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Part III and examples classes

Posted by Martin Orr on Sunday, 06 September 2009 at 16:47

Before I head off to Paris next week, I want to write a blog post about Part III. I greatly enjoyed Part III, particularly because of the friends I made and the experience of learning maths together with other people, while I have always done it as a largely solitary activity before. However when I sat down to write this, it turned out to be mainly about examples classes. These are, or should be, an important and useful part of the course, but I frequently found myself dissatisfied with my examples classes.

Through the year, I believed that the ideal class would consist of students presenting their solutions to the others. While I have never had any hesitation about presenting my solutions, it turns out that other people are more reluctant and my attempts to encourage people to take part had limited success. In part, this is sensible because it is difficult to explain to others how to solve a problem you did a few days ago and have not thought about since (a different skill from giving a prepared talk or lecture).

Furthermore audience-run examples classes have other problems. The easiest way of presenting a solution (as mentioned a difficult task) is to write up every line on the blackboard. This will probably result in the examples class taking too long, and is usually unnecessary - a summary of the key steps should be sufficient, along with pointing out any particularly fiddly bits.

At the end of the year, I had a chat with Jessica about lecturers' office hours, which are a standard part of US teaching. In Cambridge it is assumed that if you want to ask a lecturer something you will do so informally, by email or by ambushing them after a lecture. Lecturers are usually very generous in responding to such questions, but perhaps if they allocated a formal time for questions then people having difficulties would be more likely to think of using them. Also questions, even if asked privately, are usually better answered publicly because several people might have the same question (I am not sure if US office hours deal are public events).

One or two lecturers (usually with US experience) do offer office hours, but I think it would be worthwhile to conduct an experiment on a larger scale. If any Part III students are reading, I encourage you to ask your lecturers questions and, if you and your colleagues think it would be useful, suggest that they try an office hour, with an explanation of why it would provide benefits not otherwise achievable.

no comments Tags lectures, partiii

Graduation

Posted by Martin Orr on Tuesday, 30 June 2009 at 22:42

I got my BA last week, and suddenly my time in Cambridge is at an end. The best bit of the graduation was when all the graduands process in academical dress (including hoods) from the college to the Senate House where the ceremony takes place. Many people, including all the bedders and no doubt a lot of unsuspecting tourists, came out to watch. I don't know if colleges which are not as central as Trinity have such a procession; and while we got lovely weather, it wouldn't be so fun in the rain (e.g. Magdalene got rained on).

A week before that, the Part II and Part III Maths results were read out in the Senate House (as last year - the results and graduation make the only three times I have been in the building). It is likely that this will be the last year that this is the first place where people hear their results. CUSU has been campaigning for them to be sent out by email before publication, on the grounds that getting your results in public is distressing for people who did badly. This has some truth, but I think that there is a significant advantage in learning your results along with your friends, as they will be in a better position to offer support than if you learned them privately (and maybe were then embarrassed about sharing them). This advantage applies to a much greater extent to the reading out of the Maths results, which everyone really does hear simultaneously, than to publication on the Senate House noticeboards as happens for other subjects.

1 comment Tags cambridge, exams, partiii, tripos

Proof of the Nullstellensatz

Posted by Martin Orr on Friday, 19 June 2009 at 11:49

Hilbert's Nullstellensatz is an algebraic result fundamental to algebraic geometry. There are many different proofs of the Nullstellensatz. In this post I will consider the proof given in this year's Part III Commutative Algebra course, and in particular one section of the proof that seems to contain lots of magic. When I was revising for the exams, I realised that part of the mystery came from the fact that the particular theorem proved in Commutative Algebra does not require an algebraically closed field, unlike the standard statement of Hilbert's Nullstellensatz.

Thanks are due to Lloyd West for starting me thinking about this, and to Jon Nelson for giving me the courage to believe that it might be true and for supplying the proof of Lemma 4.

no comments Tags alg-geom, maths, nullstellensatz, partiii Read more...

Part III exams

Posted by Martin Orr on Monday, 08 June 2009 at 13:32

The Part III exams took place in the last couple of weeks. It was quite strange for me to do exams where you are expected to answer all or most of the questions - the undergraduate maths exams have several times more questions to choose from than anyone could possibly do and in olympiad exams completing each question is an achievement.

The most common form of a Part III exam is questions which simply ask you to write out proofs from the course. Three of my five exams took this form, and it's quite boring - except one of those, where the content was very hard so just recalling bits of the lectures is a challenge. Elliptic Curves livened things up a bit with some computation, although that is rather tedious in a different way. And my last exam, in Modular Forms, had a much better balance of bookwork and problems, although it was the hardest exam I did.

no comments Tags exams, partiii, tripos

Lent 2009

Posted by Martin Orr on Sunday, 15 March 2009 at 22:28

Another term has just finished. This term I studied:

  • Elliptic Curves (quite easy; major objects of study in algebraic number theory)
  • Modular Forms (a bit harder; more central objects in algebraic number theory)
  • Curves and Abelian Varieties (quite hard - only a handful of people did it; this is really algebraic geometry, but very relevant to number theory)
  • Complex Manifolds (differential geometry, so not my main line but handy background; I didn't try to follow this one in detail)

I have also done some work on an essay (worth the same credit as a lecture course) on Complex Multiplication (which is about a special type of elliptic curves). I gave a seminar on this last Friday but beyond the content of that I don't know much about it yet - I shall have to work on it over the holiday.

Other big news is that I have a place at the Université Paris Sud in Orsay for the second year of a French masters next year (because getting funding to go straight to a PhD in Paris would be a problem), and this week I heard I have got the Rouse Ball Travelling Studentship in Mathematics from Trinity to pay for it.

no comments Tags paris, partiii

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