Martin's Blog

Affine k-schemes

Posted by Martin Orr on Friday, 25 September 2009 at 15:45

In my last post on functors of points, I showed that functor of points of an affine k-variety is simply the functor \mathop{\mathrm{Hom}}(B, -) for a suitable k-algebra B. Only a restricted class of k-algebras could arise as B however. So in this post I generalise this to allow B to be any k-algebra, and thereby define affine k-schemes.

2 comments Tags alg-geom, maths, points-func

French Lectures

Posted by Martin Orr on Sunday, 20 September 2009 at 15:52

I had my first lectures in Paris this week. They are much longer than Cambridge lectures: 2 or 2.5 hours per lecture. And you have one lecture in the morning, then the same course again in the afternoon. (In Cambridge lectures are 1 hour, you have about 3 different lectures in a day, and they are only in the mornings.) I had this every day this week, so that was quite a lot. For the first three weeks there are some introductory courses, which take place every day. When term starts properly in October I will only have 2 or 3 days of lectures per week.

The two courses for which I had lectures this week were Algebra and Geometry, and Complex Analysis. The first is all stuff I already know from Part III. Sometimes it is interesting to see it taught from a different (more geometric) perspective, and sometimes it is just boring. The Complex Analysis one is not really relevant to what I am interested in, but I thought I would go along just to broaden my knowledge.

The lectures are all in French, but the lecturers speak reasonably clearly so that is rarely a problem. Reading what they have written can sometimes be harder, and they don't write full sentences on the board as lecturers usually do in Cambridge. That means that I'm not learning how to write maths in French as I had hoped, and I often don't know what to write to fill in the gaps in what they wrote on the board.

There are about 18 people attending the lectures. The audience at the Algebra and Geometry ones is quite international. I think I am the only native English speaker, but the majority of the audience are probably more comfortable speaking English than French. There are more French people at the analysis lectures.

no comments Tags france, languages, lectures, m2, paris

Arrival in Paris

Posted by Martin Orr on Saturday, 12 September 2009 at 15:49

On Wednesday I arrived in Paris. I am here to do the second year of a masters (French, and all other Bologna Process-compliant, masters are two years) in number theory and geometry, at the Université Paris-Sud. Getting here without flying was quite a journey - I got the ferry to Stranraer, train through Scotland and England to London, then stayed overnight with someone in London. I got the Eurostar the next morning, followed by a 40 minute journey on the RER (Paris suburban train).

I live on an island in the river Yvette. It is not very obviously an island, since the river is pretty tiny and the island is ten times the width of the river. Despite the river being so small, it is in a big valley. The university goes up the north side of this valley, and above that there is just trees. Most of Paris is somewhere over the other side of this ridge, so you wouldn't really know you are on the edge of a big city. The towns of Bures-sur-Yvette (where I live) and Orsay (at the other end of the university) go up the south side of the valley. Orsay is 15-20 minutes walk away, and has slightly more shops than Bures. There are a lot of big, quite mixed, trees everywhere.

Today I climbed up the through Bures to the top of the hill, where you come to Les Ulis, a New Town-style suburb with a big shopping centre. According to the map, this is the very southern edge of Paris - after the shopping centre you come to the countryside. It took me about an hour to come down from Les Ulis, probably longer to go up, although I didn't really know where I was going either time.

I have spent the past few days learning my way around and sorting various things - registering with the university, opening a bank account, etc. I don't understand everything that people say to me, and sometimes I have trouble explaining myself, but the people I have talked to have been very patient.

1 comment Tags france, paris

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

Functors, affine varieties and Yoneda

Posted by Martin Orr on Wednesday, 02 September 2009 at 22:51

In this article, I will examine in more detail the functor of points of an affine variety, which I defined in the last article. I shall show that this functor is the same as a Hom-functor on the category of k-algebras, and that morphisms of varieties correspond to natural transformations of functors.

no comments Tags alg-geom, maths, points-func, yoneda