Posted by martin on Thursday, 08 October 2009 at 09:58
So far in my series on functors of points, I have considered functors 
for some fixed field 
. We begin by observing that we may allow to be any ring. Then I consider whether it is possible to relate functors with base ring to functors with base ring 
, with only partial success.
Posted by martin on Thursday, 01 October 2009 at 15:45
This post will discuss the fact that 
-points of an affine -scheme 
(and more general objects) are the same as morphisms 
. James already brought this up in his comment last time. As well as proving this in the affine -scheme case, I shall attempt to give an intuitive explanation of this fact, although I don’t find this entirely satisfying.
Posted by martin 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 -variety is simply the functor 
for a suitable -algebra 
. Only a restricted class of -algebras could arise as however. So in this post I generalise this to allow to be any -algebra, and thereby define affine -schemes.
Posted by martin 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.
Posted by martin 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.