x_i + 2x_j | i != j} is monochromatic. I didn't solve this. Neither did anyone else I talked to. None of the obvious colorings work, and I couldn't really find anything else to do.

(iv) Deduce from (iii) that there is no ultrafilter on N, each member of which contains a set of the form {x_i + 2x_j | i != j}. Also really easy.

2. State and prove van der Waerden's theorem. Deduce that, if a_1,...,a_n are nonzero rationals then the matrix (a_1 ... a_n) is partition regular iff some (non-empty) subset of the a_i has sum 0. This was just incredibly annoying. Both of the parts have long proofs, neither of which has many ideas. This was my 3 page solution, and it was the most annoying thing ever. I think the prof doesn't like bookwork (yay him!) so he gives annoying bookwork problems. (Bookwork is reproducing proofs given in lecture.)

3. [Long annoying problem about ultrafilters. Easy, but long and I didn't want to write as much as I did in problem 2, so I skipped it.]

4. What does it mean to say that a subset of N^(omega) is Ramsey? Give an example of a set that is not Ramsey. Prove that every tau-open set is Ramsey.
Find, with justification, examples of each of the following:
(i) a set that is *-open but not tau-open
(ii) a set that is tau-nowhere-dense but not *-nowhere-dense
(iii) a set that is *-nowhere-dense but not tau-nowhere-dense
This problem was annoying. And part (iii) was hard; it took me half an hour. Which on a two-hour exam is a LOT. Especially for a quarter of a problem. And did I mention annoying?

Anyway, now I'm going to go veg for a while. And then study. And then try to sleep. (Maybe I'll do those in another order...)

Friday, June 01, 2007

Second Exam: Algebraic Topology

Today's exam went fine. It was a lot easier than past year's exams so it wasn't that bad. You needed to do 4 of 5 problems.

1. Compute the class of the idagonal embedding of CP^2 in CP^2 x CP^2.
I didn't do this one; I knew what the answer SHOULD be, but I couldn't figure out a good way of writing it up, so I just skipped it.

2. Describe a cell decomposition of RP^2 x RP^2. Compute the cellular chain complex of RP^2 x RP^2. Use this chain compelx to compute the integral homology of RP^2 x RP^2. I did this one, but I screwed up so many times in the computations that it took about 4 times as long as it should have.

3. Prove that Kunneth extends to fiber bundles with compact base. (It didn't actually say this, but that's what it meant. =) ) Pretty easy, once I figured out what the heck it was asking. (The question statement was about half a page, as we never discussed fiber bundles.)

4. Is the twisted circle bundle over S^2 orientable? Compute its integral cohomology and it's cohomology ring with Z/2 coefficients. I did this problem but COMPLETELY screwed up the computation. And I wrote that I was using the universal coefficient theorem when I meant Poincare duality, and I screwed up a computation of products and... pretty much I just screwed up. Hopefully I'll get partial credit since I computated at least part of things correctly, and most of my mistakes were linear algebra errors anyway. But it's kinda annoying, cause it wasn't that hard of a problem.

5. (i) Show that the symmetric group S_3 can act freely on the closed orientable surface of genus 7. This was pretty easy if you drew the right picture. =)
(ii) Define the cap product. (Don't show it's well defined.) For a continuous map f:X->Y show that f_*(x cap f^*(a)) = f_*(x) cap a. REALLY REALLY easy, and the only thing I solved in the first hour (cause I was so busy screwing up the computations in problems 2 and 4).

Maybe I should have written something for question 1... just to see if it worked out. And so I could avoid doing 4. Oh, well. I should get full credit (or almost) for 2,3,5, so I don't think I did too badly. Kinda annoyed aobut problem 4, though.